A local time scaling exponent for compact metricspaces

John Dever

School of MathematicsGeorgia Institute of Technology

Fractals 6 @ Cornell, June 15, 2017

Dever (GaTech) Exit time exponent Fractals 6 1 / 19

Local Hausdorff Dimension

Let (X ,d) a metric space.

If A ⊂ B ⊂ X then dimH(A) ≤ dimH(B)1.

So define the local Hausdorff dimension α by...

Definitionα(x) := limr→0+ dimH(Br (x)) = infr>0 dimH(Br (x)).

1dimH means the Hausdorff dimensionDever (GaTech) Exit time exponent Fractals 6 2 / 19

Caratheodory construction of a metric outer measure

For δ > 0 and A ⊂ X let Cδ(A) be the collection of all covers of Aby an at most countable number of subsets of X of diameter atmost δ.Then let µ∗(A) = supδ>0 inf

∑U∈U diam(U)dimH(U) | U ∈ Cδ(A).

If M∗ := A ⊂ X | ∀E ⊂ X [ µ∗(E) = µ∗(E ∩ A) + µ∗(E ∩ Ac) ] ,then M∗ is a σ-algebra containing the Borel measurable sets andµ := µ∗|M∗ is a complete measure.We call µ the local Hausdorff measure of variable dimension α(·).

Dever (GaTech) Exit time exponent Fractals 6 3 / 19

Variable Ahlfors Regularity

By f ≈ g we mean there exists a constant C > 0 (independent of thearguments of f and g) such that 1

C f ≤ g ≤ Cg.

DefinitionIf Q : X → (0,∞), a borel measure ν is called variable AhlforsQ(·)−regular if

ν(Br (x)) ≈ rQ(x).

Theorema Let X be a compact metric space. Then if ν is an Ahlfors Q-regularBorel measure then Q = α and ν ≈ µ.

aJ. Dever, Local Hausdorff Measure, ArXiv e-prints (2016).

Dever (GaTech) Exit time exponent Fractals 6 4 / 19

A Koch curve of continuously varying local dimension.2

Constructed by recursively varying the angles of the generator.Any constant dimensional Hausdorff measure gives measure 0 or∞. But the µ measure is positive and finite.

2Laurent Nottale Fractal space time and microphysics: towards a theory of scalerelativity, Petteri Harjulehto, Peter Hasto, and Visa Latvala, Sobolev embeddings inmetric measure spaces with variable dimension, Mathematische Zeitschrift 254(2006), no. 3, 591–609.[8]

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Time scaling exponent: discrete approach

From now on: X connected, compact, Ahlfors α(·)−regular.An ε−net on X is a set N ⊂ X such that ∪x∈NBε(x) = X and ifx , y ∈ X with x 6= y then d(x , y) ≥ ε.We way define an induced graph GN with vertex set N in manyways. Some choices for edge relations are as follows:

Given c > 0 set x ∼ y if Bcε(x) ∩ Bcε(y) 6= ∅.For x ∈ N let T (x) = z ∈ X | d(x , z) ≤ d(y , z) for all y ∈ N bethe Voronoi tile of x . Set x ∼ y it T (x) ∩ T (y) 6= ∅.

For x ∈ N let deg(x) = #y ∈ N | y ∼ x. Let DN be the degreematrix and AN the adjacency matrix of GN ...i.e.DN x ,y = deg(x)δx ,y and AN x ,y = 1 if x ∼ y and 0 otherwise.Define a random walk (Xk )∞k=1 by the transition matrixPN = DN

−1AN , PN x ,y = pN(x , y) = 1deg(x)AN x ,y .[2]

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Time scaling exponent: discrete approach

Let B = BR(x0) be a ball. Let τB,N = infk | Xk /∈ B.Let EB,N(x) = ExτB,N . Then we have the exit time equationEB,N(x) = 1 +

∑y∼x pN(x , y)EB,N(y) for x ∈ B and E(x) = 0 for

x /∈ B, i.e. (I − PB,N)EB = 1 for x ∈ B, E(x) = 0 for x /∈ B.

Example

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Time scaling exponent: discrete approach

Set E+B,N = maxx∈N EB,N(x). Then let

ωβ(B) = infδ>0 supE+B,Nε

β | N an ε− net3 with ε < δ

Note if β′ < β then E+N,Bε

β ≤ δβ−β′E+N,Bε

β′ It follows that if ωβ(B) > 0then ωβ′(B) =∞ and if ωβ′(B) <∞ then ωβ(B) = 0. Hence we maydefine

β(B) := supβ ≥ 0 | ωβ(B) =∞ = infβ ≥ 0 | ωβ(B) = 0.Then if B′ ⊂ B, β(B′) ≤ β(B).We define...

