BV FUNCTIONS AND SETS OF FINITE PERIMETER ON

CONFIGURATION SPACES

ELIA BRUE∗ AND KOHEI SUZUKI§

September 13, 2021

Abstract. This paper contributes to foundations of the geometric measure theory in the infinite

dimensional setting of the configuration space over the Euclidean space Rn equipped with the

Poisson measure π. We first provide a rigorous meaning and construction of the m-codimensional

Poisson measure —formally written as “(∞ − m)-dimensional Poisson measure”— on the con-

figuration space. We then show that our construction is consistent with potential analysis by

establishing the absolute continuity with respect to Bessel capacities. Secondly, we introduce

three different definitions of BV functions based on the variational, relaxation and the semigroup

approaches, and prove the equivalence of them. Thirdly, we construct perimeter measures and

introduce the notion of the reduced boundary. We then prove that the perimeter measure can

be expressed by the 1-codimensional Poisson measure restricted on the reduced boundary, which

is a generalisation of De Giorgi’s identity to the configuration space. Finally, we construct the

total variation measures for BV functions, and prove the Gauß–Green formula.

Contents

1. Introduction 2

1.1. Non-linear dimension reduction and overview of the main results 3

1.2. m-codimensional Poisson measure 3

1.3. Bessel capacity 4

1.4. Functions of bounded variations and Cacioppoli sets 4

1.5. Structure of the paper 5

Acknowledgement 6

2. Preliminaries 6

2.1. Notational convention 6

2.2. Configuration spaces 6

2.3. Spherical Hausdorff measure 7

2.4. Regularity of the spherical Hausdorff measures 7

2.5. Differential structure on configuration spaces 8

2.6. Product semigroups and exponential cylinder functions 11

2.7. Suslin sets 12

3. Finite-codimensional Poisson measures 13

3.1. Measurability of sections of Suslin sets 13

3.2. Localised finite-codimensional Poisson measures 14

3.3. Finite-codimensional Poisson Measures 15

3.4. Independence of ρm from the exhaustion 17

4. Bessel capacity and finite-codimensional Poisson measure 18

4.1. Localization of sets and functions 18

∗ School of Mathematics, Institute for Advanced Study, 1 Einstein Dr., Princeton NJ 05840, USA.

elia.brue@ias.edu.§ Fakultat fur Mathematik, Universitat Bielefeld, D-33501, Bielefeld, Germany.

ksuzuki@math.uni-bielefeld.de.

1

2 E. BRUE AND K. SUZUKI

4.2. Localisation of energies, resolvents and semigroups 19

4.3. Localized Bessel operators 21

4.4. Finite-dimensional counterpart 21

4.5. Proof of Theorem 4.3 22

5. BV Functions 22

5.1. Variational approach 22

5.2. Relaxation approach 26

5.3. Heat semigroup approach 27

5.4. p-Bakry-Emery inequality 27

5.5. Equivalence of BV functions 30

6. Sets of finite perimeter 32

6.1. Sets of finite perimeter in Υ(Br) 32

6.2. Sets of finite perimeter on Υ(Rn) 33

6.3. Perimeter measures 35

6.4. Perimeters and one-codimensional Poisson measures 38

7. Total variation and Gauß–Green formula 40

7.1. Total variation measures via coarea formula 40

7.2. Proof of Theorem 7.1 41

7.3. Gauß–Green formula 43

7.4. BV and Sobolev functions 44

References 45

1. Introduction

The purpose of this paper is to contribute to the foundation of the geometric measure theory in

the infinite-dimensional setting of the configuration space (without multiplicity) over the standard

Euclidean space Rn:

Υ(Rn) :=γ =

∑i∈N

δxi : γ(K) <∞ for every compact K ⊂ Rn, γ(x) ∈ 0, 1, ∀x ∈ Rn.

We equip it with the vague topology τv, the L2-transportation (extended) distance dΥ, which

stems from the optimal transport problem, and the Poisson measure π whose intensity measure

is the Lebesgue measure Ln on Rn (see Section 2 for more details). The resulting topological

(extended) metric measure structure (Υ(Rn), τv, dΥ, π) has fundamental importance for describing

dynamical systems of infinite particles, hence it has been extensively investigated in the last years

[1, 2, 23, 24, 31] (see also references therein) by a number of research groups both from the fields

of statistical physics and probability. In spite of its importance for the applications to statistical

physics and stochastic analysis, the theory of BV functions and sets of finite perimeters on Υ(Rn)has – so far – never been investigated in any literature that we are aware of, due to its nature of

infinite-dimenisonality and non-linearity.

The theory of BV functions and sets of finite perimeters beyond the standard Euclidean space

has seen a thriving development in the last years, see [3, 4, 5, 6, 7, 9, 10, 13, 18, 35] and references

therein. However, all of these results do not cover the configuration space Υ(Rn) due to its intrinsic

infinite dimensionality and the fact that the canonical (extended) distance function can be infinite

on sets of π-positive measure. This property causes several pathological phenomena (see details in

[23]):

• the extended distance dΥ is not continuous with respect the vague topology τv;

• dΥ-metric balls are negligible with respect to the Poisson measure π;

BV FUNCTIONS AND SETS OF FINITE PERIMETER ON CONFIGURATION SPACES 3

• dΥ–Lipschitz functions are not necessarily π-measurable;

• the Riesz–Markov-Kakutani’s representation theorem does not hold.

The latter makes it difficult to construct total variation measures supporting the Gauß–Green

formula by means of the standard functional analytic technique.

In the setting of infinite-dimensional spaces, the study of the geometric measure theory has

been pioneered by Feyel–de la Pradelle [26], Fukushima [28], Fukushima–Hino [29] and Hino [30]

in the Wiener space. In [26], they constructed the finite-codimensional Gauß–Hausdorff measure

in the Wiener space and investigated its relation to capacities. In [28] and [29], they developed the

theory of BV functions and constructed perimeter measures, and prove the Gauß–Green formula.

Based on these results, Hino introduced in [30] a notion of the reduced boundary and investigated

relations between the one-codimensional Hausdorff–Gauß measure and the perimeter measures.

Further fine properties were investigated by Ambrosio-Figalli [11], Ambrosio-Figalli-Runa [12],

Ambrosio-Miranda-Pallara [15, 16], Ambrosio-Maniglia-Miranda-Pallara [14]. The notion of BV

functions has been studied also in a Gelfand triple by Rockner–Zhu–Zhu [38, 39, 40]. All of the

aforementioned results rely heavily on the linear structure of the Wiener space or the Hilbert space,

which is used to perform finite-dimensional approximations. However, the configuration space does

not have a linear structure and there is no chance to apply similar techniques.

1.1. Non-linear dimension reduction and overview of the main results. To overcome

the difficulties explained above, we develop a non-linear dimensional reduction tailored for the

configuration space Υ(Rn). A key observation is that Υ(Br), the configuration space over the

Euclidean closed metric balls Br centered at the origin o with radius r > 0, is essentially finite

dimensional. More precisely, due to the compactness of Br, Υ(Br) can be written as the disjoint

union tk∈NΥk(Br) of the k-particle configuration spaces Υk(Br), each of which is isomorphic to

the quotient space of the k-product space B×kr by the symmetric group. In light of this observation,

the main task is to lift the geometric measure theory in Υ(Br) to the infinite-dimensional space

Υ(Rn) by finite-dimensional approximations.

Based on this strategy, we first construct the m-codimensional Poisson measure on the config-

uration space with much care of the measurability issues (Theorem 3.7 and Definition 3.8), and

study its relation to (1, p)-Bessel capacities (Theorem 4.3). Secondly, we introduce three different

definitions of BV functions based on the variational, relaxation and the semigroup approaches, and

prove the equivalence of them (Theorem 5.17). In the process of showing the equivalence of these

three definitions, we develop the p-Bakry–Emery inequality (Theorem 5.15) for the heat semigroup

on Υ(Rn) for 1 < p < ∞, which was previously known only for p = 2 in Erbar–Huesmann [24].

Thirdly, we construct perimeter measures and introduce the notion of the reduced boundary in

Section 6. We then prove that the perimeter measure can be expressed by the 1-codimensional

Poisson measure restricted on the reduced boundary (Theorem 6.15). Fourthly, we construct the

total variation measures for BV functions and prove the Gauß–Green formula (Theorem 7.7).

We now explain each result in more detail.

1.2. m-codimensional Poisson measure. The first achievement of this paper is the construction

of the m-codimensional Poisson measure on Υ(Rn). Since Υ(Rn) is infinite-dimensional, it is

formally written as

“(∞−m)-dimensional Poisson measure”,

which is an ill-defined object without justification. An important remark is that, in the case of

finite-dimensional spaces, usually the construction of finite-codimensional measures builds upon

covering arguments, which heavily rely on the volume doubling property of the ambient measure.

However, this property does not hold in the configuration space Υ(Rn).

4 E. BRUE AND K. SUZUKI

To overcome this difficulty we construct the m-codimensional Poisson measure on Υ(Rn) by

passing to the limit of finite dimensional approximations obtained by using the m-codimensional

Poisson measure on Υ(Br). The key step in the construction is to prove the monotonicity of these

finite dimensional approximations with respect to the radius r, allowing us to find a unique limit

measure.

More in details, based on the decomposition Υ(Br) = tk∈NΥk(Br), we build ρmΥ(Br)

, the spher-

ical Hausdorff measure of codimension m in Υ(Br), by summing the m-codimensional spheri-

cal Hausdorff measure ρm,kΥ(Br)on the k-particle configuration space Υk(Br), which is obtained

by the quotient measure of the m-codimensional spherical Hausdorff measure on the k-product

space B×kr with a suitable renormalization corresponding to the Poisson measure. The localised

m-codimensional Poisson measure ρmr of a set A ⊂ Υ(Rn) is then obtained by averaging the

ρmΥ(Br)-measure of sections of A with the Poisson measure πBc

ron Υ(Bcr), i.e.

ρmr (A) :=

ˆΥ(Bc

r)

ρmΥ(Br)(γ ∈ Υ(Br) : γ + η ∈ A)dπBc

r(η) .

We prove that ρmr is well-defined on Borel sets (indeed, we prove it for all Suslin sets), and that it

is monotone increasing with respect to r (Theorem 3.7 and Definition 3.8). In particular, we can

define the m-codimensional Poisson measure as

ρm := limr→∞

ρmr .

We refer the readers to Section 3 for the detailed construction of ρm.

1.3. Bessel capacity. In Section 4, we compare the m-codimensional Poisson measure ρm and

Capα,p, the Bessel capacity, proving that zero capacity sets are ρm negligible provided αp > m

(Theorem 4.3). This result, that is well-known in the case of finite-dimensional spaces, proves

that our m-codimensional Poisson measure ρm behaves coherently with the potential analytic

structure of Υ(Rn). To prove it, we introduce the (α, p)-Bessel capacity CapΥ(Br)α,p on Υ(Br) and

the localised (α, p)-Bessel capacity Caprα,p on Υ(Rn) based on the localisation argument of the Lp-

heat semigroup Tt on Υ(Rn). We prove that Capα,p is approximated by Caprα,p as r → ∞, hence

we can obtain the sought conclusion by lifting the corresponding result for ρmΥ(Br)and CapΥ(Br)

α,p

in Υ(Br) (see Proposition 4.14). We refer the readers to Section 4 for the detailed arguments.

A remarkable application of the result above is Corollary 7.4, which establishes a fundamental

relation between the potential theory and the theory of BV functions. It says that, if Cap1,2(A) = 0

then |DF |(A) = 0 for any F ∈ BV(Υ(Rn))∩L2(Υ(Rn), π), where |DF | is the total variation measure

(Definition 7.2) and BV(Υ(Rn)) is the space of BV functions (Definition 5.18). The latter result

will be fundamental for applications of our theory to stochastic analysis, which will be the subject

of a forthcoming paper.

1.4. Functions of bounded variations and Cacioppoli sets. In the second part of this paper

we develop the theory of BV functions and sets of finite perimeter in Υ(Rn). In Section 5 we

propose three different notions of functions with bounded variation. The first one follows the

classical variational approach, the second one is built upon the relaxation approach, while the third

one relies on the regularisation properties of the heat semigroup. It turns out that they are all

equivalent, as shown in Section 5.5, and the resulting class is denoted by BV(Υ(Rn)). For any

function F ∈ BV(Υ(Rn)) we are able to define a total variation measure |DF | and to prove a Gauß-

Green formula (see Theorem below). It is worth remarking that in our infinite dimesional setting

we cannot rely on the Riesz–Markov–Kakutani’s representation theorem due to the lack of local

compactness. In particular, the construction of the total variation measure is not straightforward

and requires several work. We follow an unusual path to show its existence: we first develop the

theory of sets with finite perimeter relying on the non-linear dimension reduction, and after we

BV FUNCTIONS AND SETS OF FINITE PERIMETER ON CONFIGURATION SPACES 5

employ the coarea formula to build the total variation measure of a BV function as a sovraposition

of perimeter measures.

Sets of finite perimeter are introduced as those Borel sets E such that χE ∈ BV(Υ(Rn)), whereχE denotes the indicator function on E. In Section 6, we study their structure by means of the

non-linear reduction approach, a part of which utilizes a strategy inspired by Hino [30] for the

study of Wiener spaces. The key result in this regard is Proposition 5.5 saying that if E is of finite

perimeter then the projection Eη,r := γ ∈ Υ(Br) : γ + η ∈ E has finite localized total variation

in Br, for πBcr-a.e. η ∈ Bcr . Hence, we can reduce the problem to the study of sections that are sets

with finite perimeter on Υ(Br). As already remarked, the latter is essentially a finite dimensional

space, so we can appeal to classical tools of the geometric measure theory to attack the problem.

The reduced boundary ∂∗E of a set of finite perimeter E ⊂ Υ(Rn) is then defined in terms of

the reduced boundary of the sections Eη,r, through a limit procedure. The resulting object allows

representing the perimeter measure as

‖E‖ = ρ1|∂∗E ,

which is a generalization of the identity proven by E. De Giorgi [21, 22] in the Euclidean setting.

It is worth stressing that our approach to the BV theory significantly deviates from the standard

one. Indeed, as already explained, we define the total variation measure |DF | of a function F ∈BV(Υ(Rn)) by imposing the validity of the coarea formula. More precisely, we show that a.e. level

set F > t is of finite perimeter and we set

|DF | :=ˆ ∞

−∞

∥∥F > t∥∥dt ,

taking advantage of the fact that the perimeter measure ‖F > t‖ has been already defined using

finite dimensional approximations. The reason for this non-standard treatment is that we are not

able to build directly |DF | through a finite dimensional approximation, since the latter does not

have a simple expression in terms of 1-codimensional Poisson measure ρ1 restricted to a suitable

subset. Our approach is, however, consistent with the standard one, as shown in Corollary 7.3 and

in Theorem 7.7.

In the statement below we summarize the main results in Section 6 and Section 7 concerning

functions of bounded variations and a sets of finite perimeter. We denote by CylV(Υ(Rn)) the

space of cylinder vector fields on Υ(Rn) and by (TΥ, 〈·, ·〉TΥ) the tangent bundle on Υ(Rn) withthe pointwise inner product 〈·, ·〉TΥ (see Section 2.5).

Theorem (Theorems 6.15, 7.7). For F ∈ L2(Υ(Rn), π)∩BV(Υ(Rn)), there exists a unique positive

finite measure |DF | on Υ(Rn) and a unique TΥ-valued Borel measurable function σ on Υ(Rn) so

that |σ|TΥ = 1 |DF |-a.e., andˆΥ(Rn)

(∇∗V )Fdπ =

ˆΥ(Rn)

〈V, σ〉TΥd|DF | , ∀V ∈ CylV(Υ) .

If, furthermore, F = χE, then

|DχE | = ρ1|∂∗E .

As indicated in [28, 29] in the case of the Wiener space, the aforementioned theorem in the

configuration space Υ(Rn) has significant applications to stochastic analysis, in particular regarding

the singular boundary theory of infinite-particle systems of Brownian motions on Rn, which will

be addressed in a forthcoming paper.

1.5. Structure of the paper. In Section 2 we collect all the needed preliminary results regard-

ing the geometric-analytic structure of the configuration space, Suslin sets and measurability of

sections. In Section 3 we construct the m-codimensional measure whose relation with the Bessel

capacity is studied Section 4. Section 5 is devoted to the study of BV functions. We introduce

6 E. BRUE AND K. SUZUKI

three different notion and prove their equivalence. In Section 6 we introduce and study sets of

finite perimeter. We build the notion of reduced boundary and the perimeter measure, and we

show integration by parts formulae. In Section 7.1 we introduce the total variation measure of

functions with bounded variations by employing the coarea formula. We also show the validity of

the Gauß-Green type integration by parts formula.

Acknowledgement. The first named author is supported by the Giorgio and Elena Petronio

Fellowship at the Institute for Advanced Study. The second named author gratefully acknowl-

edges funding by: the JSPS Overseas Research Fellowships, Grant Nr. 290142; World Premier

International Research Center Initiative (WPI), MEXT, Japan; JSPS Grant-in-Aid for Scientific

Research on Innovative Areas“Discrete Geometric Analysis for Materials Design”, Grant Number

17H06465; and the Alexander von Humboldt Stiftung, Humboldt-Forschungsstipendium.

2. Preliminaries

2.1. Notational convention. In this paper, the bold fonts Sm, Ln, Tt, . . . are mainly used for

objects in the product space Rn×k or vector-valued objects, while the serif fonts S,D, . . . are used

for objects in the quotient space Rn×k/Sk with respect to the k-symmetric group Sk or for objects

in the configuration space Υ(Rn).The lower-case fonts f, g, h, v, w, ... are used for functions on the base space Rn, while the upper-

case fonts F,G,H, V,W, ... are used for functions on the configuration space Υ(Rn).We denote by χE the indicator function on E, i.e., χE = 1 on E and χE = 0 on Ec. Let Ω ⊂ Rn

be a closed domain. We denote by C∞0 (Ω) the space of smooth functions with compact support

on Ω \ ∂Ω, while C∞c (Ω) denotes the space of compactly supported smooth functions on Ω. Notice

that f ∈ C∞c (Ω) does not necessarily vanish at the boundary, and C∞

c (Rn) = C∞0 (Rn).

2.2. Configuration spaces. Let Rn be the n-dimensional Euclidean space. LetBr := Br(o) ⊂ Rn

be the closed ball with radius r > 0 centered at the origin o. Let δx denote the point measure at

x ∈ Rn, i.e. δx(A) = 1 if and only if x ∈ A. We denote by Υ(Rn) the configuration space over Rn

without multiplicity, i.e. the set of all locally finite point measures γ on Rn so that γ(x) ∈ 0, 1for any x ∈ Rn:

Υ(Rn) :=γ =

∑i∈N

δxi: γ(K) <∞ for every compact K ⊂ Rn, γ(x) ∈ 0, 1 ∀x ∈ Rn

.

Let Υ(A) denote the configuration space over a subset A ⊂ Rn, and Υk(A) denote the space of

k-configurations on a subset A, i.e. Υk(A) = γ ∈ Υ(A) : γ(A) = k. We equip Υ(Rn) with the

vague topology τv, i.e., γn ∈ Υ(Rn) converges to γ ∈ Υ(Rn) in τv if and only if γn(f) → γ(f) for

any f ∈ Cc(Rn). For a subset A ⊂ Rn, we equip Υ(A) with the relative topology as a subset in

Υ(Rn). Let B(Υ(A), τv) denote the Borel σ-algebra associated with the vague topology τv. For a

set A ⊂ Rn, let prA : Υ(Rn) → Υ(A) be the projection defined by the restriction of configurations

on A, i.e. prA(γ) = γ|A.Given A ⊂ Rn, an open or closed domain, we denote by πA the Poisson measure on Υ(A)

whose intensity measure is the Lebesgue measure restricted on A, namely, πA is the unique Borel

probability measure so that, for all f ∈ Cc(A), the following holdsˆΥ(A)

ef⋆

dπA = exp

ˆA

(ef − 1)dLn(x)

.(2.1)

Here Ln denotes the n-dimensional Lebesgue measure. See [32] for a reference of the expression

(2.1). We write π = πRn . Note that πA coincides with the push-forwarded measure πA = (prA)#π.

Let

diagk := (x)1≤i≤m ∈ (Rn)×k : ∃i, j s.t. xi = xj ,

BV FUNCTIONS AND SETS OF FINITE PERIMETER ON CONFIGURATION SPACES 7

denote the diagonal set in (Rn)×k, and let Sk denote the k-symmetric group. For any set A ⊂ Rn,we identify

Υk(A) ∼= (A×k \ diagk)/Sk, k ∈ N.Let sk : A×k \diagk → Υk(A) be the canonical projection with respect to Sk, i.e. sk : (xi)1≤i≤k 7→∑ki=1 δxi

. We say that a function f : t∞k=1(Rn)×k → R is symmetric iff f(xσk

) = f(xk) with

xσk:= (xσk(1), . . . , xσk(k)) for any permutation σk ∈ Sk and any k ∈ N.

