Approximationofsubordinate...

Approximation of subordinate

semigroups via the Chernoff theorem

Yana A. Butko

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1. SEMIGROUPS AND MARKOV PROCESSES

Def: A family (Tt)t≥0 of bounded linear operators on a Banach space Xis called a C0-semigroup, if T0 = Id, Ts ◦ Tt = Ts+t for all t,s ≥ 0 andlimt→0‖Ttϕ−ϕ‖X = 0 for all ϕ ∈X .

Def: The generator (L,Dom(L)) of a C0-semigroup (Tt)t≥0 is defined via

Lϕ := limt→0

Ttϕ−ϕ

t, Dom(L) := {ϕ ∈X : this limit exists }.

Thm: (Tt)t≥0 is a C0-semigroup on a BS X with generator (L,Dom(L))⇐⇒ the Cauchy problem

{dfdt = Lf,f(0) = f0

is correctly posed inX for all f0 ∈Dom(L). And f(t) := Ttf0 is the uniquesolution.

Notation: Tt ≡ etL3

Rem: Let (Xt)t≥0 be a Markov process with transition kernel P (t,x,dy).Then (Tt)t≥0, given by

Ttϕ(x) :=

∫ϕ(y)P (t,x,dy)≡ E

x [ϕ(Xt)] ,

is a semigroup, can be C0 on some BS X of functions ϕ. Therefore,

To construct the C0-semigroup Tt ≡ etL on X with a given generator L

mTo solve the Cauchy problem for the evolution equation df

dt = Lf in X

mTo find the transition probability Pt(x,dy) of the corresponding Markovprocess (Xt)t≥0

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2. CHERNOFF APPROXIMATION

OF EVOLUTION SEMIGROUPS

The Chernoff theorem [1968]: Let F : [0,∞)→L(X) be such that

• F (0) = Id,

• ‖F (t)‖6 eat for some a ∈ R and all t> 0,

• the limit Lϕ := limt→0

F (t)ϕ−ϕt exists for all ϕ ∈ D, where D ⊂ X: the

closure (L,Dom(L)) of (L,D) generates a C0-semigroup (Tt ≡ etL)t≥0.

Then

etLϕ = limn→∞

[F (t/n)

]nϕ, ∀ϕ ∈X,

locally uniformly w.r.t. t≥ 0.

Notation: F (t)∼ Tt.5

Ex1: Let L be a bounded operator on X . Then F (t) := Id+tL∼ etL.Hence

etL = limn→∞

[Id+

t

nL

]n.

Ex2: Fk(t)∼ etLk , k = 1, ...,m =⇒ F1(t) ◦ ... ◦Fm(t)∼ et(L1+...+Lm).

Rem: Take k = 2 and Fk(t) := etLk to get the Daletskii–Lie–Trotter

formula:

et(L1+L2) = limn→∞

[etL1/n ◦ etL2/n

]n.

Rem: If F (t) are integral operators, one has

etLϕ= limn→∞

[F (t/n)

]nϕ=

= limn→∞

∫· · ·∫

. . .ϕ(xn)dx1 · · ·dxn ← Feynman formula=

= path integral (Feynman–Kac formula / Feynman path integral)

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Some recent results: F (t) are constructed: F (t)∼ etL on X , where

• the operatorL generates a Feller process (B., Schilling, Smolyanov 2012)

• Lϕ(x) := tr(A(x)Hessϕ(x))+ b(x) · ∇ϕ(x)+ c(x)ϕ(x)

in C∞(Rd) and in Lp(Rd), p ∈ [1,∞](B., Grothaus, Smolyanov 2010, 2016; Plyashechnik 2013)

• the same L+ Dirichlet boundary conditions(B., Grothaus, Smolyanov 2010)

• Lϕ(x) := a(x)∆Kϕ(x)+ b(x) · ∇Kϕ(x)+ c(x)ϕ(x)

in Cb(K), K is a compact Riemannian manifold

(Butko 2016, 2008 via Smolyanov, Weizsacker, Wittich 2007)

in C∞(K), K is a star graph

(Butko 2015 via Kostrykin, Potthoff, Schrader 2012)

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3. SUBORDINATE SEMIGROUPS

To construct a subordinate semigroup/process, one needs:

(1): Original (parent) C0 contraction semigroup (Tt ≡ etL)t≥0 on a BS X /

Markov process (Xt)t≥0.

(2): Subordinator, i.e. (ξt)t≥0⇐⇒ (ηt)t≥0⇐⇒ f ⇐⇒ (σ,λ,µ), where

• (ξt)t≥0 is a Levy-process with a.s. non-decreasing paths;• (ηt)t≥0 is such that P(ξt ∈ A) = ηt(A); it is a convolution semigroupsupported by [0,∞), i.e. ηt are Borel measures on [0,∞), ηt([0,∞))≤ 1,ηt ∗ ηs = ηt+s and ηt ⇀ δ0 as t→ 0;• f is a Bernstein function, such that L [ηt] = e−tf ;• (σ,λ,µ) are Levy characteristics of f , i.e. σ,λ≥ 0, µ is a Radon measureon (0,∞) with

∫∞0+

s1+sµ(ds) such that

f(z) = σ+λz+

∫ ∞

0+

(1− e−sz)µ(ds), ∀z : Re z ≥ 0.