Definitionβ(x) := infR>0 β(BR(x)) = limR→0+ β(BR(x)).

3Another (weaker) possibility is to take lim sup only along a chosen sequence ofnested ε− nets or dyadic cube approximations with the length scale going to 0.

Dever (GaTech) Exit time exponent Fractals 6 8 / 19

Example

Here β = 2α. Hence β can vary continuously as well!

ExampleOn Rn with standard euclidean lattice approximations, with the(weaker) definition of β we have β(x) = 2.

ExampleOn the Sierpinski gasket the exit time from a ball of graph distance n isof order 5n. Since length scale is 1

2n , (with weaker def.) β(x) = log(5)log(2) .

Dever (GaTech) Exit time exponent Fractals 6 9 / 19

Continuous space approach

Since we have an Ahlfors α(·)−regular measure µ we may define arandom walk at stage r by the transition kernel

pr (x , y) =1

µ(Br (x))χBr (x)(y), x , y ∈ X .

Define Pr on L2(µ) by Pr f (x) =∫

pr (x , y)dµ(y). This defines a(continuous space, discrete time) random walk (Yk )∞k=0.If B = BR(x0) is a ball let τB = infk | Yk /∈ B.Let Er ,B(x) = ExτB,r (x).Again we have the exit time equationEr ,B(x) = 1 +

∫X pr (x , y)Er ,B(y)dµ(y) for x ∈ B and Er ,B(x) = 0 for

x /∈ B. So (I − Pr )Er ,B = 1 on B and Er ,B(x) = 0 for x /∈ B.

Dever (GaTech) Exit time exponent Fractals 6 10 / 19

Time scale exponent: continuous space approach

Let E+r ,B = supx∈B Er ,B.

Then let ωβ(B) = lim supr→0+ E+r ,Brβ.

As before there is a critical exponentβ(B) = supβ > 0 | ωβ(B) =∞ = infβ > 0 | ωβ(B) = 0.

Definitionβ(x) := infR>0 β(BR(x)) = limR→0+ β(BR(x)).

ExampleLet B = BR(0) a ball about the origin in Rn with euclidean norm. Let rsmall enough so that Br (x) ⊂ B. Then we have

Er (x) =

(n + 2

n

)R2 − |x |2

r2 .

Hence β(x) = 2 on Rn.

Dever (GaTech) Exit time exponent Fractals 6 11 / 19

Continuous time re-normalization

Let τr (x) = rβ(x) for x ∈ X . Let Ω = (R+ × X )Z+ . For x = x0, r > 0define a measure on cylinder sets by

Px0r (ω ∈ (R+ × X )Z+ | ω(k) ∈ Ik × Ak for k = 1, ...,n) =(∏n

k=1∫

Ik

∫Ak

e−tk/τr (xk−1)

τr (xk−1)pr (xk−1, xk )

)dtndµ(xn)...dt1dµ(x1).4

This is Kolmogorov consistent. By the Kolmogorov Extension Theoremwe may extend to a prob. measure Px

r on Ω with product σ-algebra.For basepoint x = x0 set ω(0) = (0, x0). Then for t ≥ 0, ω ∈ Ω, let

t(ω) = infk |∑k

j=0 ω(k)1 ≥ t.

Then define (X (r)t )t≥0 by

X (r)t (ω) = ω(t(ω))2.

4A measure with local exponential waiting times in the case of a graph wasconsidered by Bellissard in [3]. A formula for the generator was also given there.

Dever (GaTech) Exit time exponent Fractals 6 12 / 19

Generator

Then (X (r)t )t≥0 has generator Lr on L2(µ) defined by

ddt

∣∣∣t=0

Ex f (Xt ) = Lr f (x) := 1rβ(x)µ(Br (x))

∫Br (x)(f (x)− f (y))dµ(y). [3]

There is an equilibrium probability measure νr with densitydνr (x)/dµ(x) := rβ(x)µ(Br (x))

Zr, where Zr is the normalization factor

Zr =∫

X rβ(z)µ(Br (z))dµ(z).

Then Lr is self-adjoint on L2(νr ).

Let τB,r := inft | X (r)t /∈ B.

For x ∈ X let φr ,B(x) := ExτB,r . Let φ+r ,B := supy∈B φr ,B(y).