For xk,yk ∈ A×k with sk(xk) = γ ∈ Υk(A) and sk(yk) = η ∈ Υk(A), define the L2-

transportation distance dΥk(γ, η) on Υk(A) by the quotient metric w.r.t. Sk:

dΥk(γ, η) = infσk∈Sk

|xσk− yk|Rnk .(2.2)

Here |xk − yk|Rnk denotes the standard Euclidean distance in Rnk.

2.3. Spherical Hausdorff measure. Let (X, d) be a metric space and n be the Hausdorff di-

mension of X. For m ≤ n, define the m-dimensional spherical Hausdorff measure SmX on X:

SmX(A) := limε→0

SmX,ε(A) := limε→0

inf∑i∈N

diam(Bi)m : Bi open ball with diam(Bi) < ε, A ⊂

∑i∈N

Bi

.

(2.3)

Here diam(Bi) = supd(x, y) : x, y ∈ Bi denotes the diameter of Bi. We call SmX,ε the m-

dimensional ε-Hausdorff measure. If X = Rn, we simply write Sm and Smε instead of SmRn and

SmRn,ε respectively.

Remark 2.1 (Comparison with the standard Hausdorff measure). In the case of m < n, the

spherical Hausdorff measure SmX does not coincide with the standard Hausdorff measure in general,

the latter is smaller since it is defined allowing all non-empty coverings instead of open balls. In

the case of m = n and X = Rn, however, Sm coincides with the standard n-dimensional Hausdorff

measure and also with the n-dimensional Lebesgue measure ([25, 2.10.35]). Note that Sm is a

Borel measure, but not σ-finite for m < n.

For a bounded set A ⊂ Rn, let Sn|A be the spherical Hausdorff measure restricted on A. The

spherical Hausdorff measure (Sn|A)⊗k on A×k can be push-forwarded to the k-configuration space

Υk(A) by the projection map sk, i.e.

SkA :=1

k!(sk)#(S

n|A)⊗k .

It is immediate by construction to see that SkA is the spherical Hausdorff measure on Υk(A)

induced by the L2-transportation distance dΥk up to constant multiplication. We introduce the m-

codimensional spherical Hausdorff measure and them-codimensional ε-spherical Hausdorff measure

on Υk(A) as follows

Sm,kA =1

k!(sk)#(S

nk−m|A×k), Sm,kA,ε =1

k!(sk)#(S

nk−mε |A×k).(2.4)

One can immediately see that Sm,kA,ε is (up to constant multiplication) the m-codimensional ε-

spherical Hausdorff measure on Υk(A) associated with the L2-transportation distance dΥk .

2.4. Regularity of the spherical Hausdorff measures. In this section, we prove the upper

semi-continuity of the ε-spherical Hausdorff measure on sections of compact sets, which will be of

use in Section 3.

Proposition 2.2. Let (X, dX), (Y, dY ) be metric spaces, and K ⊂ X×Y be a compact set. Then,

the map Y 3 y 7→ SmX,ε(Ky) is upper semi-continuous. Here, Ky := x ∈ X : (x, y) ∈ K.

Proof. Let us fix y ∈ Y and a sequence yn → y. The family of compact sets (Kyn ×yn)n∈N ⊂ K

is precompact with respect to the Hausdorff topology in K endowed with the product metric

8 E. BRUE AND K. SUZUKI

(e.g., [19, Theorem 7.3.8]). In particular, we can take a (non-relabeled) subsequence so that

Kyn × yn → K × y ⊂ K, as n→ ∞ in the Hausdorff topology, and K ⊂ Ky by the definition

of Ky.

Let us fix δ > 0 and a family of open balls B1, . . . , Bℓ ⊂ X with radius smaller than ε(1−δ) > 0

so that

K ⊂ℓ⋃i=1

Bi,

and

(2.5) SmX,ε(K) ≥ c(m)

ℓ∑i=1

rmi − δ.

Here c(m) denotes the constant depending on m such that volm(Bi) = c(m)rmi . Note that we can

always take ℓ = ℓ(δ) to be finite for any δ > 0 by the compactness of K. Let r = r(δ) := minri :1 ≤ i ≤ l(δ) > 0 be the minimum radius among Bi1≤i≤l.

We claim that there exists k = k(δ) ∈ N so that Kynk ⊂ ∪ℓi=1B(xi,1

1−δ ri) for any k ≥ k. Here

xi and ri are the centre and the radius of Bi.

Indeed, by the Hausdorff convergence of Kynk to K, there exists k := k(δ) ∈ N such that, for

any k > k, it holds that Kynk ⊂ Brδ(K). Here, Brδ(K) denotes the rδ-tubular neighbourhood of

K in X, i.e., Brδ(K) := x ∈ X : d(x, K) < rδ. Hence, for any z ∈ Kynk , we can always find

x ∈ K such that d(x, z) < rδ. By noting x ∈ Bi for some i = 1, . . . , ℓ, we conclude z ∈ B(xi,1

1−δ ri)

by noting 11−δ ri − ri =

δ1−δ ri ≥ δr.

By using the claim in the previous paragraph, the monotonicity SmX,a ≥ SmX,b whenever a ≤ b,

and (2.5), we obtain that

(2.6) SmX,ε(Kynk ) ≤ c(m)(1 + δ)m

ℓ∑i=1

rmi ≤ (1 + δ)mSmX,ε(1−δ)(K) + δ(1 + δ)mc(m),

for any k ≥ k(δ). By taking δ → 0 after taking k → ∞, we conclude that

(2.7) lim supk→∞

SmX,ε(Kynk ) ≤ SmX,ε(K) ≤ SmX,ε(K

y),

which is the sought conclusion.

2.5. Differential structure on configuration spaces. In this section Ω ⊂ Rn will be denoting

either a closed domain with smooth boundary or the whole Euclidean space Rn. Below we review

the natural differential structure of Υ(Ω), obtained by lifting the Euclidean one on Ω. We follow

closely the presentation in [2].

Cylinder functions, vector fields and divergence. We introduce a class of test functions on the

configuration space Υ(Rn).

Definition 2.3 (Cylinder functions). We define the class of cylinder functions as

(2.8) CylF(Υ(Ω)) := Φ(f⋆1 , . . . , f⋆k ) : Φ ∈ C∞b (Rk), fi ∈ C∞

c (Ω), k ∈ N ,

where f⋆(γ) :=´Ωfdγ for any γ ∈ Υ(Ω). We call fi inner function and Φ outer function.

The tangent space TγΥ(Ω) at γ ∈ Υ(Ω) is identified with the Hilbert space of measurable

γ-square-integrable vector fields V : Ω → T (Rn) equipped with the scalar product

〈V 1, V 2〉TΥ =

ˆΩ

〈V 1(x), V 2(x)〉TRndγ(x) .

We define the tangent bundle of Υ(Ω) by TΥ(Ω) := tγ∈Υ(Ω)TγΥ(Ω).

BV FUNCTIONS AND SETS OF FINITE PERIMETER ON CONFIGURATION SPACES 9

Definition 2.4 (Cylinder vector fields). We define two classes of cylinder vector fields as

CylV(Υ(Ω)) :=V (γ, x) =

k∑i=1

Fi(γ)vi(x) : Fi ∈ CylF(Υ(Ω)), vi ∈ C∞c (Ω;Rn), k ∈ N

,

CylV0(Υ(Ω)) :=V (γ, x) =

k∑i=1

Fi(γ)vi(x) : Fi ∈ CylF(Υ(Ω)), vi ∈ C∞0 (Ω;Rn), k ∈ N

.

Notice that CylV0(Υ(Ω)) ⊂ CylV(Υ(Ω)), and CylV0(Υ(Ω)) = CylV(Υ(Ω)) when Ω = Rn.Given p ∈ [1,∞), we introduce the space of p-integrable vector fields Lp(TΥ(Ω), π) as the

completion of CylV(Υ(Ω)) with respect to the norm

‖V ‖pLp(TΥ) :=

ˆΥ(Ω)

|V (γ)|pTγΥdπ(γ) .

The latter is well defined because ‖V ‖Lp(TΥ) < ∞ for any V ∈ CylV(Υ(Ω)) as the following

proposition shows.

Proposition 2.5. For any 1 ≤ p <∞ it holds

(2.9) ‖V ‖Lp(TΥ) <∞ for any V ∈ CylV(Υ(Ω)) .

Moreover, CylV0(Υ(Ω)) is dense in Lp(TΥ(Ω), π).

Proof. Let V (γ, x) =∑ki=1 Fi(γ)vi(x). Then, we have that

ˆΥ(Ω)

|V |pTγΥ(Ω)dπ(γ) ≤ max1≤i≤k

‖F‖L∞

k∑i,j=1

ˆΥ(Ω)

(ˆΩ

|vi||vj |dγ)p/2

dπ(γ).

By the exponential integrability implied by (2.1), we obtain that the function γ 7→ G(γ) :=´Ω|vi||vj |dγ is Lp(Υ(Ω), πΩ) for any 1 ≤ p <∞, which concludes the first assertion.

The density of CylV0(Υ(Ω)) in Lp(TΥ(Ω), π) follows from the density of C∞0 (Ω;Rn) in Lp(Ω;Rn).

More precisely, we check that for any V ∈ CylV(Υ(Rn)) and ε > 0 there exists W ∈ CylV0(Υ(Ω))

such that´Υ(Ω)

|V −W |pTγΥdπ ≤ ε. To this aim we write V =

∑ki=1 Fivi and pick wi ∈ C∞

0 (Ω) such

that∑ki=1 ‖vi − wi‖Lp < ε and set W :=

∑ki=1 Fiwi. It is straightforward to see that W satisfies

the needed estimate. Noting that Lp(TΥ(Ω), π) is defined as the completion of CylV(Υ(Ω)) with

respect to the norm ‖V ‖Lp(TΥ), the proof is complete.

Definition 2.6 (Directional derivatives. [2, Def. 3.1]). Let F = Φ(f∗1 , . . . , f∗k ) ∈ CylF(Υ(Ω)) and

v ∈ C∞0 (Ω,Rn). We denote by ϕ the flow associated to v, i.e.

d

dtϕt(x) = v(ϕt(x)), ϕ0(x) = x ∈ Ω .

The directional derivative ∇vF (γ) ∈ TγΥ(Ω) is defined by

∇vF (γ) :=d

dtF (ϕt(γ))

∣∣∣t=0

,

where ϕt(γ) :=∑x∈γ δϕt(x).

Definition 2.7 (Gradient of cylinder functions. [2, Def. 3.3]). The gradient ∇Υ(Ω)F of F ∈CylF(Υ(Ω)) at γ ∈ Υ(Ω) is defined as the unique vector field ∇Υ(Ω)F so that

∇vF (γ) = 〈∇Υ(Ω)F, v〉TγΥ(Ω), ∀γ ∈ Υ(Ω), ∀v ∈ C∞0 (Ω,Rn).

By the expression (2.8), the gradient ∇Υ(Ω)F can be written as

(2.10) ∇Υ(Ω)F (γ) =

k∑i=1

∂iΦ(f⋆1 , . . . , f

⋆k )(γ)∇fi ∈ TγΥ(Ω) .

10 E. BRUE AND K. SUZUKI

We set ∇ := ∇Υ(Rn).

Notice that ∇Υ(Ω)F ∈ CylV(Υ(Ω)) for any F ∈ CylF(Υ(Ω)) by (2.10). In particular, for any

F ∈ CylF(Ω), it holds that ∇Υ(Ω)F ∈ Lp(TΥ(Ω), π) for any 1 ≤ p <∞ by Proposition 2.5.

Remark 2.8 (Ampleness of L∞-vector fields). By Proposition 2.5, CylV0(Υ(Ω)) ⊂ Lp(TΥ(Ω), π)

for any p ∈ [1,∞), while the inclusion is false for p = ∞. See [23, Example 4.35] for a counterex-

ample. However, CylV0(Υ(Ω)) can be approximated by the subspace of bounded cylinder vector

fields with respect to the pointwise convergence and the Lp(TΥ) convergence for 1 ≤ p < ∞.

Indeed, given ε > 0 and V =∑ki=1 Fi(γ)vi(x) ∈ CylV0(Υ(Ω)) it holds

|V |2TγΥ =

k∑i,j=1

Fi(γ)Fj(γ)

ˆΥ(Ω)

vi(x) · vj(x)dγ(x) ,1

1 + ε|V |2TγΥ

∈ CylF(Υ(Ω)) ,

hence

Vε :=1

1 + ε|V |2TγΥ(Ω)

V ∈ CylV(Υ(Ω)) .

Finally, notice that for any γ ∈ Υ(Ω) it holds

|V − Vε|TγΥ(Ω) = ε|V |3TγΥ(Ω)

1 + ε|V |2TγΥ(Ω)

≤ ε|V |3TγΥ(Ω) → 0 , as ε→ 0 .

Moreover, for any 1 ≤ p <∞ we have

‖V − Vε‖Lp(TΥ) ≤ ε‖V ‖3L3p(TΥ) → 0 , as ε→ 0 .

We now define the adjoint operator of the gradient ∇Υ(Ω).

Definition 2.9 (Divergence. [2, Def. 3.5]). Let p > 1. We say that V ∈ Lp(TΥ(Ω), π) is in the

domain D(∇∗Υ(Ω)) of the divergence if there exists a unique function ∇∗

Υ(Ω)V ∈ Lp(Υ(Ω), π) such

that

(2.11)

ˆΥ(Ω)

〈V,∇Υ(Ω)F 〉TγΥdπ(γ) = −ˆΥ(Ω)

F (∇∗Υ(Ω)V )dπ , ∀F ∈ CylF(Υ(Ω)).

We set ∇∗ = ∇∗Υ(Rn).

Proposition 2.10. Any V (γ, x) =∑ki=1 Fi(γ)vi(x) ∈ CylV0(Υ(Ω)) belongs to the domain D(∇∗

Υ(Ω))

of the divergence and the following identity holds

∇∗Υ(Ω)V (γ) =

k∑i=1

∇viFi(γ) +

k∑i=1

Fi(γ)(∇∗vi)⋆γ .(2.12)

In particular, ∇∗Υ(Ω)V ∈ Lp(Υ(Ω), π) for any p ∈ [1,∞).

Proof. Let r > 0 be such that supp(vi) ⊂ Ωr := x ∈ Ω : d(x, ∂Ω) > r. For any ε < r/2 we

define ϕε ∈ C∞0 (Ω) satisfying ϕ = 1 on Ωε. For any i = 1, . . . , k we write Fi = Φi(f

∗1,i, . . . , f

∗mi,i

)

and set F εi := Φi((ϕεf1,i)∗, . . . , (ϕεfmi,i)

∗). Observe that Vε :=∑ki=1 F

εi (γ)vi ∈ CylV0(Υ(Ω)) and

Vε ∈ CylV(Υ(Rn)). By [2, Prop. 3.1] it holds

∇∗Υ(Ω)Vε(γ) = ∇∗

Υ(Rn)Vε(γ) =

k∑i=1

∇viFεi (γ) +

k∑i=1

F εi (γ)(∇∗vi)⋆γ

=

k∑i=1

∇viFi(γ) +

k∑i=1

F εi (γ)(∇∗vi)⋆γ .

Here we used that Fi = F εi on any γ concentrated on the support of vi. The sought conclusion

(2.12) follows from the observation that F εi → Fi in Lp(Υ(Ω), π) and Vε → V in Lp(TΥ(Rn), π)

combined with (2.11). The last assertion is then a direct consequence from Proposition 2.5 and

(2.12).

BV FUNCTIONS AND SETS OF FINITE PERIMETER ON CONFIGURATION SPACES 11

Sobolev spaces. We now introduce the (1, p)-Sobolev space. The operator

(2.13) ∇Υ(Ω) : CylF(Υ(Ω)) ⊂ Lp(Υ(Ω), π) → CylV(Υ(Ω))

is densely defined and closable. The latter fact is a direct consequence of the integration by

parts formula (2.12). Indeed, we observe that, if Fn ∈ CylF(Υ(Ω)), Fn → 0 in Lp(Υ(Ω), π), and

∇Υ(Ω)Fn →W in Lp(TΥ(Ω)), π), then for any V ∈ CylV0(Υ(Ω)), it holdsˆΥ(Ω)

〈V,W 〉TγΥdπ(γ) = limn→∞

ˆΥ(Ω)

〈V,∇Υ(Ω)Fn〉TγΥdπ(γ) = − limn→∞

ˆΥ(Ω)

(∇∗Υ(Ω)V )Fndπ(γ) = 0 ,

yielding W = 0 as a consequence of the density of CylV0(Υ(Ω)) in Lp(TΥ(Ω)), π) by Proposi-

tion 2.5. The above argument justifies the following definition.

Definition 2.11 (H1,p-Sobolev spaces). Let 1 < p < ∞. We define H1,p(Υ(Ω), π) as the closure

of CylF(Υ(Ω)) in Lp(Υ(Ω), π) with respect to the following (1, p)-Sobolev norm:

‖F‖pH1,p(Υ(Ω)) := ‖F‖pLp(Υ(Ω)) + ‖∇Υ(Ω)F‖pLp(Υ(Ω)) .

We set ‖F‖H1,p := ‖F‖H1,p(Υ(Rn)). When p = 2, we write the corresponding Dirichlet form (i.e., a

closed form satisfying the unit contraction property [34, Def. 4.5]) by

EΥ(Ω)(F,G) :=

ˆΥ(Ω)

⟨∇Υ(Ω)F,∇Υ(Ω)G

⟩TγΥ(Ω)

dπ(γ), F,G ∈ H1,2(Υ(Ω), π) .

We set E := EΥ(Rn).

Remark 2.12 (The case of p = 1). As is indicated by (2.12), it is not true in general that

∇∗Υ(Ω)V ∈ L∞(Υ(Ω), π) since arbitrarily many finite particles can be concentrated on the supports

of inner functions of F ∈ CylF(Υ(Ω)) and vector fields vi. See [23, Example 4.35] for more details.

Due to this fact, the standard integration by part argument for the closability of the operator

∇Υ(Ω) : CylF(Υ(Ω)) → CylV(Υ(Ω)) ⊂ Lp(TΥ(Ω)) does not work in the case of p = 1. For this

reason, we restricted the definition of the H1,p-Sobolev spaces to the case 1 < p <∞ in Definition

2.11.

Once the closed form EΥ(Ω) on L2(Υ(Ω), π) is constructed, one can define the infinitesimal

generator on L2(Υ(Ω), π) as the unique non-positive definite self-adjoint operator by general theory

of functional analysis.

Definition 2.13 (Laplace operator [2, Theorem 4.1]). The L2(Υ(Ω), π)-Laplace operator ∆Υ(Ω)

with domain D(∆Υ(Ω)) is defined as the unique non-positive definite self-adjoint operator ∆Υ(Ω)

so that

EΥ(Ω)(F,G) = −ˆΥ(Ω)

(∆Υ(Ω)F )Gdπ, F ∈ D(∆Υ(Ω)), G ∈ D(EΥ(Ω)) .

In the case of Ω = Rn, employing (2.10) and (2.12), one can compute that

∆Υ(Rn)F := ∇∗Υ(Rn)∇Υ(Rn)F, ∀F ∈ CylF(Υ(Rn)) .

Let TΥ(Ω)t and GΥ(Ω)

α be the L2-contraction strongly continuous Markovian semigroup and

the resolvent, respectively, corresponding to the energy EΥ(Ω). We set Gα := GΥ(Rn)α and Tt :=

TΥ(Rn)t . By the Markovian property and Riesz–Thorin Interpolation Theorem, the L2-operators

TΥ(Ω)t and GΥ(Ω)

α can be uniquely extended to the Lp-contraction strongly continuous Markovian

semigroup and resolvent, respectively, for any 1 ≤ p <∞ (see e.g. [42, Section 2, p. 70]).

2.6. Product semigroups and exponential cylinder functions. In this section, we relate the

finite-product semigroup on Ω×k and the semigroup on Υk(Ω) when Ω ⊂ Rn is a bounded closed

domain with smooth boundary. To this aim we introduce a class of test functions, which is suitable

to compute the semigroups.

12 E. BRUE AND K. SUZUKI

Definition 2.14 (Exponential cylinder functions. [2, (4.12)]). Let Ω ⊂ Rn be a bounded closed

domain with smooth boundary. A class ECyl(Υ(Ω)) of exponential cylinder functions is defined as

the vector space spanned byexplog(1 + f)⋆

: f ∈ D(∆Ω) ∩ L2(Ω) ,∆Ωf ∈ L1(Ω), −δ ≤ f ≤ 0 for some δ ∈ (0, 1)

.

Here (∆Ω,D(∆Ω)) denotes the L2-Neumann Laplacian on Ω.