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Def: The family (T ft )t≥0 defined on the Banach space X by

T ft ϕ :=

∫ ∞

0

Tsϕηt(ds), ∀ϕ ∈X,

is called subordinate to (Tt)t≥0 w.r.t. (ηt)t≥0.

• (T ft )t≥0 is a C0 contraction semigroup corresponding to the subordinate

process (Xξt)t≥0.

• Let (Lf ,Dom(Lf)) be the generator of (T ft )t≥0, then Dom(L) is a core

for Lf and

Lfϕ=−σϕ+λLϕ+

∫ ∞

0+

(Tsϕ−ϕ)µ(ds)≡−f(−L)ϕ, ∀ϕ∈Dom(L),

where (L,Dom(L)) generates the parent semigroup (Tt)t≥0.

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4. CHERNOFF APPROXIMATION

OF SUBORDINATE SEMIGROUPS

Let the parent semigroup (Tt)t≥0 be unknown explicitly. Then:

T ft ϕ :=

∫ ∞

0

Tsϕηt(ds) =⇒ unknown

and

Lfϕ=−σϕ+λLϕ+

∫ ∞

0+

(Tsϕ−ϕ)µ(ds) =⇒ unknown

Task: To find F(t)∼ T ft if F (t)∼ Tt is given.

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Idea 1:

Lfϕ=−σϕ+λLϕ+

∫ ∞

0+

(Tsϕ−ϕ)µ(ds)

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Idea 1:

Lfϕ=−σϕ+λLϕ+

∫ ∞

0+

(Tsϕ−ϕ)µ(ds)

Therefore,

T ft ≡ etL

f

= et(L2+L1+L0)

with

L2 : L2ϕ=−σϕ =⇒ etL2ϕ= e−tσϕ,

L1 : L1ϕ= λLϕ =⇒ etL1ϕ= et(λL)ϕ= e(tλ)Lϕ≡ Ttλϕ =⇒ F (tλ)∼ etL1,

L0 : L0ϕ=∫∞0+(Tsϕ−ϕ)µ(ds) = Lf0, where f0⇐⇒ (0,0,µ)⇐⇒ (η0t )t≥0.

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Idea 1:

Lfϕ=−σϕ+λLϕ+

∫ ∞

0+

(Tsϕ−ϕ)µ(ds)

Therefore,

T ft ≡ etL

f

= et(L2+L1+L0)

with




Assume F0(t)∼ etL0 is constructed. Then

F(t) := e−tσ ◦F (tλ) ◦F0(t)∼ T ft .

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Idea 1:

Lfϕ=−σϕ+λLϕ+

∫ ∞

0+

(Tsϕ−ϕ)µ(ds)

Therefore,

T ft ≡ etL

f

= et(L2+L1+L0)

with





F(t) := e−tσ ◦F (tλ) ◦F0(t)∼ T ft .

But how to construct F0(t)?

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Idea 1:

Lfϕ=−σϕ+λLϕ+

∫ ∞

0+

(Tsϕ−ϕ)µ(ds)

Therefore,

T ft ≡ etL

f

= et(L2+L1+L0)

with





F(t) := e−tσ ◦F (tλ) ◦F0(t)∼ T ft .

But how to construct F0(t)?

Note, F0(t) :=∫∞0 F (s)η0t (ds)≁ etL0 =

∫∞0 Tsη

0t (ds) since F ′0(0) 6= L0.

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Idea 2. Case (a): (η0t )t≥0 is known explicitly (IG, Gamma, e.t.c.).

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Thm: Let m : (0,∞)→ N0 be monotone and m(t)→∞ as t→ 0, e.g.,m(t) := [1/t]. Then F0(t)∼ etL0, where

F0(t)ϕ :=

∫ ∞

0

[F (s/m(t))]m(t)ϕη0t (ds), ϕ ∈X.

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F0(t)ϕ :=

∫ ∞

0


Idea 2. Case (b): (η0t )t≥0 is unknown, but µ is known and bounded.

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F0(t)ϕ :=

∫ ∞

0



Then L0 : L0ϕ=∫∞0+(Tsϕ−ϕ)µ(ds) is bounded =⇒ Id+tL0 ∼ etL0.

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F0(t)ϕ :=

∫ ∞

0



Then L0 : L0ϕ=∫∞0+(Tsϕ−ϕ)µ(ds) is bounded =⇒ Id+tL0 ∼ etL0.