Dever (GaTech) Exit time exponent Fractals 6 13 / 19

Variable Time Regularity Condition

For β > 0 let T(B) := lim supr→0+ φ+r ,B.

Definition(Time regularity, general case) For β(x) the local time exponent, wesay X satisfies E(β) if β is bounded and for all x ∈ X ,0 < r < diam(X)

2 ,we have

T(Br (x)) ≈ rβ(x).a

a[7] considers a similar condition under assumption that a heat kernelexists.

ExampleLet B the open ball of radius R about the origin in Rn under theeuclidean norm | · |. If Br (x) ⊂ B we have φr (x) =

(n+2n

)(R2 − |x |2).

Hence T(Br (x)) = (n+2n )r2.

Dever (GaTech) Exit time exponent Fractals 6 14 / 19

Dirichlet spectrum lower bound

Let B = BR(x0) and assume µ((Bc)) > 0. Then define LBr on L2(B, νr )

by LBr f (x) = χB(x)Lr (χBf )(x) where we define χBf outside B to be 0.

We have the exit time equation LBr φB,r (x) = 1.5 Then one can show

LBr has a νr -symmetric Green’s function gB(x , y) such that∫

gB(x , y)dνr (y) = φB,r (x).

TheoremIf λB

r is the bottom of the spectrum of LBr on L2(B, νr ) then

λBr ≥

cRβ(x0)

for some constant c > 0 independent of r ,R, and x0.

5Compare to “torsion function” and “torsional rigidity” on a Riemannian manifold.Dever (GaTech) Exit time exponent Fractals 6 15 / 19

Γ−Convergence6

If X is a separable second countable topological space withneighborhood basis at x B(x) then(Γ- lim infn→∞ fn)(x) := supU∈B(x) lim infn→∞ infy∈U fn(y) and(Γ- lim supn→∞ fn)(x) := supU∈B(x) lim supn→∞ infy∈U fn(y). We say fnΓ−converges to f if the two are equal.

Theorema Every sequence of non-negative Markovian forms forms on aseparable Hilbert space L2(X , µ) has a subsequence Γ−converging toa closed, non-negative, Markovian form (i.e. a Dirichlet form).

aproven in Mosco Composite media and asymptotic Dirichlet forms[4].

This Theorem was applied in the Kumagai-Sturm construction in [1]. Inthat paper Γ−limits of approximate Dirichlet formsEr (u) =

∫X

∫X |u(x)− u(y)|2kr (x , y)dµ(x)dµ(y) are considered.

Particular attention is paid to the choice kr (x , y) = 1h(r)µ(Br (x))χBr (x).

6For more information see book by Dal Maso [5]Dever (GaTech) Exit time exponent Fractals 6 16 / 19

In our case the process (X (r)t )t≥0 leads us to consider approximate

forms given byEr (f ) = 〈f ,Lr f 〉L2(X ,νr ) = 1

Zr

∫X

∫Br (x) |f (y)− f (x)|2dµ(y)dµ(x), where

Zr = 〈vr rβ〉µ =∫

X µ(Br (x))rβ(x)dµ(x),dν(x) = µ(Br (x))rβ(x)dµ(x).

TheoremA sequence (rn)∞n=1 of positive numbers decreasing to 0 can bechosen such that E = Γ- limn→∞ Ern exists, and D(E) contains analgebra of bounded measurable functions separating pointsa.

athis algebra is generated by pointwise limits of exit time functions fromballs in a neighborhood basis.

Dever (GaTech) Exit time exponent Fractals 6 17 / 19

[1]Takashi Kumagai, Karl-Theodor Sturm, Construction of diffusionprocesses on fractals, d-sets, and general metric measurespaces(2005).

[2]Andras Telcs, The art of random walks, 2006.

[3]Jean Bellissard, Diffusion on compact metric spaces,Unpublished private notes.

[4]Umberto Mosco, Composite media and asymptotic Dirichletforms, 1994.

[5]Gianni Dal Maso, An introduction to Γ-convergence, 2012.

[6]J. Dever, Local Hausdorff measure, ArXiv (2016).

[7]Alexander Grigor’yan, Heat kernels on manifolds, graphs andfractals, 2001.

[8]Harjulehto, Hasto, and Latvala, Sobolev embeddings in metricmeasure spaces with variable dimension(2006).

Dever (GaTech) Exit time exponent Fractals 6 18 / 19

Thank you.

Dever (GaTech) Exit time exponent Fractals 6 19 / 19