The space ECyl(Υ(Ω)) is dense in Lp(Υ(Ω)) for any 1 ≤ p <∞ (see [2, p. 479]). Noting that ∆Ω

is essentially self-adjoint on the core C∞c (Ω)∩ ∂f∂n = 0 in ∂Ω, where ∂

∂n is the normal derivative on

∂Ω, and the corresponding L2-semigroup TΩt is conservative, we can apply the same argument

in the proof of [2, Prop. 4.1] to obtain the following: TΥ(Ω)t ECyl(Υ(Ω)) ⊂ ECyl(Υ(Ω)) and

TΥ(Ω)t exp

log(1 + f)⋆

= exp

log(1 + (TΩ

t f))⋆

.(2.14)

Let TΩ,⊗kt be the k-tensor semigroup of TΩ

t , i.e. the unique semigroup in Lp(Ωk) satisfying

(2.15) TΩ,⊗kt f(x1, . . . , xk) := TΩ

t f1(x1) · · · TΩt fk(xk)

whenever f(x1, . . . , xk) = f1(x1) · . . . · fk(xk) where fi ∈ L∞(Ω) for any i = 1, . . . , k.

Proposition 2.15. Let Ω ⊂ Rn be a bounded closed domain with smooth boundary and 1 ≤ p <∞.

For F ∈ Lp(Υk(Ω), πΩ|Υk(Ω)), it holds

TΩ,⊗kt (F sk) = (T

Υk(Ω)t F ) sk, SknΩ×k -a.e.(2.16)

Proof. Since ECyl(Υk(Ω)) is dense in Lp(Υk(Ω)) for any 1 ≤ p <∞, it suffices to show (2.16) only

for F ∈ ECyl(Υk(Ω)). Furthermore, we can reduce the argument to the case F = explog(1+f)⋆

by using the linearity of semigroups. From (2.15) and (2.14) we get

TΩ,⊗kt (F sk)(x1, . . . , xk) = TΩ,⊗k

t (explog(1 + f)⋆ sk)(x1, . . . , xk)

= TΩ,⊗kt

( k∏i=1

(1 + f)(·i))(x1, . . . , xk)

=

k∏i=1

(1 + TΩ

t f(xi))

= explog(1 + (TΩ

t f)⋆)

sk(x1, . . . , xk)

= TΥ(Ω)t exp

log(1 + f)⋆

sk(x1, . . . , xk) .

The proof is complete.

2.7. Suslin sets. Let X be a set. We denote by NN the space of all infinite sequences nii∈N of

natural numbers. For ϕ ∈ NN, we write ϕ|l ∈ Nl for the restriction of ϕ to the first l elements, i.e.,

ϕl := (ϕi : 1 ≤ i ≤ l). Let S := ∪l∈NNl, and for σ ∈ S, we denote the length of the sequence σ

by #σ := #σi. Let E ⊂ 2X be a family of subsets in X. We write S(E ) for the family of sets

expressible in the following form: ⋃ϕ∈NN

⋂l≥1

Eϕ|l ,

for some family Eσσ∈S in E . A family Eσσ∈S is called Suslin scheme; the corresponding set

∪ϕ∈NN ∩l≥1 Eϕ|l is its kernel; the operation

Eσσ∈S 7→⋃ϕ∈NN

⋂l≥1

Eϕ|l ,

BV FUNCTIONS AND SETS OF FINITE PERIMETER ON CONFIGURATION SPACES 13

is called Suslin’s operation. Thus, S(E ) is the family of sets generated from sets in E by Suslin’s

operation, whose element is called an E -Suslin set. It is well-known that S(E ) is closed under

Suslin’s operation ([43], and e.g., [27, 421D Theorem]).

Assume that

(X, τ) is a Polish space, c is a Choque capacity on X, µ is a bounded Borel measure,

E := C(X) := C : closed set in X,(2.17)

and A ∈ S(E ) is simply called a Suslin set. We refer readers to, e.g., [27, 432I Definition] for the

definition of Choque capacity. If Eσ is compact for all σ ∈ S, we call Eσσ∈S a compact Suslin

scheme. We say that Eσσ∈S is regular if Eσ ⊂ Eτ whenever #τ ≤ #σ and σi ≤ τi for any

i < #σ ([27, 421X (n) & 422H Theorem (b)]).

Remark 2.16. Under the assumption (2.17), the following hold:

(i) Every Borel set is a Suslin set, i.e., B(τ) ⊂ S(E ) (e.g., [17, 6.6.7. Corollary]);

(ii) Every Suslin set is µ-measurable, i.e., S(E ) ⊂ Bµ(τ) (e.g., [27, 431B Corollary]);

(iii) Let A be a Suslin set in X. Then, A is the kernel of a compact regular Suslin scheme

Eσσ∈S . Furthermore, it holds that

c(A) = supψ∈NN

c(Aψ), Aψ =⋃ϕ≤ψ

⋂l≥1

Eϕ|l ,(2.18)

whereby ϕ ≤ ψ means that ϕ(l) ≤ ψ(l) for all l ∈ N (e.g., [27, 423B Theorem & the proof

of 432J Theorem]). By the regularity of Eσσ∈S , (2.18) can be reduced to the following

form:

(2.19) c(A) = supψ∈NN

c(Aψ), Aψ =⋂l≥1

Eψ|l , ψ ∈ NN;

(iv) A subset A ⊂ X is Suslin iff A is analytic iff A is K-analytic ([27, 423E Theorem (b)]).

3. Finite-codimensional Poisson measures

In this section, we construct finite-codimensional Poisson measures on Υ(Rn). As a first step

we prove measurability results for sections of Suslin subsets of the configuration space.

3.1. Measurability of sections of Suslin sets. Let B ⊂ Rn. For A ⊂ Υ(Rn) and η ∈ Υ(B),

the section Aη,B ⊂ Υ(Bc) of A at η is defined as

Aη,B = γ ∈ Υ(Bc) : γ + η ∈ A.(3.1)

The subset of Aη,B consisting of k-particle space Υk(Bc) is denoted by Akη,B := Aη,B ∩ Υk(Bc).

To shorten notation we often write Aη,r in place of Aη,Bcr, where Br is the closed ball centered at

the origin.

Lemma 3.1. Let B ⊂ Rn be a Borel set. If A is Suslin in Υ(Rn) then Akη,B is Suslin in Υk(Bc)

for any η ∈ Υ(B), k ∈ N and r > 0.

Proof. We can express Aη,B = prBc

(pr−1B (η)∩A

). The set pr−1

B (η)∩A is Suslin in Υ(Rn) wheneverA is Suslin. Set Υη,B(Rn) = pr−1

B (η)∩Υ(Rn), which is Suslin. The map prBc : Υη,B(Rn) → Υ(Bc)

is continuous by definition. Thus, Aη,B is the continuous image prBc

(pr−1B (η)∩A

)of the Suslin set

pr−1B (η)∩A in the Suslin Hausdorff space Υη,B(Rn). Hence, Aη,B is Suslin ([27, 423B Proposition

(b) & 423E Theorem (b)]). By noting that Akη,B = Aη,B ∩ Υk(Bc), and that Υk(Bc) is Borel in

Υ(Bc), we conclude that Akη,B is Suslin.

Lemma 3.2. Let B ⊂ Rn be an open set. Let A ⊂ Υ(Rn) be the kernel of compact Suslin’s scheme

Eσσ∈S , i.e., A = ∪ϕ∈NN ∩l≥1 Eϕ|l with Eσ compact for any σ ∈ S. Then, Aη,B is the kernel of

compact Suslin’s scheme (Eσ)η,rσ∈S .

14 E. BRUE AND K. SUZUKI

Proof. By expressing (Eσ)η,B = prBc

(Υη,B(Rn) ∩ Eσ

), where Υη,B(Rn) = pr−1

B (η) ∩ Υ(Rn), wesee that (Eσ)η,B is compact since Υη,B(Rn) is closed, Eσ is compact by the hypothesis, prBc is

continuous on Υη,B(Rn) and every continuous image of compact sets is compact. To see that Aη,Bis the kernel of (Eσ)η,rσ∈S ,

Aη,B = pr(Υη,B(Rn) ∩A

)= pr

(Υη,B(Rn) ∩

⋃ϕ∈NN

⋂l≥1

Eϕ|l

)= pr

( ⋃ϕ∈NN

⋂l≥1

Υη,B(Rn) ∩ Eϕ|l

)

=⋃ϕ∈NN

⋂l≥1

pr(Υη,B(Rn) ∩ Eϕ|l

)=⋃ϕ∈NN

⋂l≥1

(Eσ)η,B .

The proof is complete.

3.2. Localised finite-codimensional Poisson measures. In this section, we construct a lo-

calised version of the m-codimensional Poisson measure ρmr , which will be used to construct the

m-condimensional Poisson measure by taking the limit for r → ∞. We also show that Suslin sets

are contained in the domain of the finite-codimensional Poisson measure.

Let A ⊂ Υ(Rn) be a Suslin subset. By Lemma 3.1, the set Akη,r = Akη,Bcris Suslin, thus also

analytic by (iv) of Remark 2.16. Since Sm,kBris an outer measure on Υk(Br) by construction, Sm,kBr

is a Choque capacity. Therefore, all analytic sets are contained in the completion of the domain

of Sm,kBr(see e.g. [27, 432J]), hence Sm,kBr

(Akη,r) is well-defined. We define the domain Dm of the

m-codimensional measures by

(3.2) Dm :=⋂r>0

Dmr ,

where the localised domain Dmr is defined by

Dmr := A ⊂ Υ(Rn) : the map Υ(Bcr) 3 η 7→ Sm,kBr

(Akη,r) is πBcr-measurable for every k .

We first introduce the m-codimensional Poisson measure on the configuration space Υ(Br) over

the ball Br.

Definition 3.3. The m-codimensional Poisson measure ρmΥ(Br)on Υ(Br) is defined as

(3.3) ρmΥ(Br)(A) := e−Sn(Br)

∞∑k=1

Sm,kBr(Ak) for any Suslin set A in Υ(Br) ,

where Ak = A ∩Υk(Br).

Remark 3.4. Notice that ρ0Υ(Br)= πBr

, in other words the 0-codimension Poisson measure ρ0Υ(Br)

on Υ(Br) is the Poisson measure πBron Υ(Br). It can be shown by noting that the m-dimensional

spherical Hausdorff measure Sm and the n-dimensional Lebesgue measure Ln coincide whenm = n

(see Remark 2.1).

We introduce the localised m-codimensional Poisson measure on Υ(Rn) by averaging the m-

codimensional Poisson measure ρmΥ(Br)by means of πBc

r.

Definition 3.5. The localised m-codimensional Poisson measure ρmr on Υ(Rn) is defined by

ρmr (A) =

ˆΥ(Bc

r)

ρmΥ(Br)(Aη,r)dπBc

r(η), ∀A ∈ Dm.(3.4)

Before investigating the main properties of ρmr , we check that sufficiently many sets are contained

in Dm, i.e. we show that all Suslin sets are contained in the domain Dm for m ≤ n.

Proposition 3.6. Any Suslin set in Υ(Rn) is contained in Dm for m ≤ n.

BV FUNCTIONS AND SETS OF FINITE PERIMETER ON CONFIGURATION SPACES 15

Proof. Let A ⊂ Υ(Rn) be a Suslin set. Let Eσσ∈S be a Suslin scheme whose kernel is A. Noting

that Υ(Bcr) is Polish, by applying (i) of Remark 2.16 with X = Υ(Bcr) and µ = πBcr, any Suslin

set is πBcr-measurable. Hence, it suffices to show that every super-level set η : Sm,kBr

(Akη,r) > a is

Suslin for any a ∈ R, r > 0, k ∈ N and m ≤ n. Note that Akη,r is Suslin by Lemma 3.1, whence the

expression η : Sm,kBr(Akη,r) > a is well-defined as was discussed in the paragraph before (3.2).

Since Υ(Rn) is Polish, by using (iii) in Remark 2.16, we may assume that Eσσ∈S is a compact

regular Suslin scheme. By Lemma 3.2 and Υ(Br) = tk∈NΥk(Br), we see that A

kη,r ⊂ Υk(Br) is the

kernel of the compact regular Suslin scheme (Eσ)kη,rσ∈S , whereby (Eσ)kη,r := (Eσ)η,Bc

r∩Υk(Br).

Since Sm,kBris an outer measure on Υk(Br) by construction, Sm,kBr

is a Choque capacity on Υk(Br).

Hence, by applying (2.19) in (iii) of Remark 2.16 with X = Υk(Br) and c = Sm,kBr, we obtain that

Sm,kBr(Akη,r) = sup

ψ∈NNSm,kBr

((Akη,r)ψ), (Akη,r)ψ =⋂l≥1

(Eψ|l)kη,r, ψ ∈ NN.

Thus, noting the monotonicity Sm,kBr,ε≤ Sm,kBr,δ

(δ ≤ ε) of the ε-Hausdorff measure defined in (2.3),

the super-level set η : Sm,kr (Akη,r) > a can be expressed in the following way:

η : Sm,kBr(Akη,r) > a =

⋃ε>0

⋃ψ∈NN

η : Sm,kBr,ε

((Akη,r)ψ

)> a.

Since the space S(E ) of Suslin sets is closed under Suslin’s operation, it suffices to show that

η : Sm,kBr,ε

((Akη,r)ψ

)> a is Suslin.

We equip Υk(Br) with the L2-transportation distance dΥk as defined in (2.2), and equip Υ(Bcr)

with some distance d generating the vague topology. By Proposition 2.2 and noting that (Akη,r)ψ is

compact and that Sm,kBr,εis (up to constant multiplication) the m-codimensional ε-spherical Haus-

dorff measure on Υk(Br) associated with dΥk , we conclude that η : Sm,kBr,ε

((Akη,r)ψ

)> a is open

in Υ(Bcr) for any a ∈ R, r > 0, k ∈ N and m ≤ n. The proof is complete.

3.3. Finite-codimensional Poisson Measures. In this section, we construct them-codimensional

Poisson measure on Υ(Rn), which is the first main result of this paper. By Proposition 3.6, the

set function ρmr given in (3.4) turned out to be well-defined in the sense that the space S(E ) of all

Susline sets in Υ(Rn) are contained in its domain Dm. We show the following monotonicity result

which allows us to pass to the limit of ρmr as r → ∞.

Theorem 3.7. The map r 7→ ρmr (A) is monotone non-decreasing for any A ∈ S(E ).

The proof of Theorem 3.7 is given at the end of this section. We can now introduce the m-

codimensional Poisson measure on Υ(Rn) as the monotone limit of ρmr on the space S(E ) of Suslin

sets:

ρm(A) = limr→∞

ρmr (A), ∀A ∈ S(E ).(3.5)

Definition 3.8 (m-codimensional Poisson Measure). Let Dm be the completion of S(E ) with

respect to ρm. The measure (ρm,Dm) is called the m-codimensional Poisson measure on Υ(Rn).

Remark 3.9. We give two remarks below:

(i) Note ρ0 = π, i.e. 0-codimensional Poisson measure ρ0 on Υ(Rn) is the Poisson measure

π on Υ(Rn) by noting that the m-dimensional spherical Hausdorff measure Sm and the

n-dimensional Lebesgue measure Ln coincide when m = n (see Remark 2.1).

(ii) The construction of ρm, a priori, depends on the choice of the exhaustion Br ⊂ Rn.However, in Proposition 3.13, we will see that it is not the case.

The rest of this section is devoted to the proof of Theorem 3.7. Let us begin with a definition.

16 E. BRUE AND K. SUZUKI

Definition 3.10 (Section of Functions, Multi-Section). Let M,N ⊂ Rn be two disjoint sets and

L =M tN . For every F : Υ(L) → R and ξ ∈ Υ(M), define Fξ,M : Υ(N) → R as

Fξ,M (ζ) := F (ζ + ξ), ζ ∈ Υ(N).(3.6)

For a set A ⊂ Υ(Rn), let Aξ,η,M,N denote the multi-section both at ξ ∈ Υ(M) and ζ ∈ Υ(N):

(3.7) Aξ,ζ,M,N := γ ∈ Υ(Lc) : γ + ξ + ζ ∈ A, and Akξ,ζ,M,N = Aξ,ζ,M,N ∩Υk(Lc).

Lemma 3.11. Let A be a Suslin set in Υ(Rn). Let M,N ⊂ Rn be two disjoint Borel sets. Set

L =M tN . Let F : Υ(L) → R be defined by γ 7→ F (γ) := Sm,kLc (Akγ,L). Then,

Fξ,M (ζ) = Sm,kLc (Akζ,ξ,N,M ), ∀ξ ∈ Υ(M), ∀ζ ∈ Υ(N).(3.8)

Proof. The set Akζ,ξ,N,M is Suslin by the same argument as in Lemma 3.1. Thus, Sm,kLc (Akζ,ξ,N,M )

is well-defined. By Definition 3.10, we have that

Fξ,M (ζ) = F (ζ + ξ) = Sm,kLc (Akζ+ξ,L) = Sm,kLc

(γ ∈ Υ(Lc) : γ + ξ + ζ ∈ A

)= Sm,kLc (Akζ,ξ,N,M ) .

The proof is complete.

The next lemma is straightforward by the independence of the Poisson measures πM and πN .

Lemma 3.12. Suppose the same notation M,N and L as in Lemma 3.11. For any bounded

measurable function G on Υ(L),ˆΥ(L)

G(η)dπL(η) =

ˆΥ(N)

ˆΥ(M)

Gξ,M (ζ)dπM (ξ)dπN (ζ).(3.9)

Proof of Theorem 3.7. Let Ar,ε := Br+ε\Br be the annulus of width ε and radius r. Fix A ∈ S(E ),

r > 0, ε > 0 and ζ ∈ Υ(Bcr+ε). We claim that

(3.10) Sm,kBr+ε(Akζ,Bc

r+ε) ≥

k∑j=0

ˆΥ(Ar,ε)

Sm,kBr(Ak−jζ,ξ,Bc

r+ε,Ar,ε)dSjAr,ε

(ξ).

Let us first show how (3.10) concludes the proof. For simplicity of notation, we set M = Ar,ε,

N = Bcr+ε and L =M tN . Then, (3.10) is reformulated as follows:

Sm,kNc (Akζ,N ) ≥k∑j=0

ˆΥ(M)

Sm,kLc (Ak−jζ,ξ,N,M )dSjM (ξ).

Then, by using Lemma 3.12 and Lemma 3.11 we deduce

ρmr (A) = e−Sn(Lc)∞∑k=0

ˆΥ(L)

Sm,kLc (Akη,L)dπL(η)

= e−Sn(Lc)∞∑k=0

ˆΥ(N)

ˆΥ(M)

(Sm,kLc (Akζ,L)

)ξ,M

dπM (ξ)dπN (ζ)

= e−Sn(Lc)∞∑k=0

ˆΥ(N)

ˆΥ(M)

Sm,kLc (Akζ,ξ,N,M )dπM (ξ)dπN (ζ)

= e−Sn(Lc)e−Sn(M)∞∑k=0

k∑j=0

ˆΥ(N)

ˆΥ(M)

Sm,k−jLc (Ak−jζ,ξ,N,M )dSjM (ξ)dπN (ζ)

≤ e−Sn(Nc)∞∑k=0

ˆΥ(N)

Sm,kNc (Akζ,N )dπN (ζ)

= ρmr+ε(A) .

BV FUNCTIONS AND SETS OF FINITE PERIMETER ON CONFIGURATION SPACES 17

To show (3.10), it is enough to verify that, for any bounded measurable function F on Υ(Rn),

(3.11)

ˆΥ(Nc)

Fζ,N (γ)dSm,kNc (γ) ≥k∑j=0

ˆΥ(M)

ˆΥ(Lc)

(Fζ,N )ξ,M (γ)dSm,k−jLc (γ)dSjM (ξ).

By the definition of Sm,kNc , the L.H.S. of (3.11) can be deduced as follows:ˆΥ(Nc)

Fζ,N (γ)dSm,kNc (γ) =1

k!

ˆ(Nc)⊗k

(Fζ,N sk)(xk)dSnk−mNc (xk),

whereby xk := (x0, . . . , xk−1) and x0 = x0. Furthermore, by the definition of (Fζ,N )ξ,M , the R.H.S.

of (3.11) can be deduced as follows:ˆΥ(M)

ˆΥ(Lc)

(Fζ,N )ξ,M (γ)dSm,k−jLc (γ)dSjM (ξ)

=

ˆΥ(M)

ˆΥ(Lc)

(Fζ,N )(γ + ξ)dSm,k−jLc (γ)dSjM (ξ)

=1

j!(k − j)!

ˆM×j

ˆ(Lc)×(k−j)

(Fζ,N sk)(xk−j ,yj)dSn(k−j)−mLc (xk−j)dSnjM (yj),

whereby (xk−j ,yj) = (x0, . . . , xk−j−1, y0, . . . , yj−1). Hence, in order to conclude (3.11), it suffices

to show the following inequality: for any bounded measurable symmetric function f on (Rn)×k,ˆB×k

r+ε

f(xk)dSnk−mBr+ε

(xk) ≥k∑j=0

k!

j!(k − j)!