Thm: Let m : (0,∞)→ N0 be monotone and m(t)→∞ as t→ 0, e.g.,m(t) := [1/t]. Then Fµ

0 (t)∼ etL0, where

Fµ0 (t)ϕ := ϕ+ t

∫ ∞

0+

([F (s/m(t))]m(t)ϕ−ϕ

)µ(ds), ϕ ∈X.

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Example: L : Lϕ(x) = 12tr(A(x)Hessϕ(x))+B(x) ·∇ϕ(x)−C(x)ϕ(x).

In case (a):

F(t)ϕ(q) := e−tσ∞∫

0+

∫

Rd(m(t)+1)

e−tλC(qm(t)+2)− s

m(t)

m(t)∑k=1

C(qk+1)×

× e−

m(t)+1∑k=1

A−1(qk+1)B(qk+1)·(qk+1−qk)e− s

m(t)

m(t)∑k=1

A−1(qk+1)B(qk+1)·B(qk+1)×× e−

tλ2 A

−1(qm(t)+2)B(qm(t)+2)·B(qm(t)+2)ϕ(q1)×

×

pA(tλ,qm(t)+1, qm(t)+2)

m(t)∏

k=1

pA(s/m(t), qk, qk+1)

m(t)+1∏

k=1

dqkη0t (ds)

qm(t)+2 := q and pA(t,x,y) :=1√

detA(x)(2πt)dexp

(− A−1(x)(x−y)·(x−y)

2t

);

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In case (b):

Fµ(t)ϕ(q) :=

=exp(−t(σ+λC(q)))√

detA(q)(2πtλ)d

∫

Rd

e−A−1(q)(q−qm(t)+1+tλB(q))·(q−qm(t)+1+tλB(q))

2tλ ×

×(ϕ(qm(t)+1)+ t

∞∫

0+

[ ∫

Rdm(t)

e− s

m(t)

m(t)∑k=1

C(qk+1)m(t)∏

k=1

(detA(qk+1)(2πtλ)

d

)−1/2

× e−

m(t)∑k=1

A−1(qk+1)

(qk+1−qk+

sm(t)

B(qk+1)

)·

(qk+1−qk+

sm(t)

B(qk+1)

)

2s/m(t)

×ϕ(q1)dq1 . . .dqm(t)−ϕ(qm(t)+1)

]µ(ds)

)dqm(t)+1

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THANKS FOR THE ATTENTION!

1. Ya.A. Butko. Chernoff approximation of subordinate semigroups and

applications. Preprint. http://arxiv.org/pdf/1512.05258.pdf.

2. Ya.A. Butko, M. Grothaus and O.G. Smolyanov. Feynman formulae

and phase space Feynman path integrals for tau-quantization of some

Levy-Khintchine type Hamilton functions. J. Math. Phys. 57 023508

(2016), 22 p.

3. Ya.A. Butko. Description of quantum and classical dynamics via

Feynman formulae. Mathematical Results in Quantum Mechanics:

Proceedings of the QMath12 Conference, p.227-234. World Scientific,

2014. ISBN: 978-981-4618-13-7 (hardcover), ISBN: 978-981-4618-15-

1 (ebook).

4. Ya.A. Butko. Feynman formulae for evolution semigroups (in Russian).

Electronic scientific and technical periodical ”Science and education”,

DOI: 10.7463/0314.0701581 , N 3 (2014), 95-132.23

5. Ya.A. Butko, R.L. Schilling and O.G. Smolyanov. Lagrangian and

Hamiltonian Feynman formulae for some Feller semigroups and their

perturbations, Inf. Dim. Anal. Quant. Probab. Rel. Top., 15N 3 (2012),

26 p.

6. B. Bottcher, Ya.A. Butko, R.L. Schilling and O.G. Smolyanov. Feynman

formulae and path integrals for some evolutionary semigroups related to

τ -quantization, Rus. J. Math. Phys. 18 N4 (2011), 387–399.

7. Ya.A. Butko, M. Grothaus and O.G. Smolyanov. Lagrangian Feynman

formulae for second order parabolic equations in bounded and unbounded

domains, Inf. Dim. Anal. Quant. Probab. Rel. Top. 13 N3 (2010),

377-392.

8. Ya.A. Butko. Feynman formulas and functional integrals for diffusion

with drift in a domain on a manifold, Math. Notes 83 N3 (2008), 301–

316.

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9. V. Kostrykin, J. Potthoff, R. Schrader. Construction of the paths of

Brownian motions on star graphs II, Commun. Stoch. Anal., 6 N 2

(2012), 247-261.

10. A. S. Plyashechnik. Feynman formulas for second-order parabolic

equations with variable coefficients, Russ. J. Math. Phys. 20 N 3

(2013), 377-379.

11. O. G. Smolyanov, H. v. Weizsacker, O. Wittich. Chernoff’s theorem

and discrete time approximations of Brownian motion on manifolds.

Potential Anal., 26 N 1 (2007), 1-29.

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