ˆB

×(k−j)r

ˆA×j

r,ε

f(xk−j ,yj)dSn(k−j)−mAr,ε

(xk−j)dSnjBr

(yj).

By using the symmetry of f and a simple combinatorial argument, we obtain

ˆB×k

r+ε

f(xk)dSnk−mBr+ε

(xk) =

k∑j=0

k!

j!(k − j)!

ˆB

×(k−j)r

ˆA×j

r,ε

f(xk−j ,yj)dSnk−mBr+ε

(xk−j ,yj),

while [25, 2.10.27, p. 190] impliesˆB

×(k−j)r

ˆA×j

r,ε

f(xk−j ,yj)dSnk−mBr+ε

((xk−j), (yj))

≥ˆB

×(k−j)r

ˆA×j

r,r+ε

f(xk−j ,yj)dSn(k−j)−mAr,ε

((xk−j))dSnjBr

(yj).

The proof is complete.

3.4. Independence of ρm from the exhaustion. So far we have built the m-codimensional

measure ρm by passing to the limit a sequence of finite dimensional measures ρmr . The latter have

been constructed by relying on the exhaustion Br : r > 0 of Rn. Hence, a priori, ρm depends

on the chosen exhaustion. In this subsection we make a remark that actually it is not the case.

Let Ω ⊂ Rn be a compact set. Following closely the proof in section 3.3 we can prove that

ρmΩ (A) := e−Sn(Ω)∞∑k=1

ˆΥ(Ωc)

Sm,kΩ (Akη,Ωc) dπBcr(η).(3.12)

is well defined for any Suslin set A.

The next proposition can be proven by arguing as in Theorem 3.7. We omit the proof.

Proposition 3.13 (Independence of Exhaustion). Let 0 < r < R <∞ and Ω ⊂ Rn be a compact

subset satisfying Br ⊂ Ω ⊂ BR. Then

(3.13) ρmr (A) ≤ ρmΩ (A) ≤ ρmR (A) , for any Suslin set A .

In particular ρm does not depend on the choice of the exhaustion.

18 E. BRUE AND K. SUZUKI

4. Bessel capacity and finite-codimensional Poisson measure

In this section, we discuss an important relation between Bessel capacities and finite-codimensional

Poisson measures ρm. This will play a significant role to develop fundamental relations between

potential theory and theory of BV functions in Section 5 and Section 7.

Definition 4.1 (Bessel operator). Let α > 0 and 1 ≤ p <∞. We set

Bα,p :=1

Γ(α/2)

ˆ ∞

0

e−ttα/2−1T(p)t dt ,(4.1)

where T(p)t is the Lp-heat semigroup, see Section 2.5.

Notice that Bα,p is well defined for F ∈ Lp(Υ(Rn), π) and satisfies

(4.2) ‖Bα,pF‖Lp ≤ ‖F‖Lp ,

due to the contractivity of T(p)t in Lp(Υ(Rn), π).

Definition 4.2 (Bessel capacity). Let α > 0 and 1 ≤ p <∞. The (α, p)-Bessel capacity is defined

as

Capα,p(E) := inf‖F‖pLp : Bα,pF ≥ 1 on E, F ≥ 0 ,(4.3)

for any E ⊂ Υ(Rn).

We are now ready to state the main theorem of this section.

Theorem 4.3. Let αp > m. Then, Capα,p(E) = 0 implies ρm(E) = 0 for any E ∈ S(E ).

Let us briefly explain the heuristic idea of proof. In view of the identities

ρm(E) = limr→∞

ρmr (E) ,

ρmr (E) = e−Sn(Br)∞∑k=1

ˆΥ(Bc

r)

Sm,kBr(Ekη,r)dπBc

r(η) ,

it is enough to prove that Sm,kBr(Ekη,r) = 0 for πBc

r-a.e. η, all k ∈ N and r > 0. This, along with the

implication

Capα,p(E) = 0 =⇒ Capη,rα,p(Ekη,r) = 0, for πBc

r-a.e. η and all k ∈ N and r > 0 ,(4.4)

where Capη,rα,p is the Bessel (α, p)-capacity on Υk(Br), reduces the problem to a finite dimensional

one. More precisely we will show that

Capη,rα,p(Ekη,r) = 0 =⇒ Sm,kBr

(Ekη,r) = 0 ,

which is a standard result of finite dimensional spaces.

In the rest of this section, we make the aforementioned idea rigorous. The key point is to show

(4.4), for which we introduce localisations of functional analytic objects in Section 4.1 and Sec-

tion 4.2. We then introduce localised Bessel operators and localised Bessel capacities in Section 4.3.

4.1. Localization of sets and functions.

Lemma 4.4. Let A ⊂ Υ(Rn) be a π-measurable set. Let B ⊂ Rn be a Borel set. Then, Aη,Bis πBc-measurable for πB-a.e. η ∈ Υ(B). Moreover, if π(A) = 0, then πBc(Aη,B) = 0 for a.e.

η ∈ Υ(B).

Proof. By hypothesis, there exist Borel sets A ⊂ A ⊂ A so that π(A\A) = 0. By (i) in Remark 2.16,

A and A are Suslin. By Lemma 3.1, Aη,B and Aη,B are Suslin. By the standard disintegration

argument as in Lemma 3.12, it holds that

0 = π(A \A) =ˆΥ(B)

πBc((A \A)η,B)dπB(η).(4.5)

BV FUNCTIONS AND SETS OF FINITE PERIMETER ON CONFIGURATION SPACES 19

Therefore, there exists a πB-measurable set Ω ⊂ Υ(B) so that πBc((A \A)η,B) = 0 for any η ∈ Ω.

By noting that Aη,B ⊂ Aη,B ⊂ Aη,B , we concluded that Aη,B is πB-measurable since, up to

πB negligible sets, it coincides with a Suslin set and every Suslin set is πB-measurable by (ii) in

Remark 2.16. The proof of the first assertion is complete.

If π(A) = 0 the disintegration

(4.6) 0 = π(A) =

ˆΥ(B)

πBc(Aη,B)dπB(η) ,

immediately gives the second assertion.

Corollary 4.5. Let A ⊂ Υ(Rn) be a π-measurable set, B ⊂ Rn a Borel set, and let g be a

π-measurable function on Υ(Rn) with g ≥ 1 π-a.e. on A. Then, for πB-a.e. η it holds

gη,B ≥ 1, πBc-a.e. on Aη,B .(4.7)

Proof. Taking A = A \ g ≥ 1 and applying Lemma 4.4 with A in place of A, we obtain the

conclusion.

Lemma 4.6. Let 1 ≤ p < ∞ and r > 0. Let Fn, F ∈ Lp(Υ(Rn), π) such that Fn → F in

Lp(Υ(Rn), π) as n→ ∞. Then, there exists a subsequence (non-relabelled) of (Fn) and a measur-

able set Ar ⊂ Υ(Rn) so that πBcr(Ar) = 1 and

Fnη,r → Fη,r, in Lp(πBr ), for any η ∈ Ar.

Note that Fη,r := Fη,Bcrwas defined in Definition 3.10.

Proof. By Lemma 3.12, we have thatˆΥ(Bc

r)

(ˆΥ(Br)

|Fnη,r − Fη,r|pdπBr

)dπBc

r(η) =

ˆΥ(Rn)

|Fn − F |pdπ → 0, as n→ ∞ .(4.8)

In particular, up to subsequence´Υ(Br)

|Fnη,r −Fη,r|pdπBr → 0 for πBcr-a.e. η, which completes the

proof.

4.2. Localisation of energies, resolvents and semigroups. In this section, we localise differ-

ential operators and related objects introduced in Section 2.5.

Let r > 0. The localised energy (Er,D(Er)) is defined by the closure of the following pre-Dirichlet

form

(4.9) Er(F ) =ˆΥ(Rn)

|∇rF |2TΥdπ F ∈ CylF(Υ),

where

(4.10) ∇rF (γ, x) := χBr(x)∇F (γ, x) .

The closability follows from [36, Lemma 2.2], or [23, the first paragraph of the proof of Thm. 3.48].

We denote by Grαα>0 and T rt t>0 the L2-resolvent operator and the semigroup associated with

(Er,D(Er)), respectively. The next result is taken from [36, Lemma 2.2] or [23, the second paragraph

of the proof of Thm. 3.48].

Proposition 4.7. The form (Er,D(Er)) is monotone non-decreasing in r, i.e. for any s ≤ r,

D(Er) ⊂ D(Es), Es(F ) ≤ Er(F ), F ∈ D(Er).

Furthermore, limr→∞ Er(F ) = E(F ) for F ∈ H1,2(Υ(Rn), π).

The next proposition shows the monotonicity property for the resolvent operator Grα and the

semigroup T rt .

20 E. BRUE AND K. SUZUKI

Proposition 4.8. The resolvent operator Grαα and the semigroup T rt t are monotone non-

increasing on non-negative functions, i.e.,

GrαF ≤ GsαF, T rt F ≤ T st F, for any nonnegative F ∈ L2(Υ, π), s ≤ r.(4.11)

Furthermore, limr→∞GrαF = GαF and limr→∞ T rt F = TtF for F ∈ L2(Υ, π) and α, t > 0.

Proof. Thanks to the identity

Grα =

ˆ ∞

0

e−αtT rt dt ,

it suffices to show (4.11) only for T rt . By a direct application of [37, Theorem 3.3] and the

monotonicity of the Dirichlet form in Proposition 4.7, we obtain the monotonicity of the semigroup.

The second part of the statement follows from the monotone convergence Er ↑ E combined with

[33, S.14, p.372].

The next proposition is taken from [36, (4.1), Prop. 4.1] or [23, Proposition 3.45].

Proposition 4.9. For any F ∈ CylF(Υ), η ∈ Υ(Bcr) and r > 0, it holds

Er(F ) =ˆΥ(Bc

r)

EΥ(Br)(Fη,r)dπBcr(η) ,(4.12)

where EΥ(Br) is the energy on Υ(Br) defined in Definition 2.11, and Fη,r = Fη,Bcrwas introduced

in Definition 3.10.

Note that (EΥ(Br),CylF(Υ(Br))) is closable and the closure is denoted by (EΥ(Br),H1,2(Υ(Br), π)),

see Definition 2.11. Recall that GΥ(Br)α α and TΥ(Br)

t denote the L2-resolvent operator and the

semigroup corresponding to (EΥ(Br),H1,2(Υ(Br), π)).

We now provide the relation between Grαα>0, T rt t>0 and GΥ(Br)α α, TΥ(Br)

t .

Proposition 4.10. Let α > 0, t > 0, and r > 0 be fixed. Then, for any bounded measurable

function F , it holds that

GrαF (γ) = GΥ(Br)α Fγ|Bc

r,r(γ|Br

) ,(4.13)

T rt F (γ) = TΥ(Br)t Fγ|Bc

r,r(γ|Br

) ,(4.14)

for π-a.e. γ ∈ Υ(Rn).

Proof. Let F,H ∈ CylF(Υ). Set RrαF (·) := GΥ(Br)α F·|Bc

r,r(·|Br

). By definition,(RrαF

)η,r

(·) = GΥ(Br)α Fη,r(·), πBr

-a.e. on Υ(Br) for πBcr-a.e. η.(4.15)

By Proposition 4.9, we get

Er(RrαF,H) =

ˆΥ(Rn)

EΥ(Br)((RrαF )η,r,Hη,r)dπ(η)

=

ˆΥ(Rn)

EΥ(Br)(GΥ(Br)α Fη,r,Hη,r)dπ(η)

=

ˆΥ(Rn)

(ˆΥ(Br)

(Fη,r − αGΥ(Br)α Fη,r)Hη,rdπBr

)dπBc

r(η)

=

ˆΥ(Rn)

(F − αRrαF )H dπ ,(4.16)

where the second line follows from (4.15), and in the third line we used the fundamental equality

−∆GΥ(Br)α Fη,r + αG

Υ(Br)α Fη,r = Fη,r. Since the resolvent operator Grα is the unique L2-bounded

operator satisfying

Er(GrαF,H) =

ˆΥ(Rn)

(F − αGrαF )H dπ ,

BV FUNCTIONS AND SETS OF FINITE PERIMETER ON CONFIGURATION SPACES 21

we conclude that GrαF (γ) = RrαF (γ) = GΥ(Br)α Fγ|Bc

r,r(γ|Br ) π-a.e. γ. Thus the proof of (4.13) is

complete.

The second conclusion (4.14) follows from the identityˆ ∞

0

e−αtT rt F (γ)dt = GrαF (γ)dt = GΥ(Br)α Fγ|Bc

r,r(γ|Br

) =

ˆ ∞

0

e−αtTΥ(Br)t Fγ|Bc

r,r(γ|Br

)dt,

and the injectivity of the Laplace transform on continuous functions in t.

Remark 4.11. Although Proposition 4.10 provides the statement only for the L2-semigroups and

resolvents, it is straightforward to extend it to the Lp-semigroups and resolvents for any 1 ≤ p <∞.

4.3. Localized Bessel operators. Let Brα,p and BΥ(Br)α,p be the (α, p)-Bessel operators corre-

sponding to T rt t>0 and TΥ(Br)t t>0, respectively defined in the analogous way as in (4.1). The

corresponding (α, p)-Bessel capacities are denoted by Caprα,p and CapΥ(Br)α,p defined in the analogous

way as in (4.3)

Lemma 4.12. Caprα,p(E) ≤ Capα,p(E) for any E ⊂ Υ(Rn) and r > 0.

Proof. It suffices to show that Brα,pF ≤ Bα,pF for any F ≥ 0 with F ∈ Lp(Υ, π), which immediately

follows from Proposition 4.8 and (4.1).

Lemma 4.13. If Capα,p(E) = 0, then CapΥ(Br)α,p (Eη,r) = 0 for πBc

r-a.e. η and any r > 0.

Proof. By Lemma 4.12 we may assume Caprα,p(E) = 0 for any r > 0. Let Fn ⊂ Lp(Υ, π) be a

sequence so that Fn ≥ 0, Brα,pFn ≥ 1 on E, and ‖Fn‖pLp → 0. By Lemma 4.5, (Fn)η,r ≥ 0 for πBcr-

a.e. η. Furthermore, by Lemma 4.6, there exists Ar ⊂ Υ(Bcr) and a (non-relabelled) subsequence

(Fn)η,r so that πBcr(Ar) = 1, and for every η ∈ Ar,

(Fn)η,r → 0, in Lp(Υ(Br), πBr) .(4.17)

By Proposition 4.10 and Remark 4.11, we have that

(Brα,pFn)η,r =

(1

Γ(α/2)

ˆ ∞

0

e−ttα/2−1T rt Fndt

)η,r

=1

Γ(α/2)

ˆ ∞

0

e−ttα/2−1(T rt Fn

)η,rdt

=1

Γ(α/2)

ˆ ∞

0

e−ttα/2−1TΥ(Br)t (Fn)η,rdt

= BΥ(Br)α,p (Fn)η,r.(4.18)

Note that we dropped the specification of p in the semigroups for the notational simplicity in

(4.18).

Since Brα,pFn ≥ 1 on E, by applying Corollary 4.5, we obtain that (Brα,pFn)η,r ≥ 1 on Eη,r for

πBcr-a.e. η. Thus, by (4.18), B

Υ(Br)α,p (Fn)η,r ≥ 1 on Eη,r for πBc

r-a.e. η. By (4.17), we conclude that

CapΥ(Br)α,p (Eη,r) = 0 for πBc

r-a.e. η and any r > 0. The proof is complete.

4.4. Finite-dimensional counterpart. In this section, we develop the finite-dimensional coun-

terpart of Theorem 4.3. The goal is to prove the following proposition.

Proposition 4.14. Let αp > m. If CapΥk(Br)

α,p (E) = 0, then Sm,kBr(E) = 0 for any k ∈ N.

Proof. Recall that TΩ,⊗kt is the k-tensor semigroup of TΩ

t as defined in (2.15). Let BB×k

rα,p be the

corresponding Bessel operator defined analogously as in (4.1), and CapB×kr

α,p be the corresponding

(α, p)-capacity.

22 E. BRUE AND K. SUZUKI

Let Fm ⊂ Lp(Υ(Br), πBr) be a sequence so that Fm ≥ 0 and B

Υk(Br)α,p Fm ≥ 1 on E ⊂ Υk(E),

and ‖Fm‖Lp → 0. By Proposition 2.15 and the definition of Bessel operator, we have

BΥk(Br)α,p Fm sk = B

B×kr

α,p (Fm sk) ,

hence Fm sk ≥ 0, BB×k

rα,p (Fm sk) ≥ 1 on s−1

k (E). Furthermore,

‖Fm sk‖Lp(B×kr ) = C(k, n, r)‖Fm‖Lp(Υk(Br)) → 0 , as m→ ∞ ,

where C(k, n, r) > 0 comes from the constant appearing in front of the Hausdorff measure in

the definition of πBr . This implies that CapB×kr

α,p (s−1k (E)) = 0. We can now rely on standard

capacity estimates in the Euclidean setting (see, e.g. [44, Theorem 2.6.16]) to conclude that

Snk−m(s−1k (E)) = 0. Recalling (2.4), we have that

Sm,kBr(E) =

1

k!(sk)#S

nk−m(E) =1

k!Snk−m(s−1

k (E)) = 0 .

The proof is complete.

4.5. Proof of Theorem 4.3. Let E ∈ S(E ) such that Capα,p(E) = 0. Thanks to Lemma 4.13

we have CapΥ(Br)α,p (E) = 0 for any r > 0, hence Sm,kBr

(Ekη,r) = 0 for any k ∈ N as a consequence of

Proposition 4.14. It implies

ρmr (E) = e−Sn(Br)∞∑k=1

ˆΥ(Bc

r)

Sm,kBr(Ekη,r)dπBc

r(η) = 0 ,

for any r > 0. Recalling that ρmr (E) ↑ ρm(E) by (3.5), we obtain the sought conclusion.

5. BV Functions

In this section, we introduce functions of bounded variations (called BV functions) on Υ(Rn) fol-lowing three different approaches: the variational approach (§5.1), the relaxation approach (§5.2),and the semigroup approach (§5.3). In Section 5.5, we prove that they all coincide.

5.1. Variational approach. Let us begin by introducing a class of BV functions through inte-

gration by parts. We then discuss localisation properties.

Definition 5.1 (BV functions I: variational approach). Let Ω ⊂ Rn be either a closed domain

with smooth boundary or Rn. For F ∈ ∪p>1Lp(Υ(Ω), π), we define the total variation as

VΥ(Ω)(F ) := sup

ˆΥ(Rn)

(∇∗Υ(Ω)V )Fdπ : V ∈ CylV0(Υ(Ω)), |V |TΥ(Ω) ≤ 1

.(5.1)

We set V(F ) := VΥ(Rn)(F ). We say that F is BV in the variational sense if V(F ) <∞.

Remark 5.2. The assumption F ∈ ∪p>1Lp(Υ(Ω), π) plays an important role in Definition 5.1,

ensuring that´Υ(Ω)

(∇∗V )Fdπ is well defined for any V ∈ CylV(Υ(Ω)). Indeed, one can easily

prove that ∇∗Υ(Ω)V ∈ ∪1≤p<∞L

p(Υ(Ω), π) for any V ∈ CylV0(Υ(Ω)), but it is not L∞(Υ(Ω), π) in

general.

Remark 5.3. As was shown in Remark 2.8, the set of V ∈ CylV0(Υ(Ω)) with |V |TΥ ≤ 1 is not

empty and dense in CylV0(Υ(Ω)) with respect to the point-wise topology and the Lp(TΥ) topology

for 1 ≤ p <∞.

In order to localize the total variation we employ a family of cylinder vector fields concentrated

on Br, for some r > 0.

BV FUNCTIONS AND SETS OF FINITE PERIMETER ON CONFIGURATION SPACES 23

Definition 5.4. For F ∈ ∪p>1Lp(Υ(Rn), π), we define the localized total variation as

Vr(F ) := sup

ˆΥ(Rn)

(∇∗V )Fdπ : V ∈ CylVr0(Υ(Rn)), |V |TΥ(Rn) ≤ 1

,(5.2)

where

CylVr0(Υ(Rn)) :=V (γ, x) =

k∑i=1

Fi(γ)vi(x) : Fi ∈ CylF(Υ(Rn)), vi ∈ C∞0 (Br;Rn), k ∈ N

.

The next result shows that VΥ(Br)(Fη,r) <∞ for πBcr-a.e. η whenever Vr(F ) <∞. It is the key

step to perform our nonlinear dimension reduction. Indeed it allows to reduce the study of BV

functions on Υ(Rn) to their sections, which live on the finite dimensional space Υ(Br).

Proposition 5.5. Let r > 0 and p > 1. For F ∈ Lp(Υ(Rn), π) with Vr(F ) <∞, it holdsˆΥ(Bc

r)

VΥ(Br)(Fη,r)dπBcr(η) = Vr(F ) .(5.3)

Let us begin with a simple technical lemma.

Lemma 5.6. Let r > 0. For all V ∈ CylVr0(Υ(Rn)), and F ∈ CylF(Υ(Rn)) it holdsˆΥ(Bc

r)

(ˆΥ(Br)

Fη,r(γ)∇∗Υ(Br)

Vη,r(γ)dπBr (γ)

)dπBc

r(η) =

ˆΥ(Rn)

F∇∗V dπ .(5.4)

Proof of Lemma 5.6. Recall that for r > 0 and η ∈ Υ(Bcr) we have Vη,r ∈ CylV0(Br). By the

divergence formula (2.12) and the disintegration Lemma 3.12, we have thatˆΥ(Bc

r)

(ˆΥ(Br)

Fη,r(γ)∇∗Υ(Br)

Vη,r(γ)dπBr(γ)

)dπBc

r(η)

= −ˆΥ(Bc

r)

(ˆΥ(Br)

Fη,r(γ)( k∑i=1

∇vi(Fi)η,r(γ) +

k∑i=1

(Fi)η,r(γ)(∇∗vi)⋆(γ)

)dπBr

(γ)

)dπBc

r(η)

= −ˆΥ(Bc

r)

ˆΥ(Br)

(F( k∑i=1

∇viFi +

k∑i=1

Fi(∇∗vi)⋆))

η,r

(γ)dπBr(γ)dπBc

r(η)

= −ˆΥ(Bc

r)

F( k∑i=1

∇viFi +

k∑i=1

Fi(∇∗vi)⋆)dπ

=

ˆΥ(Rn)

F∇∗V dπ .

The proof is complete.

Proof of Proposition 5.5. We first prove thatˆΥ(Bc

r)

VΥ(Br)(Fη,r)dπBcr(η) ≥ Vr(F ) .(5.5)

Let Vi ∈ CylVr0(Υ(Rn)) with |Vi|TΥ ≤ 1 so that

Vr(F ) = limi→∞

ˆΥ(Rn)

(∇∗Vi)Fdπ.

Observe that (Vi)η,r ∈ CylV0(Υ(Br)), then by definition of VΥ(Br)(Fη,r) we getˆΥ(Br)

((∇∗Vi)F )η,rdπBr=

ˆΥ(Br)

(∇∗Υ(Br)

(Vi)η,r)Fη,rdπBr≤ VΥ(Br)(Fη,r), ∀i ∈ N .

24 E. BRUE AND K. SUZUKI

Therefore, by Lemma 5.6,

Vr(F ) = limi→∞

ˆΥ(Rn)

(∇∗Vi)Fdπ

= limi→∞

ˆΥ(Bc

r)

ˆΥ(Br)

(∇∗Υ(Br)

(Vi)η,r)Fη,rdπBrdπBcr(η)

≤ˆΥ(Bc

r)

VΥ(Br)(Fη,r)dπBcr(η) ,

which completes the proof of (5.5).

Let us now pass to the proof of the opposite inequalityˆΥ(Bc

r)

VΥ(Br)(Fη,r)dπBcr(η) ≤ Vr(F ) .(5.6)

We divide it into three steps.

Step 1. We show the existence of Vi : i ∈ N ⊂ CylV0(Υ(Br)) such that |Vi|TΥ ≤ 1 and

(5.7) VΥ(Br)(G) = supi∈N

ˆΥ(Br)

(∇∗Υ(Br)

Vi)GdπBr,

for any G ∈ ∪p>1Lp(Υ(Br), π).

First we observe that there exists FF := Gi : i ∈ N ⊂ CylF(Υ(Br)) such that any cylinder

function can be approximated strongly in H1,q(Υ(Br)) for any q < ∞, by elements of FF . Let

D ⊂ C∞0 (Br;Rn) be a countable dense subset, w.r.t. the C1-norm: ‖v‖C1 := ‖∇v‖L∞ + ‖v‖L∞ .

We define the countable family

FV :=

βV (γ, x)ϕα(|V |TγΥ) : V (γ, x) =

m∑i=1

wi(x)Gi(γ), α, β ∈ Q+, m ∈ N, wi ∈ D, Gi ∈ FF

,

where ϕα ∈ C∞([0,∞)) satisfies 0 ≤ ϕα ≤ 1, |ϕ′α| ≤ 2/α and ϕα(t) = 1 on [0, 1 + α], ϕ(t) = 0 on

[1 + 2α,∞).

Fix δ > 0, q ∈ [1,∞) and V ∈ CylV0(Υ(Br)) with |V |TγΥ ≤ 1. To prove (5.7) it suffices to

show that there exists W ∈ FV with |W |TΥ ≤ 1 such that ‖∇∗(V −W )‖Lq ≤ δ.

Fix t ∈ (q, 2q) and ε ∈ (0, 1/9). Letting V =∑mi=1 Fivi ∈ CylV0(Υ(Br)), we pick Gi ∈ FF and

wi ∈ D such that

(5.8)

m∑i=1

(‖vi − wi‖C1 + ‖Fi −Gi‖Lt(Υ(Br)) + ‖∇(Fi −Gi)‖Lt(Υ(Br))

)< ε,

and consider W :=∑mi=1 wiGi. By using the divergence formula (2.12), we can obtain that

(5.9)

ˆΥ(Br)

|∇∗(W − V )|tdπBr+

ˆΥ(Br)

∣∣|W |TγΥ − |V |TγΥ

∣∣tdπBr≤ Cεt ,

where C = max‖wi‖C1 , ‖Gi‖Lt(Υ(Br)), ‖∇Gi‖Lt(Υ(Br)) : 1 ≤ i ≤ m does not depend on ε. We

assume without loss of generality that ε, ε1

10t ∈ Q and set

(5.10) W := (1− 2ε1

10t )ϕε

110t

(|W |2TγΥ

)W ∈ FV ,

which satisfies

|W |TγΥ = (1− 2ε1

10t )ϕε

110t

(|W |2TγΥ

)|W |TγΥ ≤ (1− 2ε

110t )(1 + 2ε

110t ) ≤ 1 .

We now check that ‖∇∗(V −W )‖Lq ≤ δ. From the identity

∇∗W = (1− 2ε1

10t )ϕε

110t

(|W |2TγΥ

)(∇∗W )− 2(1− 2ε

110t )ϕ′

ε1

10t

(|W |2TγΥ

)|W |2TγΥ ,

BV FUNCTIONS AND SETS OF FINITE PERIMETER ON CONFIGURATION SPACES 25

and the inequality∣∣ϕ′ε

110t

(|W |2TγΥ

)∣∣|W |2TγΥ ≤ 2ε−1

10tχ1+ε

110t ≤|W |2TγΥ≤1+2ε

110t

|W |2TγΥ ≤ 5ε−1

10tχ|W |2TγΥ≥1+ε

110t

,

we obtain

‖∇∗(W − W )‖Lq ≤∥∥∥((1− 2ε

110t )ϕ

ε1

10t(|W |2TγΥ)− 1

)(∇∗W )

∥∥∥Lq

+ 5ε−1

10t

∥∥∥χ|W |2TγΥ≥1+ε1

10t

∥∥∥Lq

≤ 5ε1

10t ‖∇∗W‖Lq +∥∥∥χ|W |2TγΥ≥1+ε

110t

(∇∗W )∥∥∥Lq

+ 5ε−1

10t

∥∥∥χ|W |2TγΥ≥1+ε1

10t

∥∥∥Lq

≤ C(‖∇∗W‖Lt , t, q

)(ε

110t + ε−

110t

∥∥∥χ|W |2TγΥ≥1+ε1

10t

∥∥∥Lt

),(5.11)

where we estimated ‖χ|W |2TγΥ≥1+ε

110t

(∇∗W )‖Lq by means of the Holder inequality and using that

t < 2q. The Chebyshev inequality and (5.9) give∥∥∥χ|W |2TγΥ≥1+ε1

10t

∥∥∥Lt

≤∥∥∥χ|W |TγΥ≥1+ε

13t

∥∥∥Lt

≤∥∥∥χ||W |TγΥ−|V |TγΥ|≥ε

13t

∥∥∥Lt

≤(ε−

13t

∥∥|W |TγΥ − |V |TγΥ

∥∥L1

)1/t≤ Cε

1t−

13t2 ≤ Cε

12t (ε < 1),

where C = max‖wi‖C1 , ‖Gi‖L1(Υ(Br)), ‖∇Gi‖L1(Υ(Br)) : 1 ≤ i ≤ m is independent of ε. There-

fore, we conclude

‖∇∗(W − V )‖Lq ≤ ‖∇∗(W − W )‖Lq + ‖∇∗(W − V )‖Lq

≤ C(ε1

10t + ε15t ) + ε ≤ δ ,

provided ε is small enough. The proof of (5.7) is complete.

Step 2. We conclude the proof of (5.6).

Note that the map γ 7→ F (γ)∇∗Υ(Br)

V (γ|Br) is π-measurable. Furthermore, by Lemma 4.4,

Fη,Bcris πBr

-measurable and the map

Υ(Bcr) 3 η 7→ˆΥ(Br)

(∇∗Υ(Br)

V )Fη,BcrdπBr

is πBcr-measurable. Therefore, the map η 7→ VΥ(Br)(Fη,Bc

r) is πBc

r-measurable.

Fix now ε > 0 and define a sequence Cj : j ∈ N of subsets in Υ(Bcr) so that C0 = ∅, and

Cj :=η ∈ Υ(Bcr) : Fη,r is πBr

-measurable and,

ˆΥ(Br)

(∇∗Υ(Br)

Vj)Fη,rdπBr ≥ (1− ε)VΥ(Br)(Fη,r) ∧ ε−1\j−1⋃i=1

Ci ,

where the family Vi : i ∈ N has been built in Step 1.

Then, Cj is πBcr-measurable for any j and πBc

r(Υ(Bcr) \ ∪∞

j=1Cj) = 0. Set

W ηn (γ) :=Wn(γ + η) :=

n∑j=1

Vj(γ)χCj(η), γ ∈ Υ(Br), η ∈ Υ(Bcr) .

We approximate χCj by F iji∈N ⊂ CylF(Υ(Bcr)) with |F ij | ≤ 1 in the strong Lp′(Υ(Bcr), πBc

r)

topology, where 1p′ +

1p = 1. Thus, setting W i

n(γ + η) :=∑nj=1 Vj(γ)F

ij (η), we see thatˆ

Υ(Bcr)

‖∇∗Υ(Br)

(Wn −W in)(·+ η)‖Lp′ (Υ(Br))

dπBcr(η) → 0 as i→ ∞ .

26 E. BRUE AND K. SUZUKI

Notice that W in ∈ CylVr0(Υ(Rn)), hence

limi→∞

ˆΥ(Bc

r)

(ˆΥ(Br)

(∇∗Υ(Br)

W in(·+ η))fη,rdπBr

)dπBc

r(η)(5.12)

=

ˆΥ(Bc

r)

ˆΥ(Br)

(∇∗Υ(Br)

W ηn )fη,rdπBr

πBcr(η)

=

ˆΥ(Bc

r)

n∑j=1

χCj (η)

ˆΥ(Br)

(∇∗Υ(Br)

Vj)fη,rdπBr

dπBcr(η)

≥ (1− ε)

ˆΥ(Bc

r)

n∑j=1

χCj(η)VΥ(Br)(fη,r) ∧ ε

−1

dπBcr

= (1− ε)

ˆ∪n

j=1Cj

VΥ(Br)(fη,r) ∧ ε−1dπBc

r(η).

By Lemma 5.6,

(5.13)

ˆΥ(Rn)

(∇∗W in)fdπ =

(ˆΥ(Br)

(∇∗Υ(Br)

W in(·+ η))fη,rdπBr

)dπBc

r(η) ,

which along with (5.12) gives the claimed inequality by letting i→ ∞ and n→ ∞.

5.2. Relaxation approach. In this subsection we introduce a second notion of functions with

bounded variations. We rely on a relaxation approach.

Definition 5.7 (BV functions II: Relaxation). Let F ∈ L1(Υ(Rn)), we define the total variation

of F by

(5.14) |D∗F |(Υ(Rn)) := inflim infn→∞

‖∇Fn‖L1(TΥ) : Fn → F in L1(Υ(Rn)) , Fn ∈ CylF(Υ(Rn)) .

If |D∗F |(Υ(Rn)) <∞, we say that F has finite relaxed total variation.

Definition 5.8 (Total variation pre-measure). If |D∗F |(Υ(Rn)) <∞, we define a map

|D∗F | : G ∈ CylF(Υ(Rn)) : G is nonnegative → R ,

(5.15)

|D∗F |[G] := inf

lim infn→∞

ˆΥ(Rn)

G|∇Fn|TΥdπ : Fn → F in L1(A, π) , Fn ∈ CylF(Υ(Rn))

.

Notice that |D∗F |[G] ≤ ‖G‖L∞ |D∗F | and |D∗F |[G1 + G2] ≥ |D∗F |[G1] + |D∗F |[G2]. By con-

struction, |D∗F |[G] is the lower semi-continuous envelope of the functional CylF(Υ(Rn)) 3 F 7→´Υ(Rn)

G|∇F |TΥdπ. Therefore, the map F 7→ |D∗F |(G) is lower semi-continuous with respect to

the L1-convergence for any nonnegative G ∈ CylF(Υ(Rn)).

It will be shown in Corollary 7.4 that |D∗F | is represented by a finite measure |DF |, i.e.

|D∗F |[G] =ˆΥ(Rn)

Gd|DF | for any nonnegative G ∈ CylF(Υ(Rn)) .

Remark 5.9. Our definitions above deviate slightly from the ones considered in the literature of

BV functions on metric measure spaces [10, 35].

Given a metric measure structure (X, d,m) and F ∈ L1(X,m), usually the total variation

|D∗F |(X) is defined by considering regularization sequences Fn ∈ Lipb(X) that are Lipschitz and

bounded. The supports of the latter are fine enough to prove that, for any open set A ⊂ X, the

BV FUNCTIONS AND SETS OF FINITE PERIMETER ON CONFIGURATION SPACES 27

map

(5.16) |D∗F |(A) := inflim infn→∞

‖∇Fn‖L1(A) : Fn → F in L1(A,m) , Fn ∈ Lipb(X) ,

is a restriction to open set of a finite measure, provided |D∗F |(X) <∞.

To capture the geometric-analytic structure of the configuration space Υ(Rn), which is an ex-

tended metric measure structure, is natural to replace Lipschitz functions by Cylinder functions

as a class of test objects. The supports of the latter, however, are coarser than open sets, hence it

is not natural to define the total variation measure via (5.16). Definition 5.8 provides the natural

replacement of (5.16) in our setting.

5.3. Heat semigroup approach. In this subsection we present the third approach to BV func-

tions. We employ the heat semigroup to define the total variation of a function F ∈ Lp(Υ(Rn)),p > 1.

Proposition 5.10. Let F ∈ ∪p>1Lp(Υ(Rn), π). Then ‖∇TtF‖L1 < ∞ for any t > 0 and the

following limit exists

(5.17) T (F ) := limt→0

‖∇TtF‖L1 .

Definition 5.11 (BV functions III: Heat semigroup). A function F ∈ ∪p>1Lp(Υ(Rn), π) is BV in

the sense of the heat semigroup if T (F ) <∞. We define the total variation of F by T (F ).

To prove Proposition 5.10, we need the Bakry-Emery inequality with exponent q = 1, i.e. for

any t, s > 0, F ∈ ∪p>1Lp(Υ(Rn), π), it holds

(5.18)

ˆΥ(Rn)

|∇TtF |dπ <∞ , |∇Tt+sF | ≤ Tt|∇TsF | π-a.e. .

The inequality (5.18) will be proven in Corollary 5.15 in Section 5.4. Let us now use it to show

Proposition 5.10.

Proof of Proposition 5.10. Let F ∈ Lp(Υ, π) for p > 1. By (5.18), we see that

‖∇TtF‖L1 ≤ lim infs→0

‖∇Tt+sF‖L1 ≤ lim infs→0

‖∇TsF‖L1 .

By taking lim supt→0, we obtain lim supt→0 ‖∇TtF‖L1 ≤ lim infs→0 ‖∇TsF‖L1 , which concludes

the proof.

5.4. p-Bakry-Emery inequality. In order to complete the proof of Proposition 5.10, we show

the p-Bakry-Emery inequality for the Hodge heat flow, which implies in turn the scalar version

(5.18). It will play a significant role also in the proof of Theorem 5.17.

Definition 5.12 (Hodge Laplacian). Let us consider V =∑mk=1 Fkvk ∈ CylV(Υ(Rn)) with Fk =

Φk((fk1 )⋆, . . . , (fkℓ )

⋆). Define Hodge Laplacian of V as

∆HV (γ, x) :=

m∑k=1

ℓ∑i,j=1

∂2Φk∂xi∂xj

((fk1 )

⋆γ, . . . , (fkℓ )⋆γ)⟨∇fki ,∇fkj

⟩TγΥ

vk(x)(5.19)

+

m∑k=1

ℓ∑i=1

∂Φk∂xi

((fk1 )

⋆γ, . . . , (fkℓ )⋆γ)(∆fk)

⋆γvk(x)

+

m∑k=1

Φk((fk1 )

⋆γ, . . . , (fkℓ )⋆γ)∆Hvk(x),

where ∆Hvk is the Hodge Laplacian of vk ∈ C∞(Rn;Rn). It turns out that ∆HV does not depend

on the choice of both the representative of V and the inner and outer functions of Fk (see [1,

Theorem 3.5]).

28 E. BRUE AND K. SUZUKI

We define the corresponding energy functional:

(5.20) EH(V,W ) := 〈−∆HV,W 〉L2(TΥ,π) =

ˆΥ(Rn)

ΓΥ(V,W )dπ , V,W ∈ CylV(Υ(Rn)) ,

where ΓΥ denotes the square field operator associated with ∆H . By [1, Theorem 3.5], the form EH is

closable on CylV(Υ(Rn)) and the corresponding closure is denoted by D(EH) and the corresponding

(Friedrichs) extension of CylV(Υ(Rn)) is denoted by D(∆H). Let Tt denote the corresponding

L2-semigroup. By a general theory of functional analysis, it holds that

(5.21) TtV ∈ D(EH) , for any t ≥ 0 and V ∈ CylV(Υ(Rn)) .

The following intertwining property holds.

Theorem 5.13. ∇TtF = Tt∇F for any t ≥ 0 and for any F ∈W 1,2(Υ(Rn), π).

Proof. Using the commutation ∇∆ = ∆H∇ of the operators at the level of the base space Rn, wecan show ∇∆F = ∆H∇F for any F ∈ CylF(Υ(Rn)) by the representation (5.19). By the essential

self-adjointness of (∆,CylF(Υ(Rn))) (see [2, Theorem 5.3]), and applying [41, Theorem 2.1] with

D = CylF(Υ(Rn)), A = ∆, A = ∆H , Tt = Tt, R = 0, we conclude the sought conclusion.

Theorem 5.14. Let F ∈ D(EH). Then |TtF |TΥ ≤ Tt|F |TΥ π-a.e. for any t ≥ 0. In particular Tt

can be extended to the Lp-velocity fields Lp(TΥ, π) for any 1 ≤ p <∞.

Proof. By the Weitzenbock formula [1, Theorem 3.7] on Υ(Rn), we can express ∆H = ∇∗∇+RΥ,

where RΥ is the lifted curvature tensor from the base space Rn. Using the flatness Rn we can

easily deduce RΥ = 0.

Now, setting Γ(V,W ) := ΓΥ(V,W ) + 2RΥ(V,W ) = ΓΥ(V,W ) we can apply [42, Theorem 3.1]

(see the proof of [42, Theorem 3.1] for p = 1) and [42, Proposition 3.5], to get the sought conclusion

of the first assertion.

We now prove the second assertion. Let V ∈ Lp(TΥ), the density of cylinder vector fields gives

the existence of a sequence Vn ∈ CylF(Υ) ⊂ D(EH) such that |Vn − V |TΥ → 0 in Lp(Υ(Rn), π) asn→ ∞. We can define

(5.22) TtV := limn→∞

TtVn .

The existence of the limit follows from

(5.23) |TtVn −TtVm|TΥ ≤ Tt|Vn − Vm|TΥ ,

as well as the independence of the limit from the approximating sequence (Vn)n∈N.

Theorem 5.15 (p-Bakry-Emery estimate). Let p > 1. The following assertions hold:

(i) Tt : H1,p(Υ(Rn), π) → H1,p(Υ(Rn), π) is a continuous operator for any t > 0.

(ii) For any F ∈ H1,p(Υ(Rn), π),

(5.24) |∇TtF |pTΥ ≤ Tt|∇F |pTΥ π-a.e. .

(iii) Let 1 < p ≤ 2. For any F ∈ Lp(Υ(Rn), π) it holds that

‖∇TtF‖Lp ≤ C(p)t−1/2‖F‖Lp , ∀t > 0.(5.25)

In particular, Tt : Lp(Υ(Rn), π) → H1,p(Υ(Rn), π) is a continuous operator for any t > 0.

(iv) For any t, s > 0, F ∈ Lp(Υ(Rn), π), it holds that ‖∇TtF‖L1(Υ(Rn)) <∞ and

(5.26) |∇Tt+sF |TΥ ≤ Tt|∇TsF |TΥ π-a.e. .

Proof. (i). By Theorem 5.13 and Theorem 5.14, for any F ∈ CylF(Υ(Rn)) it holds that

(5.27) |∇TtF |TΥ = |Tt∇F |TΥ ≤ Tt|∇F |TΥ π-a.e. .

BV FUNCTIONS AND SETS OF FINITE PERIMETER ON CONFIGURATION SPACES 29

A simple application of Jensen’s inequality to (5.27) gives

(5.28) |∇TtF |pTΥ ≤ Tt|∇F |pTΥ , for any F ∈ CylF(Υ) and p ≥ 1 .

Let Fn ∈ CylF(Υ) be a H1,p(Υ(Rn), π)-Cauchy sequence. Then, by (5.28) and the invariance

π(Ttf) = π(f),

ˆΥ(Rn)

|∇Tt(Fn − Fm)|pTΥdπ ≤ˆΥ(Rn)

Tt|∇(Fn − Fm)|pTΥdπ =

ˆΥ(Rn)

|∇(Fn − Fm)|pTΥdπ → 0.

(5.29)

Since H1,p(Υ(Rn), π) is the closure of CylF(Υ) w.r.t. the norm ‖∇ · ‖Lp(Υ,π) + ‖ · ‖Lp(Υ,π), by

(5.29), the operator Tt is extended to H1,p(Υ(Rn), π) continuously. The proof of the first assertionis complete.

(ii). Let F ∈ H1,p(Υ(Rn), π) and take Fn ∈ CylF(Υ) converging to F in H1,p(Υ(Rn), π).Then, by the lower semi-continuity of |∇ · |pTΥ w.r.t. the Lp-strong convergence, the continuity of

the Lp-semigroup Tt and the inequality (5.28), we obtain

|∇Tt+sF |pTΥ = |∇TtTsF |pTΥ ≤ lim infn→∞

|∇TtTsFn|pTΥ ≤ lim infn→∞

Tt|∇TsFn|pTΥ ≤ Tt|∇TsF |pTΥ.

Here the last equality follows from the assertion (i).

(iii). Let p > 1 be fixed. For any F ∈ CylF(Υ(Rn)) satisfying F ≥ 0, it holds

p(p− 1)

ˆ t

0

ˆΥ(Rn)

|∇TsF |2TΥ|TsF |p−2dπds =

ˆΥ(Rn)

|F |pdπ −ˆΥ(Rn)

|TtF |pdπ ≤ˆΥ(Rn)

|F |pdπ .

It follows by studying first ddt

´Υ(Rn)

|Tt(F + ε)|dπ, and after letting ε→ 0.

By the contraction property of Tt, we obtain

ˆ t

0

ˆΥ(Rn)

|∇TsF |pTΥdπds ≤

(ˆ t

0

ˆΥ(Rn)

|TsF |pdπds

) 2−p2(ˆ t

0

ˆΥ(Rn)

|∇TsF |2TΥ|TsF |p−2dπdsds

) p2

≤ Ct2−p2 ‖F‖pLp .

We now employ the Bakry-Emery inequality (5.28) combined with the contraction property of Ttto show that s→

´Υ(Rn)

|∇TsF |pTΥdπ is nonincreasing, which yields

(5.30) t

ˆΥ(Rn)

|∇TtF |pTΥdπ ≤ˆ t

0

ˆΥ(Rn)

|∇TsF |pTΥdπds ≤ Ct2−p2 ‖F‖pLp .

This implies our conclusion for cylinder functions. We extended it to any F ∈ Lp(Υ(Rn), π) by

means of a density argument. Indeed, given F ∈ Lp(Υ(Rn), π), we can find Fn ∈ CylF(Υ(Rn)) suchthat Fn → F in Lp. The continuity of the semigroup Tt gives TtFn → TtF in Lp, while the lower

semicontinuity of the functional G→´Υ(Rn)

|∇G|pTΥ(Rn)dπ with respect to the Lp convergence for

p > 1 yields ˆΥ(Rn)

|∇TtF |pTΥdπ ≤ lim infn→∞

ˆΥ(Rn)

|∇TtFn|pTΥdπ ≤ Ct−1/2‖F‖Lp .

(iv). Note that the assertion in the case of 1 < p ≤ 2 implies the one in the case of p > 2 by

Lp(Υ(Rn), π) ⊂ Lq(Υ(Rn), π) whenever 1 ≤ q ≤ p. Thus, we only need to prove it in the case

of 1 < p ≤ 2. Let F ∈ Lp(Υ(Rn), π). Then, by the assertion (iii), TsF ∈ H1,p(Υ(Rn), π). Take

Gn converging to TsF in H1,p(Υ(Rn), π). Then, up to taking a subsequence from Gn, and by

making use of (5.27), we conclude that

|∇Tt+sF |TΥ = |∇TtTsF |TΥ = limn→∞

|∇TtGn|TΥ ≤ limn→∞

Tt|∇Gn|TΥ = Tt|∇TsF |TΥ .

The proof is complete.

30 E. BRUE AND K. SUZUKI

Remark 5.16. In [24], the 2-Bakry-Emery estimate was proved in the case of the configuration

space over a complete Riemannian manifold with Ricci curvature bound. For the purpose of the

current paper, however, we need a stronger estimate, i.e., the p-Bakry-Emery estimate (5.24) for

arbitrary 1 < p <∞ and also the regularity estimate (5.25) of the heat semigroup, both of which

do not follow only from the 2-Bakry-Emery inequality.

5.5. Equivalence of BV functions. In Section 5, we introduced the three different definitions

(the variational/the relaxation/the semigroup approaches) of BV functions. In this section we show

that the three different definitions of BV functions are equivalent and we introduce a universal

definition of BV functions for L2(Υ(Rn), π)-functions.

Theorem 5.17 (Equivalence of BV functions). Let F ∈ L2(Υ(Rn), π). Then,

V(F ) = |D∗F |(Υ(Rn)) = T (F ) .

The proof of Theorem 5.17 will be given later in this section. Thanks to Theorem 5.17, we can

introduce a universal definition of BV functions for L2(Υ(Rn), π)-functions.

Definition 5.18 (BV functions). A function F ∈ L2(Υ(Rn)) belongs to BV(Υ(Rn)) if

V(F ) = |D∗F |(Υ(Rn)) = T (F ) <∞ .

We prepare several lemmas for the proof of Theorem 5.17.

Lemma 5.19. For any V ∈ CylV(Υ(Rn)) and t ≥ 0 it holds

(5.31) (∇∗TtV ) = Tt(∇∗V ) .

In particular (∇∗TtV ) ∈ Lp(Υ(Rn)) for any 1 < p <∞.

Proof. Let F ∈ CylF(Υ). By the π-symmetry of Tt and Theorem 5.13, we have thatˆΥ(Rn)

F Tt(∇∗V )dπ =

ˆΥ(Rn)

TtF (∇∗V )dπ = −ˆΥ(Rn)

〈V (γ, ·),∇TtF (γ)〉TΥdπ

= −ˆΥ(Rn)

〈V (γ, ·),Tt∇F (γ)〉TΥdπ = −ˆΥ(Rn)

〈TtV (γ, ·),∇F (γ)〉TΥdπ

=

ˆΥ(Rn)

F (∇∗TtV )dπ,

which immediately implies (5.31). The proof is complete.

Let us now introduce Dp(TΥ(Rn), π), the space of vector fields with divergence in Lp(Υ(Rn), π),as the closure of CylV(Υ(Rn)) ⊂ Lp(TΥ(Rn)) with respect to the norm ‖V ‖Lp + ‖∇∗V ‖Lp .

In the case p = 2, we have the following inclusion

(5.32) D(EH) ⊂ D2(TΥ(Rn), π) ,

as a consequence of the inequality ‖∇∗V ‖L2 ≤ EH(V, V ) for any V ∈ CylV(Υ(Rn)).

Lemma 5.20. Let 1 < p < ∞ and 1 < p′ < ∞ such that 1/p + 1/p′ = 1. If F ∈ Lp′(Υ(Rn), π)

then

(5.33) V(F ) = sup

ˆΥ(Rn)

(∇∗V )Fdπ : V ∈ Dp(TΥ(Rn), π), |V |TΥ ≤ 1

.

Proof. Let V ∈ Dp(TΥ(Rn), π) with |V |TΥ ≤ 1, to conclude the proof we just need to build a

sequence (Wn)n∈N ⊂ CylV(Υ(Rn)) such that |Wn| ≤ 1 and ‖∇∗V −∇∗Wn‖Lp → 0 as n→ ∞. To

that aim we first consider a sequence Vn ∈ CylV(Υ(Rn)) such that ‖V −Vn‖Lp+‖∇∗V −∇∗Vn‖Lp →0 as n→ ∞, which exists by definition. We now define Wn by cutting Vn of as we did in (5.10) in

the proof of Proposition 5.5.

BV FUNCTIONS AND SETS OF FINITE PERIMETER ON CONFIGURATION SPACES 31

Proof of Theorem 5.17. We first show the inequality V(F ) ≤ |D∗F |(Υ(Rn)) in the general case of

F ∈ Lp(Υ(Rn), π) with 1 < p ≤ ∞.

Let F ∈ Lp(Υ(Rn), π) for some p > 1 and |D∗F |(Υ(Rn)) < ∞. Let Fn ∈ CylF(Υ) such that

Fn → F in L1(Υ(Rn)) and ‖∇Fn‖L1(TΥ) → |D∗F |(Υ(Rn)). Let Fn,M := (Fn ∨ −M) ∧M and

FM := (F ∨−M) ∧M . Then, Fn,M → FM in L1(Υ(Rn), π) and ‖∇Fn,M‖L1(TΥ) ≤ ‖∇Fn‖L1(TΥ).

Thus, lim supn→∞ ‖∇Fn,M‖L1(TΥ) ≤ |D∗F |(Υ(Rn)). By the integration by parts formula (2.11),

it holdsˆΥ(Rn)

Fn,M∇∗V dπ = −ˆΥ(Rn)

〈V,∇Fn,M 〉TΥdπ ≤ ‖∇Fn,M‖L1(TΥ) ≤ ‖∇Fn‖L1(TΥ) ,

for any V ∈ CylV(Υ(Rn)) with |V |TΥ ≤ 1. By taking a (non-relabelled) subsequence from

Fn,M so that Fn,M → FM π-a.e., and using the dominated convergence theorem (notice that

|Fn,M∇∗V | ≤M |∇∗V | ∈ L1(Υ(Rn), π) uniformly in n), we obtain thatˆΥ(Rn)

FM∇∗V dπ = limn→∞

ˆΥ(Rn)

Fn,M∇∗V dπ ≤ lim infn→∞

‖∇Fn‖L1(TΥ) ≤ |D∗F |(Υ(Rn)),

for any V ∈ CylV(Υ(Rn)) with |V |TΥ ≤ 1. Since FM → F in Lp(Υ(Rn), π) as M → ∞ by the

hypothesis F ∈ Lp(Υ(Rn), π), we conclude V(F ) ≤ |DF |(Υ(Rn)).

We now show the inequality |D∗F |(Υ(Rn)) ≤ V(F ) < ∞. Let F ∈ L2(Υ(Rn), π). Set Fn =

T1/nF ∈ H1,2(Υ(Rn), π). By the symmetry of Tt in L2(TΥ, π) and Lemma 5.19, we have that, for

any V ∈ CylV(Υ(Rn)) with |V |TΥ ≤ 1, it holdsˆΥ(Rn)

Fn∇∗V dπ =

ˆΥ(Rn)

T1/n(∇∗V )Fdπ =

ˆΥ(Rn)

∇∗(T1/nV )Fdπ.(5.34)

The inclusion (5.21) and (5.32) imply that T1/nV ∈ D2(TΥ(Rn), π), while Theorem 5.14 ensures

that |T1/nV |TΥ ≤ T1/n|V |TΥ ≤ 1. Therefore, we can apply Lemma 5.20 to (5.34) to obtain

‖∇Fn‖L1 ≤ V(F ). Since Fn ∈ H1,2(Υ(Rn), π) and CylF(Υ(Rn)) is dense in H1,2(Υ(Rn), π), wehave |D∗Fn|(Υ(Rn)) ≤ ‖∇Fn‖L1 by definition. By the lower semi-continuity |D∗F |(Υ(Rn)) with

respect to the L2-convergence, it holds

|D∗F |(Υ(Rn)) ≤ lim infn→∞

|D∗Fn|(Υ(Rn)) ≤ lim infn→∞

‖∇Fn‖L1(TΥ,π) ≤ V(F ).

This conclude the proof of V(F ) = |D∗F |(Υ(Rn)).

We now prove T (F ) ≤ |D∗F |(Υ(Rn)). Let Fn ∈ CylF(Υ) such that Fn → F in L1(Υ(Rn), π)and ‖∇Fn‖L1(TΥ) → |D∗F |(Υ(Rn)). Then, by the 1-Bakry–Emery inequality (5.27) on cylinder

functions,

‖∇TtF‖L1 ≤ lim infn→∞

‖∇TtFn‖L1 ≤ lim infn→∞

‖∇Fn‖L1 = |D∗F |(Υ(Rn)).

Thus, T (F ) ≤ |D∗F |(Υ(Rn)).

Finally we prove V(F ) ≤ T (F ). For F ∈ Lp(Υ(Rn), π) and V ∈ CylV(Υ(Rn)) with |V |TΥ ≤ 1,

we have that ˆΥ(Rn)

TtF∇∗V dπ =

ˆΥ(Rn)

〈∇TtF, V 〉dπ ≤ˆΥ(Rn)

|∇TtF |TΥdπ.

Since TtF → F in Lp(Υ(Rn), π), we obtain thatˆΥ(Rn)

F∇∗V dπ ≤ limt→0

ˆΥ(Rn)

|∇TtF |TΥdπ.

Thus, we conclude V(F ) ≤ T (F ). The proof is complete.

32 E. BRUE AND K. SUZUKI

Remark 5.21. The proof of all the inequalities except |D∗F |(Υ(Rn)) ≤ V(F ) remains true for

any 1 < p <∞. In order to prove the inequality |D∗F |(Υ(Rn)) ≤ V(F ) in full generality following

the same strategy we need show that TtV ∈ Dp(TΥ(Rn), π) for 1 < p < ∞ and V ∈ CylV(TΥ).

This should follow, for instance, from the Lp–boundedness of vector-valued Riesz transforms, and

will be addressed in a future work.

6. Sets of finite perimeter

In this section we introduce and study the notion of set with finite perimeter. Let us begin with

a definition

Definition 6.1 (Sets of finite perimeters). Let Ω ⊂ Rn be either a closed domain or the Euclidean

space Rn. A Borel set E ⊂ Υ(Ω) is said to have finite perimeter if VΥ(Ω)(χE) <∞.

We refer the reader to Definition 5.1 for the introduction of the total variation VΥ(Ω)(·).

6.1. Sets of finite perimeter in Υ(Br). We first develop the necessary theory in the configura-

tion space Υ(Br), in which every argument essentially comes down to finite-dimensional geometric

analysis since only finitely many particles are allowed to belong to Br.

Let us recall the decomposition Υ(Br) =⊔k≥0 Υ

k(Br), where (Υk(Br), dΥk , πkBr) is the k-

particle configuration space Υk(Br) over Br equipped with the L2-transportation distance dΥk

and πkBr:= πBr

|Υk(Br). We introduce the reduced boundary in Υ(Br).

Definition 6.2 (Reduced boundary in Υ(Br)). Fix r > 0. Given E ⊂ Υ(Br), set Ek := E∩Υk(Br)

and define

∂∗Υ(Br)E :=

⊔k≥0

∂∗Υk(Br)Ek ,

∂∗Υk(Br)Ek :=

γ ∈ Υk(Br) : lim sup

s→0

πkBr(Bks(γ) ∩ Ek)πkBr

(Bks(γ))> 0, lim sup

s→0

πkBr(Bks(γ) \ Ek)πkBr

(Bks(γ))> 0

,

where Bks(γ) denotes the metric ball of radius s > 0 centered at γ ∈ Υk(Br) w.r.t. dΥk .

We can readily show that the m-codimensional Hausdorff measure ρmΥk(Br)w.r.t. dΥk coincides

with the push-forwarded measure of the m-codimensional spherical Hausdorff measure ρmB×k

ron

B×kr w.r.t. the quotient map sk:

ρmΥk(Br)= (sk)#ρ

mB×k

r= (sk)#S

nk−mB×k

r,(6.1)

where Snk−mB×k

ris the m-codimensional spherical Hausdorff measure on B×k

r and sk is the quotient

map B×kr → Υk(Br) as defined in Section 2. Having this in mind, we prove the following Gauß–

Green formula in Υ(Br).

Proposition 6.3 (Gauß–Green formula in Υ(Br)). Fix r > 0. If E ⊂ Υ(Br) is a set of finite

perimeter then there exists a vector field σE : Υ(Br) → TΥ(Br) such that |σE |TΥ(Br) = 1 ρ1Υ(Br)-

a.e. on ∂∗Υ(Br)E, and

(6.2)

ˆE

(∇∗V )dπBr=

ˆ∂∗Υ(Br)

E

〈V, σE〉dρ1Υ(Br)for any V ∈ CylV(Υ(Br)).

Moreover VΥ(Br)(χE) = ρ1Υ(Br)(∂∗Υ(Br)

E).

Proof. Exploiting the decomposition Υ(Br) =⊔k≥0 Υ

k(Br), where each Υk(Br) is a connected

component, we reduce our analysis to the study of Ek := E ∩Υk(Br).

Set Ek := s−1k (Ek). Given

V =

m∑k=1

Φ(f∗1,k, . . . , f∗nk,k

)vk ∈ CylV(Υ(Br)) ,

BV FUNCTIONS AND SETS OF FINITE PERIMETER ON CONFIGURATION SPACES 33

we can define V ∈ C∞0 (B×k

r ;Rnk) as

V(x1, . . . , xk) =

m∑k=1

Φ(f1,k(x1) + . . .+ fnk,k(xk), . . . , fnk,k(x1) + . . .+ fnk,k(xk))vk(x1, . . . , xk) .

Notice that |V|Rnk ≤ 1 whenever |V |TΥ ≤ 1. It is now immediate to see that Ek is of finite

perimeter on B×kr . Thus, standard results og geometric measure theory on the Euclidean space

Rnk (see e.g., [44, Thm. 5.8.2]), we obtain

(6.3)

ˆEk

(∇∗V)dSnkB×k

r=

ˆ∂∗B

×kr

Ek

〈V, σEk〉dρ1B×k

r. for any V ∈ C∞

0 (B×kr ;Rnk) .

Here σEk is a vector field σEk : B×kr → Rnk such that |σEk |Rnk = 1 ρ1

B×kr

-a.e. on ∂∗B×k

rEk. By

passing to the quotient by means of the map sk in both sides of (6.3) and using (6.1), we get the

sought conclusion.

Remark 6.4. An alternative proof of Proposition 6.3 can be given by employing the theory of

RCD spaces (see [8] and references therein). Indeed (Υk(Br), dk, πkBr

) is an RCD(0, kn) and Ek is

of finite perimeter. Hence we can apply [18, Theorem 2.2] to get the integration by parts formula,

written in terms of the total variation measure |DχEk |. From [9, Corollary 4.7] we deduce the

identity |DχEk | = ρ1Υ(Br)|∂∗

Υk(Br)Ek .

Let us now put a measurability statement. The proof follows by arguing exactly in the same

way as in the proof of Proposition 3.6, thus, we omit the proof.

Lemma 6.5. Fix r > 0. If F : Υ(Rn) → R is a Borel function, then

Υ(Bcr) 3 η →ˆΥ(Br)

Fη,rdρ1Υ(Br)

is πBcr-measurable .

6.2. Sets of finite perimeter on Υ(Rn). We now study sets of finite perimeter on the config-

uration space Υ(Rn) by employing the already developed theory for the space Υ(Br). The main

idea is to reduce a set E ⊂ Υ(Rn) to its sections Eη,r ⊂ Υ(Br) and apply the results for sets of

finite perimeter in Υ(Br), combined with the disintegration argument. We finally let r → ∞ to

recover the information on the perimeter of the original set E.

Let us begin by introducing the definition of the reduced boundary in Υ(Rn).

Definition 6.6 (Reduced boundary in Υ(Rn)). Let E ⊂ Υ(Rn) be a Borel set. For any r > 0 we

set

(6.4) ∂∗rE := γ ∈ Υ(Rn) : γ|Br∈ ∂∗Υ(Br)

Eγ|Bcr,r .

The reduced boundary of E is defined as follows

(6.5) ∂∗E := lim infi→∞, i∈N

∂∗i E =⋃i>0

⋂j>i, j∈N

∂∗jE .

Remark 6.7. We defined ∂∗E by taking the liminf along the sequence ∂∗i Ei∈N. This choice is

completely arbitrary and, as we will see in the sequel (cf. Theorem 6.15), if we change the defining

sequence, then the reduced boundary can change, but only up to an ‖E‖-negligible set, where ‖E‖is the perimeter measure that will be defined later. Thus, the reduced boundary is well-defined up

to ‖E‖-negligible sets.

Notice that, for any η ∈ Υ(Bcr) it holds

(6.6) (∂∗rE)η,r = ∂∗Υ(Br)Eη,r .

Lemma 6.8. If E is a Borel subset of Υ(Rn), then ∂∗rE and ∂∗E are Borel.

34 E. BRUE AND K. SUZUKI

Proof. Since ∂∗E = lim infr→∞ ∂∗rE, it suffices to show the Borel measurability of ∂∗rE for any

r > 0.

Step 1: We prove the following statement: for any k ∈ N and s > 0 the function

(6.7) γ ∈ Υ(Rn) : γ(Br) = k 3 γ 7→πkBr

(Bks(γ|Br) ∩ Ekγ|Bc

r,r)

πkBr(Bks(γ|Br

))

is Borel.

Since the Borel measurability of the map γ 7→ πkBr(Bks(γ|Br

)) is easy, we only give a proof of

the Borel measurability of the map γ 7→ πkBr(Bks(γ|Br

) ∩ Ekγ|Bcr,r).

Let us identify γ ∈ Υ(Rn) : γ(Br) = k ' Υk(Br) × Υ(Bcr). It allows us to introduce the

product topology τp on γ ∈ Υ(Rn) : γ(Br) = k, that is coarser than the vague topology τvas a consequence of the following observation: since Bcr is open, the vague topology τv on Υ(Bcr)

coincides with the relative topology induced by Υ(Rn). Thus, it suffices to see that the vague

topology on Υ(Br) is coarser than the relative topology induced by Υ(Rn). For this purpose,

we only need to show that, for any ϕ ∈ Cc(Br) (note that ϕ does not necessarily vanish at the

boundary of Br), there exists an extension ϕ ∈ Cc(Rn) so that ϕ = ϕ on Br. Given ϕ ∈ Cc(Br),

we take Φ ∈ C(Rn) which is the extension of ϕ to Rn given by the Tietz extension theorem. Let us

now pick κ ∈ Cc(Rn) such that κ = 1 on Br and κ = 0 on Bc2r. Then, it holds ϕ := κΦ ∈ Cc(Rn)and ϕ = ϕ in Br, which concludes the sought statement.

By the inclusion τp ⊂ τv of the topologies, we have the inclusion of the corresponding Borel

σ-algebras B(τp) ⊂ B(τv). Since the map

(6.8) Υk(Br)×Υk(Br)×Υ(Bcr) 3 (γ1, γ2, η) → χE(γ1 + η)χBks (γ2)

(γ1) ,

is B(τp)-measurable, it is also B(τv)-measurable. Hence, Fubini’s theorem gives that

(6.9) Υk(Br)×Υ(Bcr) 3 (γ2, η) →ˆΥk(Br)

χE(γ1+η)χBks (γ2)

(γ1)dπkBr(γ1) = πkBr

(Bks(γ2|Br )∩Ekη,r) ,

is B(τv)-measurable as well.

Step 2: Fix k ∈ N and set

Ak,r1 :=

γ ∈ Υ(Rn) : lim sup

s→0

πkBr(Bks(γ|Br

) ∩ Ekγ|Bcr,r)

πkBr(Bs(γ|Br

))> 0

,

Ak,r2 :=

γ ∈ Υ(Rn) : lim sup

j→∞

πkBr(Bk2−j (γ|Br

) ∩ Ekγ|Bcr,r)

πkBr(Bk2−j (γ|Br

))> 0

.

Then Ak,r1 = Ak,r2 .

Observe that Ak,r2 ⊂ Ak,r1 . The converse inequality follows from the following observation. If

2−j ≤ s ≤ 2−j+1 then

πkBr(Bks(γ|Br ) ∩ Ekγ|Bc

r,r)

πkBr(Bks(γ|Br

))≥πkBr

(Bk2−j (γ|Br ) ∩ Ekγ|Bcr,r)

πkBr(Bk2−j (γ|Br

))

πkBr(Bk2−j (γ|Br

))

πkBr(Bks(γ|Br

))

≥ C(k, n)πkBr

(Bk2−j (γ|Br ) ∩ Ekγ|Bcr,r)

πkBr(Bk2−j (γ|Br

)),

where we used the estimate C(n, k)−1e−Ln(Br)snk ≤ πkBr(Bks(γ)) ≤ C(n, k)e−Ln(Br)snk for any

s < r/5, γ ∈ Υ(Br) and some constant C(n, k) ≥ 1 depending only on n and k. Indeed, the latter

estimate can be obtained by the following observation: letting γ = x1, . . . , xk, we have

B×kr ∩ s−1

k (Bks(γ)) = B×kr ∩

⋃σk∈Sk

Bs(xσk) ,

BV FUNCTIONS AND SETS OF FINITE PERIMETER ON CONFIGURATION SPACES 35

hence

πkBr(Bks(γ)) =

e−Ln(Br)

k!Lkn(B×k

r ∩ s−1k (Bks(γ))) ≤ e−Ln(Br)C(n, k)snk ,

recall that Ln denotes the n-dimensional Lebesgue measure. The opposite inequality follows from

πkBr(Bks(γ)) =

e−Ln(Br)

k!Lkn(B×k

r ∩ s−1k (Bks(γ)))

≥ e−Ln(Br)

k!Lkn(B×k

r ∩Bs(xσk)) ≥ e−Ln(Br)C(n, k)snk .

Step 3: Let us conclude the proof. Thanks to Step 1 and Step 2 we know that Ak,r1 is Borel for

any k ∈ N and r > 0. The same arguments as in Step 1 and Step 2 apply to the Borel measurability

for the following set:

(6.10)

γ ∈ Υ(Rn) : lim sup

s→0

πkBr(Bks(γ|Br

) \ Ekr,γ|Bcr

)

πkBr(Bks(γ|Br

))> 0

,

hence ∂∗rE is a Borel set.

6.3. Perimeter measures. In this subsection, based on the variational approach, we introduce

the perimeter measure ‖E‖ for a set E ⊂ Υ(Rn) satisfying V(χE) < ∞. In order to construct

‖E‖, we first introduce a localised perimeter measure ‖E‖r on Υ(Rn), and show the monotonicity

of ‖E‖r as r → ∞.

Definition 6.9. For any Borel set E ⊂ Υ(Rn) with Vr(χE) <∞, we define

(6.11) ‖E‖r := ρ1Υ(Br)|(∂∗

rE)η,r⊗ πBc

r(η) on Υ(Rn) ,

which is equivalently defines as follows: for any bounded Borel measurable function F on Υ(Rn),ˆΥ(Rn)

Fd‖E‖r :=ˆΥ(Bc

r)

(ˆΥ(Br)

Fη,rdρ1Υ(Br)

|∂∗Υ(Br)

Eη,r

)dπBc

r(η).(6.12)

Lemma 6.10. Let r > 0. For any Borel set E ⊂ Υ(Rn) with V(χE) <∞, ‖E‖r is a well-defined

finite Borel measure.

Proof. Let us first show that ‖E‖r is well-defined. The map γ 7→ Fη,r(γ) is ρ1Υ(Br)|∂∗Eη,r

-

measurable by Lemma 3.1. On account of the definition (6.12), we only need to show that the

map

(6.13) Υ(Bcr) 3 η →ˆΥ(Br)

Fη,rdρ1Υ(Br)

|∂∗Υ(Br)

Eη,r,

is πBcr-measurable for any Borel function F : Υ(Rn) → R. To show it, we use (6.6) and rewriteˆ

∂∗Υ(Br)

Eη,r

Fη,r dρ1Υ(Br)

=

ˆ(∂∗

rE)η,r

Fη,r dρ1Υ(Br)

=

ˆΥ(Br)

(χ∂∗rEF )η,r dρ

1Υ(Br)

.

Now, the claimed conclusion follows from Lemma 6.5 by observing that χ∂∗rEF is a Borel function.

The finiteness of the measure ‖E‖r is immediate by Proposition 6.3 and Proposition 5.5, indeed

‖E‖r(Υ(Rn)) =ˆΥ(Bc

r)

VΥ(Br)((χE)η,r)dπBcr(η) = Vr(χE) ≤ V(χE) <∞.

Lemma 6.11. Let r > 0. For any Borel set E ⊂ Υ(Rn) with Vr(χE) < ∞, there exists a vector

field σE,r : Υ(Rn) → TΥ(Rn) such that

(i) σE,r(γ) ∈ TγΥ(Rn) satisfies σE,r(γ, x) = 0 for x ∈ Bcr;

36 E. BRUE AND K. SUZUKI

(ii) |σE,r|TΥ = 1, ‖E‖r-a.e.;(iii) for any V ∈ CylVr0(Υ(Rn)),ˆ

E

(∇∗V )dπ =

ˆΥ(Rn)

〈V, σE,r〉TΥd‖E‖r .(6.14)

(iv) Vr(χE) = ‖E‖r(Υ(Rn)), and for any nonnegative function F ∈ CylF(Υ(Rn)) it holds

(6.15)

ˆΥ(Rn)

Fd‖E‖r = sup

ˆE

(∇∗FV )dπ : V ∈ CylVr0(Υ(Rn)), |V |TΥ ≤ 1

.

Proof. By Proposition 5.5, there exists a measurable set Ωr ⊂ Υ(Bcr) so that πBcr(Ωr) = 1 and

VΥ(Br)(χEη,r ) < ∞ for every η ∈ Ωr. By Proposition 6.3, for every η ∈ Ωr, there exists a unique

TΥ(Br)-valued Borel measurable map ση,r on Υ(Br) so that |ση,r|TΥ(Br) = 1 ρ1Υ(Br)|∂∗Eη,r

-a.e.,

and ˆEη,r

(∇∗Vη,r)dπBr =

ˆ∂∗Υ(Br)

Eη,r

〈Vη,r, ση,r〉TΥ(Br)dρ1Υ(Br)

, ∀V ∈ CylVr0(Υ(Rn)) .(6.16)

Note that, in the above, we used Vη,r ∈ CylV0(Υ(Br)) whenever V ∈ CylVr0(Υ(Rn)). By taking

the integral with respect to πBcr, and arguing as in Proposition 4.9 we obtainˆ

E

(∇∗V )dπ =

ˆΥ(Bc

r)

ˆEη,r

(∇∗rVη,r)dπBrdπBc

r(η)(6.17)

=

ˆΥ(Bc

r)

ˆ∂∗Υ(Br)

Eη,r

〈Vη,r, ση,r〉TΥ(Br)dρ1Υ(Br)

dπBcr(η) .

Note that the map η 7→´∂∗Υ(Br)

Eη,r〈Vη,r, ση,r〉TΥ(Br)dρ

1Υ(Br)

is πBcr-measurable since, in view of

(6.16), it is equal to a πBcr-measurable function, and therefore, the argument (6.17) is justified.

For γ ∈ Υ(Rn) we define

(6.18) σE,r(γ) :=

σγ|Bc

r,r(γ|Br ) if γ|Bc

r∈ Ωr,

σr(γ) = 0 otherwise .

Let us now observe that, for any V ∈ CylV(Υ(Rn)), we have

(6.19) (〈V, σE,r〉TΥ(Rn))η,r = 〈Vη,r, ση,r〉TΥ(Br) .

By combining the definition (6.11) of ‖E‖r with (6.17), (6.18) and (6.19), we deduce the assertion

(iii).

The assertion (i) follows from the definition (6.18), and the assertion (ii) follows from

(|σE,r|TΥ(Rn))η,r = |ση,r| = 1, ρ1Υ(Br)|∂∗Eη,r -a.e..

We now prove (iv). We first prove the equality Vr(χE) = ‖E‖r(Υ(Rn)). From (iii) and (ii) we

deduce

Vr(χE) = sup

ˆΥ(Rn)

(∇∗V )fdπ : V ∈ CylVr(Υ(Rn)), |V |TΥ(Rn) ≤ 1

= sup

ˆΥ(Rn)

〈V, σE,r〉TΥd‖E‖r : V ∈ CylVr(Υ(Rn)), |V |TΥ(Rn) ≤ 1

≤ ‖E‖r(Υ(Rn)) .

Furthermore, Proposition 5.5 and Lemma 6.3 imply

(6.20)

Vr(χE) ≥ˆΥ(Bc

r)

VΥ(Br)((χE)η,r)dπBcr(η) =

ˆΥ(Bc

r)

ρ1Υ(Br)(∂∗Υ(Br)

Eη,r)dπBcr(η) = ‖E‖r(Υ(Rn)) .

Thus, the proof of the equality Vr(χE) = ‖E‖r(Υ(Rn)) is complete.

BV FUNCTIONS AND SETS OF FINITE PERIMETER ON CONFIGURATION SPACES 37

Let us finally address (6.15). From the equality Vr(χE) = ‖E‖r(Υ(Rn)), we deduce the existenceof a sequence Vk ∈ CylVr0(Υ(Rn)) such that |Vk|TΥ ≤ 1, and

limk→∞

ˆΥ(Rn)

〈Vk, σE,r〉TΥd‖E‖r =ˆΥ(Rn)

d‖E‖r ,

hence,

limk→∞

ˆΥ(Rn)

|Vk − σE,r|2TΥd‖E‖r = limk→∞

ˆΥ(Rn)

(|Vk|2TΥ + |σE,r|2TΥ − 2〈Vk, σE,r〉TΥ)d‖E‖r

≤ limk→∞

2

ˆΥ(Rn)

(1− 〈Vk, σE,r〉TΥ)d‖E‖r = 0 .

Therefore, for any F ∈ CylF(Υ)

limk→∞

ˆΥ(Rn)

F 〈Vk, σE,r〉TΥd‖E‖r =ˆΥ(Rn)

F d‖E‖r ,

in particular, by making use of (6.14) with V = FVk, it holds that

(6.21)

ˆΥ(Rn)

F d‖E‖r ≤ sup

ˆE

(∇∗FV )dπ : V ∈ CylVr0(Υ(Rn)), |V |TΥ ≤ 1

.

The converse inequality follows form |σE,r|TΥ = 1 ‖E‖r-a.e. and the fact that F is nonnegative:

(6.22)

ˆE

(∇∗FV )dπ =

ˆΥ(Rn)

F 〈Vk, σE,r〉TΥd‖E‖r ≤ˆΥ(Rn)

|F |d‖E‖r =ˆΥ(Rn)

Fd‖E‖r .

The proof is complete.

Corollary 6.12. If V(χE) < ∞, then r 7→ ‖E‖r(A) is monotone non-decreasing for every Borel

measurable set A.

Proof. In view of the density of cylinder functions on L2(Υ(Rn), π) it is enough to check that

r →ˆΥ(Rn)

Fd‖E‖r is non-decreasing ,

for any nonnegative F ∈ CylF(Υ(Rn)). The latter conclusion easily follows from (6.15) and the

inclusion CylVs0(Υ(Rn)) ⊂ CylVr0(Υ(Rn)) for s ≤ r.

By the monotonicity of r 7→ ‖E‖r in Corollary 6.12, we may define the limit measure as follows:

Definition 6.13 (Perimeter measure). Given E ⊂ Υ(Rn) with V(χE) <∞, we define the perime-

ter measure as

(6.23) ‖E‖(A) := limr→∞

‖E‖r(A) for any Borel set A .

We finally obtain the Gauß–Green formula for the perimeter measure ‖E‖.

Theorem 6.14 (Gauß–Green formula for ‖E‖). For any Borel set E ⊂ Υ(Rn) with V(χE) <∞,

there exists a unique element σE ∈ L2(TΥ, ‖E‖) such that |σE |TΥ = 1 ‖E‖-a.e. andˆE

∇∗V dπ =

ˆΥ(Rn)

〈V, σE〉TΥd‖E‖ ∀V ∈ CylV(Υ(Rn)).(6.24)

Proof. We assume without loss of generality that ‖E‖(Υ(Rn)) = 1. Note that, for any V ∈CylV(Υ(Rn)), there exists r > 0 so that V ∈ CylVr0(Υ(Rn)). Thus, by (iii) in Lemma 6.11, for

38 E. BRUE AND K. SUZUKI

any V ∈ CylV(Υ(Rn)), there exists r > 0 and σE,r : Υ(Rn) → TΥ so that |σE,r| = 1 ‖E‖-a.e., andˆE

∇∗V dπ =

ˆΥ(Rn)

〈V, σE,r〉TΥd‖E‖r

≤ ‖V ‖L2(TΥ,∥E∥r)

≤ ‖V ‖L2(TΥ,∥E∥).(6.25)

The last inequality followed from the monotonicity in Corollary 6.12. In particular, denoting by

H ⊂ L2(TΥ, ‖E‖) the closure of CylV(Υ) in L2(TΥ, ‖E‖), the linear operator L defined below

(6.26) L : H → R , H 3 V 7→ L(V ) :=

ˆE

∇∗V dπ ,

is a well-defined continuous operator on the Hilbert space H and satisfies ‖L‖ ≤ 1. Therefore, the

Riesz representation theorem in the Hilbert space H gives the existence of σE ∈ H so that

(6.27) ‖σE‖L2(TΥ,∥E∥) ≤ 1 ,

ˆE

∇∗V dπ =

ˆΥ(Rn)

〈V, σ〉d‖E‖ ∀V ∈ CylV(Υ(Rn)) .

It suffices to show that |σ|TΥ = 1 ‖E‖-a.e. By (iv) in Lemma 6.11 and Corollary 6.12, we deduce

that

1 = ‖E‖(Υ(Rn)) = limr→∞

‖E‖r(Υ(Rn)) = limr→∞

Vr(χE)

= limr→∞

supV ∈CylVr

r(Υ(Rn)) ,|V |TΥ≤1

ˆE

∇∗V dπ ≤ˆΥ(Rn)

|σ|TΥd‖E‖ ≤ ‖σ‖L2(TΥ,∥E∥) ≤ 1 ,

which yields ‖σ‖L1(TΥ,∥E∥) = ‖σ‖L2(TΥ,∥E∥) = 1. Therefore |σ|TΥ = 1 ‖E‖-a.e. as a consequence

of the characterisation of the equality in Jensen’s inequality. The proof is complete.

6.4. Perimeters and one-codimensional Poisson measures. In this subsection, we prove one

of the main results in this paper. Namely, the perimeter measure ‖E‖ based on the variational

approach (Definition 6.13) coincides with the 1-codimensinal Poisson measure ρ1 (Definition 3.8)

restricted on the reduced boundary ∂∗E of E (Definition 6.6).

Theorem 6.15. Let E ⊂ Υ(Rn) be a set with V(χE) <∞. Then,

‖E‖ = ρ1|∂∗E .

Before giving the proof, we prove a lemma.

Lemma 6.16. Let E ⊂ Υ(Rn) be a set with V(χE) <∞. Then, for any r > 0, ε > 0, it holds

(6.28) (∂∗rE)η,r ⊂ (∂∗r+εE)η,r up to ρ1Υ(Br)-negligible sets for πBc

r-a.e. η.

Namely, there exists a measurable set Ωr,ε ⊂ Υ(Rn) so that πBcr(Ωr,ε) = 1 and for any η ∈ Ωr,ε, it

holds that

ρ1Υ(Br)

((∂∗rE)η,r \ ∂∗r+εE)η,r

)= 0 .(6.29)

Proof. By (6.6) and the definition (6.11) of the perimeter measure ‖E‖r, we see that

∞ > ‖E‖(A) ≥ ‖E‖r+ε(A) =ˆΥ(Bc

r+ε)

ρ1Υ(Br+ε)(∂∗Υ(Br+ε)

Eη,r+ε ∩Aη,r+ε)dπBcr+ε

(η)

=

ˆΥ(Bc

r+ε)

ρ1Υ(Br+ε)

((∂∗r+εE)η,r+ε ∩Aη,r+ε

)dπBc

r+ε(η)

=

ˆΥ(Bc

r+ε)

ρ1Υ(Br+ε)

((∂∗r+εE ∩A)η,r+ε

)dπBc

r+ε(η) .(6.30)

BV FUNCTIONS AND SETS OF FINITE PERIMETER ON CONFIGURATION SPACES 39

By the monotonicity ‖E‖r+ε(A) ≥ ‖E‖r(A) in Corollary 6.12, we obtain thatˆΥ(Bc

r+ε)

ρ1Υ(Br+ε)

((∂∗r+εE ∩A)η,r+ε

)dπBc

r+ε(η) ≥

ˆΥ(Bc

r)

ρ1Υ(Br)

((∂∗rE ∩A)η,r

)dπBc

r(η).

Taking A = Υ(Rn) \ ∂∗r+εE, we have that

0 =

ˆΥ(Bc

r+ε)

ρ1Υ(Br+ε)

((∂∗r+εE ∩A)η,r+ε

)dπBc

r+ε(η)

≥ˆΥ(Bc

r)

ρ1Υ(Br)

((∂∗rE ∩A)η,r

)dπBc

r(η) .

Thus, ρ1Υ(Br)

((∂∗rE ∩A)η,r

)= 0 for πBc

r-a.e. η, which implies that(

∂∗rE ∩ (Υ(Rn) \ ∂∗r+εE))η,r

= (∂∗rE)η,r \((∂∗r+εE)η,r ∩ (∂∗rE)η,r

)is ρ1Υ(Br)

-negligible for πBcr-a.e. η. The proof is complete.

Proof of Theorem 6.15. Fix r > 0 and η ∈ Υ(Bcr). It holds

(6.31) (∂∗E)η,r :=

⋃i>0

⋂j>i

∂∗jE

η,r

=⋃i>0

⋂j>i

(∂∗jE)η,r .

The monotonicity formula (6.29) in Lemma 6.16 gives the existence of Ωr,j ⊂ Υ(Rn) so that

πBcr(Ωr,j) = 1, and for any η ∈ Ωr,j

(∂∗rE)η,r ⊂ (∂∗jE)η,r j ≥ r up to a ρ1Υ(Br)-negligible set .

Take Ωr = ∩j≥r,j∈NΩr,j . Then πBcr(Ωr) = 1, and by using (6.6), we obtain that for any η ∈ Ωr,

∂∗Υ(Br)Eη,r = (∂∗rE)η,r ⊂ (∂∗E)η,r up to a ρ1Υ(Br)

-negligible set .

This implies that for any Borel set A ⊂ Υ(Rn),

ρ1Υ(Br)(∂∗Υ(Br)

Eη,r ∩Aη,r) ≤ ρ1Υ(Br)((∂∗E ∩A)η,r), ∀η ∈ Ωr.

Thus, by noting that πBcr(Ωr) = 1 and recalling Definition 6.13, Definition 3.8, we obtain

‖E‖(A) := limr→∞

‖E‖r(A)

= limr→∞

ˆΥ(Bc

r)

ρ1Υ(Br)(∂∗Υ(Br)

Eη,r ∩Aη,r)dπBcr(η)

≤ limr→∞

ˆΥ(Bc

r)

ρ1Υ(Br)((∂∗E ∩A)η,r)dπBc

r(η)

= ρ1(A ∩ ∂∗E) .

In order to conclude the proof, it is enough to check that

(6.32) ‖E‖(Υ(Rn)) ≥ ρ1(∂∗E) .

Indeed, given any Borel set A, by making use of the already proven inequality ‖E‖ ≤ ρ1|∂∗E , we

obtain

‖E‖(Υ(Rn)) = ‖E‖(A) + ‖E‖(Ac) ≤ ρ1(A ∩ ∂∗E) + ρ1(Ac ∩ ∂∗E) = ρ1(∂∗E) ≤ ‖E‖(Υ(Rn)) .

Thus, ‖E‖(A) + ‖E‖(Ac) = ρ1(A ∩ ∂∗E) + ρ1(Ac ∩ ∂∗E) for any Borel set A. Assume that there

exists a Borel set A so that ‖E‖(A) < ρ1(A ∩ ∂∗E). Since ‖E‖ ≤ ρ1(· ∩ ∂∗E), it implies

‖E‖(A) + ‖E‖(Ac) < ρ1(A ∩ ∂∗E) + ρ1(Ac ∩ ∂∗E),

which is a contradiction.

40 E. BRUE AND K. SUZUKI

We now prove (6.32). Let s < r. By recalling Definitions 6.9, 3.5 of ‖E‖r and ρ1r respectively

and using the monotonicity of ρ1r in Theorem 3.7, we have

‖E‖r(Υ(Rn)) =ˆΥ(Bc

r)

ρ1Υ(Br)((∂∗rE)η,r)dπBc

r(η) = ρ1r(∂

∗rE) ≥ ρ1s(∂

∗rE) ,

hence

‖E‖(Υ(Rn)) = limi→∞

‖E‖i(Υ(Rn)) ≥ lim infi→∞

ρ1s(∂∗i E) ≥ ρ1s(lim inf

i→∞∂∗i E) = ρs(∂

∗E) .

Passing to the limit s→ ∞, we conclude (6.32). The proof is complete.

7. Total variation and Gauß–Green formula

In this section, we prove a relation between the coarea with respect to the perimeter measure

‖E‖ and the variation |D∗F | obtained via relaxation of Cylinder functions. As an application,

we introduce the total variation measure |DF | for BV functions F , and prove the Gauß–Green

formula.

7.1. Total variation measures via coarea formula. Recall that, for F ∈ BV(Υ(Rn)), the map

CylF(Υ(Rn)) 3 G 7→ |D∗F |[G] is defined by the relaxation approach in Definition 5.7. The main

result of this subsection is the following formula:

Theorem 7.1. Let F ∈ L2(Υ(Rn), π) ∩ BV(Υ(Rn)). Then,

V(χF>t) <∞ a.e. t ∈ R,(7.1)

and the following formula holds:ˆ ∞

−∞

(ˆΥ(Rn)

Gd∥∥F > t

∥∥) dt = |D∗F |[G], for any nonnegative G ∈ CylF(Υ(Rn)) .(7.2)

The proof of Theorem 7.1 will be given later in this section. Before discussing the proof, we

study several consequences of Theorem 7.1. By (7.1), the left-hand side of (7.2) makes sense with

G ≡ 1 since the right-hand side |D∗F |[1] <∞ is finite due to F ∈ BV(Υ(Rn)) and Theorem 5.17.

This leads us to provide the following definition of the total variation measure.

Definition 7.2 (Total variation measure). For F ∈ L2(Υ(Rn), π) ∩ BV(Υ(Rn)), define the total

variation measure |DF | as follows:

|DF | :=ˆ ∞

−∞

∥∥F > t∥∥dt.(7.3)

We now investigate relations between the total variation measure |DχE | and the perimeter

measure ‖E‖ defined in Definition 6.13 and the (1, 2)-capacity Cap1,2 defined in Definition 4.3.

Corollary 7.3 (Total variation and perimeters). Let E ⊂ Υ(Rn) satisfy |DχE |(Υ(Rn)) < ∞.

Then,

|DχE | = ‖E‖ as measures .

.

Proof. By Theorem 5.17, V(χE) <∞ and ‖E‖ is well-defined. Noting that

χE > t =

Υ(Rn) t ≤ 0;

E 0 < t ≤ 1;

∅ t > 1,

BV FUNCTIONS AND SETS OF FINITE PERIMETER ON CONFIGURATION SPACES 41

and ‖Υ(Rn)‖ = 0 and ‖∅‖ = 0, we obtain that

|DχE |(A) =ˆ ∞

−∞

∥∥χE > t∥∥(A)dt = 0 + ‖E‖(A) + 0 = ‖E‖(A) for any Borel set A .

The proof is complete.

Corollary 7.4 (Total variation and capacity). Let F ∈ L2(Υ(Rn), π)∩BV(Υ(Rn)). For any Borel

set A ⊂ Υ(Rn),Cap1,2(A) = 0 =⇒ |DF |(A) = 0.

Proof. Let Cap1,2(A) = 0. By Theorem 7.1, and Theorem 6.15, we can write

(7.4) |DF |(A) =ˆ ∞

−∞‖F > t‖(A)dt =

ˆ ∞

−∞ρ1(∂∗F > t ∩A)dt ,

hence it suffices to show that ρ1(∂∗F > t ∩A) = 0. This follows from the absolute continuity of

ρ1 with respect to Cap1,2 obtained in Theorem 4.3.

7.2. Proof of Theorem 7.1. This subsection is devoted to the proof of Theorem 7.1. Let us

begin with two propositions.

Proposition 7.5. Let E ⊂ Υ(Rn) be a set with V(χE) <∞. Then, for any nonnegative function

G ∈ CylF(Υ(Rn)) it holds

(7.5)

ˆΥ(Rn)

Gd‖E‖ = sup

ˆE

(∇∗GV )dπ : V ∈ CylV(Υ(Rn)), |V |TΥ ≤ 1

.

In particular, the following hold:

(i) if Fk ∈ CylF(Υ(Rn)), and Fk → χE in L1(Υ(Rn), π) as k → ∞, then

lim infk→∞

ˆΥ(Rn)

G|∇Fk|TΥdπ ≥ˆΥ(Rn)

Gd‖E‖ , for any nonnegative G ∈ CylF(Υ(Rn)) ;

(ii) if χEk→ χE in L1(Υ(Rn), π) as k → ∞, where (Ek)k are sets of finite perimeter, then

lim infk→∞

ˆΥ(Rn)

Gd‖Ek‖ ≥ˆΥ(Rn)

Fd‖E‖ , for any nonnegative G ∈ CylF(Υ(Rn)) .

Proof. Fix ε > 0. We pick r > 0 such that´Υ(Rn)

Gd‖E‖r ≥´Υ(Rn)

Gd‖E‖ − ε. From (6.15)

we deduce the existence of V ∈ CylVr0(Υ(Rn)) with |V |TΥ ≤ 1 such that´E(∇∗GV )dπ ≥´

Υ(Rn)Gd‖E‖r − ε, yielding

ˆΥ(Rn)

Gd‖E‖ ≤ˆE

(∇∗GV )dπ+2ε ≤ sup

ˆE

(∇∗GV )dπ : V ∈ CylV(Υ(Rn)), |V |TΥ ≤ 1

+2ε .

By taking ε→ 0, the one inequality is proved.

We now prove the converse inequality. Take a representative G = Φ(f⋆1 , . . . , f⋆k ) and take r > 0

so that ∪ki=1supp[fi] ⊂ Br. By the divergence formula (2.12), we can easily see

sup

ˆE

(∇∗GV )dπ : V ∈ CylVr0(Υ(Rn)), |V |TΥ ≤ 1

= sup

ˆE

(∇∗GV )dπ : V ∈ CylV(Υ(Rn)), |V |TΥ ≤ 1

.

By combining it with the formula (6.15) and the monotonicity of r 7→ ‖E‖r in Corollary 6.12, the

converse inequality is proved.

42 E. BRUE AND K. SUZUKI

Let us now prove (i) and (ii). Fix ε > 0. By Theorem 6.14, we can take V ∈ CylV(Υ(Rn)) suchthat |V |TΥ ≤ 1 and ˆ

E

(∇∗GV )dπ ≥ˆΥ(Rn)

Gd‖E‖ − ε .

Let kj be a subsequence such that limj→∞´Υ(Rn)

G|∇Fkj |TΥdπ = lim infk→∞´Υ(Rn)

G|∇Fk|TΥdπ,

it holdsˆΥ(Rn)

Gd‖E‖ − ε ≤ˆE

(∇∗GV )dπ = limj→∞

ˆΥ(Rn)

Fkj (∇∗GV )dπ = limj→∞

ˆΥ(Rn)

G〈∇Fkj , V 〉TΥdπ

≤ lim infk→∞

ˆΥ(Rn)

G|∇Fk|TΥdπ .

Furthermore, by using Theorem 6.14 with V being GV , we deduce thatˆΥ(Rn)

Gd‖E‖ − ε ≤ˆE

(∇∗GV )dπ = limj→∞

ˆEkj

(∇∗GV )dπ = limj→∞

ˆΥ(Rn)

G〈V, σEkj〉TΥd‖Ekj‖

≤ lim infk→∞

ˆΥ(Rn)

Gd‖Ekj‖ .

The proof is complete.

Proposition 7.6. For any F ∈ CylF(Υ(Rn)) it holds(7.6)ˆ ∞

−∞

ˆΥ(Rn)

Gd∥∥F > t

∥∥dt = ˆΥ(Rn)

G |∇F |TΥdπ , for any nonnegative G ∈ CylF(Υ(Rn)) .

Proof. The map

(7.7) R 3 t→ m(t) :=

ˆF>t

G|∇F |TΥdπ

is monotone and finite since |∇F |TΥ ∈ L1(Υ(Rn)). Let t ∈ R be a point on which the map

t 7→ m(t) is differentiable and set

(7.8) gε(s) :=

1 s ≤ t

ε−1(t− s) + 1 t ≤ s ≤ t+ ε

0 s > t+ ε .

Notice that gε F → χF>t in Lp(Υ(Rn)) for any p ∈ [1,∞) as ε→ 0. Indeed,

(7.9)

ˆΥ(Rn)

|gε F − χF>t|pdπ ≤ 2pπ(t ≤ F ≤ t+ ε) → 0, as ε→ 0 .

Standard calculus rules give

(7.10)

ˆΥ(Rn)

G|∇(gε F )|TΥdπ ≤ ε−1

ˆt<F≤t+ε

G|∇F |TΥdπ ≤ m(t+ ε)−m(t)

ε,

while (7.5) in Proposition 7.5 implies

(7.11)

ˆΥ(Rn)

Gd‖F > t‖ ≤ lim infε→0

ˆΥ(Rn)

G|∇(gε F )|TΥdπ = m′(t) .

Since m is differentiable for a.e. t ∈ R, the one inequality comes by integrating (7.11).

BV FUNCTIONS AND SETS OF FINITE PERIMETER ON CONFIGURATION SPACES 43

Let us prove the converse inequality. Let V ∈ CylV(Υ(Rn)) such that |V |TΥ ≤ 1. Then, by

Theorem 6.14, we deduceˆΥ(Rn)

F (∇∗GV )dπ =

ˆ ∞

−∞

ˆF>t

(∇∗GV )dπdt

≤ˆ ∞

−∞

ˆΥ(Rn)

Gd‖F > t‖dt ,

which easily yields the sought conclusion.

Proof of Theorem 7.1. Let F ∈ L2(Υ(Rn), π) such that |DF |(Υ(Rn)) < ∞ and G ∈ CylF(Υ(Rn))be nonnegative. By definition there exists a sequence (Fn) ⊂ CylF(Υ) such that Fn → F in

L1(Υ(Rn), π) and´Υ(Rn)

G|∇Fn|TΥdπ → |D∗F |[G]. From Proposition 7.6 we get

(7.12)

ˆ ∞

−∞

ˆΥ(Rn)

Gd‖Fn > t‖dt =ˆΥ(Rn)

G|∇Fn|TΥdπ ,

and passing to the limit for n→ ∞ we deduce

(7.13)

ˆ ∞

−∞

ˆΥ(Rn)

Gd‖F > t‖dt ≤ |D∗F |[G] ,

as a consequence of (ii) in Proposition 7.5 and Fatous’ Lemma. In particular F > t is of finite

perimeter for a.e.-t ∈ R.Let us now fix ε > 0 and consider V ∈ CylV(Υ(Rn)) such that |V |TΥ ≤ 1 and V(F ) − ε ≤´

Υ(Rn)F (∇∗V )dπ. By Theorem 6.14, we have

|D∗F |(Υ(Rn))− ε = V(F )− ε ≤ˆΥ(Rn)

F (∇∗V )dπ =

ˆ ∞

−∞

ˆF>t

(∇∗V )dπdt

≤ˆ ∞

−∞

ˆΥ(Rn)

d‖F > t‖dt ,

which easily yields ˆ ∞

−∞

ˆΥ(Rn)

d‖F > t‖dt ≥ |D∗F |(Υ(Rn)) = |D∗F |[1] .

The sought conclusion follows now by recalling that |D∗F |[G1 +G2] ≥ |D∗F |[G1] + |D∗F |[G2] and

by the same argument in the paragraph after (6.32). Indeed,

|D∗F |[G] + |D∗F |[1−G] ≤ |D∗F |[1] ≤ˆ ∞

−∞

ˆΥ(Rn)

d‖F > t‖dt

=

ˆ ∞

−∞

ˆΥ(Rn)

Gd‖F > t‖dt+ˆ ∞

−∞

ˆΥ(Rn)

(1−G)d‖F > t‖dt

≤ |D∗F |[G] + |D∗F |[1−G] ,

for any 0 ≤ G ≤ 1, G ∈ CylF(Υ(Rn)). The proof is complete.

7.3. Gauß–Green formula. We prove the Gauß–Green formula, which is one of the main results

in this paper.

Theorem 7.7 (Gauß–Green formula). For F ∈ L2(Υ(Rn), π)∩BV(Υ(Rn)), there exists a unique

element σF ∈ L2(TΥ, |DF |) such that |σF |TΥ = 1 |DF |-a.e., andˆΥ(Rn)

(∇∗V )Fdπ =

ˆΥ(Rn)

〈V, σF 〉TΥd|DF |, ∀V ∈ CylV(Υ(Rn)) .(7.14)

44 E. BRUE AND K. SUZUKI

Proof. We assume without loss of generality that |DF |(Υ(Rn)) = 1. By Theorem 6.14 and Theorem

7.1, it holds that ˆΥ(Rn)

(∇∗V )Fdπ =

ˆ ∞

−∞

ˆF>t

(∇∗V )dπdt

=

ˆ ∞

−∞

ˆΥ(Rn)

〈V, σF>t〉TΥd∥∥F > t

∥∥dt≤ˆΥ(Rn)

|V |TΥd|DF |

≤ ‖V ‖L2(TΥ,|DF |),(7.15)

for any V ∈ CylV(Υ). In particular, denoting by H ⊂ L2(TΥ, |DF |) the closure of CylV(Υ) in

L2(TΥ, |DF |), the map L defined by

(7.16) L : H → R , H 3 V 7→ L(V ) :=

ˆΥ(Rn)

(∇∗V )Fdπ ,

is a well-defined continuous operator on the Hilbert space H and satisfies ‖L‖ ≤ 1. Therefore, the

Riesz representation theorem on the Hilbert space H gives the existence of σF ∈ H so that

(7.17) ‖σF ‖L2(TΥ,|DF |) ≤ 1 ,

ˆΥ(Rn)

(∇∗V )Fdπ =

ˆΥ(Rn)

〈V, σF 〉d|DF | ∀V ∈ CylV(Υ(Rn)) .

From Theorem 5.17 and Theorem 7.1, we deduce

1 = |DF |(Υ(Rn)) = |D∗F |[1] = V(F ) = supV ∈CylV ,|V |TΥ≤1

ˆΥ(Rn)

(∇∗V )Fdπ

≤ˆΥ(Rn)

|σF |TΥd|DF | ≤ ‖σF ‖L2(TΥ,|DF |) ≤ 1 ,

which yields ‖σF ‖L1(TΥ,|DF |) = ‖σF ‖L2(TΥ,|DF |) = 1, and therefore |σF |TΥ = 1 |DF |-a.e. as a

consequence of the characterization of the equality in Jensen’s inequality. The proof is complete.

7.4. BV and Sobolev functions. In this subsection, we discuss the consistency of the just de-

veloped theory of BV functions with the (1, 2)-Sobolev space H1,2(Υ(Rn), π).

Proposition 7.8. Let F ∈ L2(Υ(Rn), π)∩BV(Υ(Rn)). Suppose |DF | π with |DF | = H ·π and

H ∈ L2(Υ(Rn), π). Then F ∈ H1,2(Υ(Rn), π) and

H = |∇F | , σF =∇F|∇F |

· χ|∇F |=0 ,

where σF is the unique element in L2(TΥ, |DF |) in the Gauß–Green formula (7.14).

Proof. By Theorem 7.7 and recalling TtV ∈ D(EH) ⊂ D2(TΥ(Rn), π) for any V ∈ CylV(Υ(Rn), π)by (5.32), the approximation of TtV by CylV(Υ(Rn), π) implies thatˆ

Υ(Rn)

(∇∗G)Fdπ =

ˆΥ(Rn)

〈G, σF 〉TΥFdπ ∀G ∈ TtCylV(Υ(Rn)) ∀t > 0 ,(7.18)

where TtCylV(Υ(Rn)) := G = TtF : F ∈ CylV(Υ(Rn)) for t > 0. By Lemma 5.19 and the

π-symmetry of Tt, for any U ∈ CylV(Υ(Rn)), setting G = TtU , we obtainˆΥ(Rn)

〈U,∇TtF 〉dπ =

ˆΥ(Rn)

(∇∗U)TtFdπ =

ˆΥ(Rn)

Tt(∇∗U)Fdπ =

ˆΥ(Rn)

(∇∗G)Fdπ

=

ˆΥ(Rn)

〈G, σF 〉TΥd|DF | =ˆΥ(Rn)

〈G, σF 〉TΥHdπ =

ˆΥ(Rn)

〈U,Tt(HσF )〉TΥdπ .

BV FUNCTIONS AND SETS OF FINITE PERIMETER ON CONFIGURATION SPACES 45

Thus, Tt(HσF ) = ∇TtF . Letting t→ 0, Tt(HσF ) converges to HσF in L2(TΥ, π), which implies

that ∇TtF converges to HσF in L2(TΥ(Rn), π). Since TtF → F in L2(Υ(Rn), π), we conclude

that F ∈ H1,2(Υ(Rn), π), and ∇F = HσF . Therefore, H · π = |DF | = |∇F | · π, and

σF =∇FH

χH =0 =∇F|∇F |

χ|∇F |=0 .

The proof is complete.

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