Chernoff approximation of evolution semigroups and beyond · Cherno approximation of evolution...

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Chernoff approximation of evolution semigroups and beyond Yana A. Butko SOTA 2018, 04.10.2018 Yana A. Butko Chernoff approximation SOTA 2018, 04.10.2018 1 / 56

Transcript of Chernoff approximation of evolution semigroups and beyond · Cherno approximation of evolution...

Page 1: Chernoff approximation of evolution semigroups and beyond · Cherno approximation of evolution semigroups and beyond Yana A. Butko SOTA 2018, 04.10.2018 Yana A. Butko Cherno approximation

Chernoff approximation of evolution semigroupsand beyond

Yana A. Butko

SOTA 2018, 04.10.2018

Yana A. Butko Chernoff approximation SOTA 2018, 04.10.2018 1 / 56

Page 2: Chernoff approximation of evolution semigroups and beyond · Cherno approximation of evolution semigroups and beyond Yana A. Butko SOTA 2018, 04.10.2018 Yana A. Butko Cherno approximation

Chernoff approximation of Markov evolution

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Chernoff approximation of Markov evolution

(ξt)t⩾0 is a time homogeneous Markov process (⇒ no memory)⇒ transition kernel P(t, x ,dy) ∶= P(ξt ∈ dy ∣ ξ0 = x)

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Chernoff approximation of Markov evolution

(ξt)t⩾0 is a time homogeneous Markov process (⇒ no memory)⇒ transition kernel P(t, x ,dy) ∶= P(ξt ∈ dy ∣ ξ0 = x)

Thenf (t, x) ∶= ∫ f0(y)P(t, x ,dy) ≡ E [f0(ξt) ∣ ξ0 = x]

solves the following evolution equation:

{∂f∂t (t, x) = Lf (t, x),f (0, x) = f0(x),

whereTt f0(x) ∶= ∫ f0(y)P(t, x ,dy) ≡ etLf0.

(Tt)t⩾0 is an operator semigroup (i.e. T0 = Id, Tt ○Ts = Tt+s).

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Chernoff approximation of Markov evolution

Stochastics

To determine the transition kernel P(t, x ,dy) for a given process (ξt)t⩾0.

Functional Analysis

To construct the semigroup Tt ≡ etL with a given generator L.

PDEs

To solve a (Cauchy problem for a) given PDE ∂f∂t = Lf .

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Chernoff approximation of Markov evolution

Example:

Heat equation∂f

∂t= 1

2∆f , x ∈ Rd .

Heat semigroup

Tt f0(x) ∶= (2πt)−d/2∫Rd

f0(y) exp{− ∣x − y ∣2

2t}dy .

Transition kernel of Brownian motion

P(t, x ,dy) = (2πt)−d/2 exp{− ∣x − y ∣2

2t}dy .

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Page 7: Chernoff approximation of evolution semigroups and beyond · Cherno approximation of evolution semigroups and beyond Yana A. Butko SOTA 2018, 04.10.2018 Yana A. Butko Cherno approximation

Chernoff approximation of Markov evolution

Chernoff approximation: To find (F (t))t⩾0 (not a SG!!!) such that

Tt f0 = limn→∞

[F (t/n)]n f0.

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Page 8: Chernoff approximation of evolution semigroups and beyond · Cherno approximation of evolution semigroups and beyond Yana A. Butko SOTA 2018, 04.10.2018 Yana A. Butko Cherno approximation

Chernoff approximation of Markov evolution

Chernoff approximation: To find (F (t))t⩾0 (not a SG!!!) such that

Tt f0 = limn→∞

[F (t/n)]n f0.

⇒ discrete time approximation to the solution f (t, x):

u0 ∶= f0, uk ∶= F (t/n)uk−1, k = 1, . . . ,n, f (t, ⋅) ≈ un.

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Page 9: Chernoff approximation of evolution semigroups and beyond · Cherno approximation of evolution semigroups and beyond Yana A. Butko SOTA 2018, 04.10.2018 Yana A. Butko Cherno approximation

Chernoff approximation of Markov evolution

Chernoff approximation: To find (F (t))t⩾0 (not a SG!!!) such that

Tt f0 = limn→∞

[F (t/n)]n f0.

⇒ discrete time approximation to the solution f (t, x):

u0 ∶= f0, uk ∶= F (t/n)uk−1, k = 1, . . . ,n, f (t, ⋅) ≈ un.

⇒ Markov chain approximation to (ξt)t⩾0 (e.g., Euler scheme),

(ξnk)k=1,...,n, ∶ E [f0(ξnk) ∣ ξnk−1] = F (t/n)f0(ξnk−1)

⇒ E [f0(ξt) ∣ ξ0 = x] = limn→∞

E [f0(ξnn) ∣ ξ0 = x]

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Page 10: Chernoff approximation of evolution semigroups and beyond · Cherno approximation of evolution semigroups and beyond Yana A. Butko SOTA 2018, 04.10.2018 Yana A. Butko Cherno approximation

Chernoff approximation of Markov evolution

Chernoff approximation: To find (F (t))t⩾0 (not a SG!!!) such that

Tt f0 = limn→∞

[F (t/n)]n f0.

⇒ discrete time approximation to the solution f (t, x):

u0 ∶= f0, uk ∶= F (t/n)uk−1, k = 1, . . . ,n, f (t, ⋅) ≈ un.

⇒ Markov chain approximation to (ξt)t⩾0 (e.g., Euler scheme),

(ξnk)k=1,...,n, ∶ E [f0(ξnk) ∣ ξnk−1] = F (t/n)f0(ξnk−1)

⇒ E [f0(ξt) ∣ ξ0 = x] = limn→∞

E [f0(ξnn) ∣ ξ0 = x]

⇒ approximation of path integrals in Feynman-Kac formulae.Yana A. Butko Chernoff approximation SOTA 2018, 04.10.2018 10 / 56

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Chernoff approximation of Markov evolution

Chernoff Theorem (1968): Let (Tt)t⩾0 be a strongly continuoussemigroup on X with generator (L,Dom(L)). Let (F (t))t⩾0 be a family ofbounded linear operators on X . Assume that

● F (0) = Id,

● ∥F (t)∥ ⩽ ewt for some w ∈ R, and all t ⩾ 0,

● limt→0

F (t)ϕ − ϕt

= Lϕ

for all ϕ ∈ D, where D is a core for (L,Dom(L)).

Then it holdsTtϕ = lim

n→∞[F (t/n)]n ϕ, ∀ ϕ ∈ X ,

and the convergence is locally uniform with respect to t ⩾ 0.

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Page 12: Chernoff approximation of evolution semigroups and beyond · Cherno approximation of evolution semigroups and beyond Yana A. Butko SOTA 2018, 04.10.2018 Yana A. Butko Cherno approximation

Chernoff approximation of Markov evolution

Chernoff Theorem (1968): Let (Tt)t⩾0 be a strongly continuoussemigroup on X with generator (L,Dom(L)). Let (F (t))t⩾0 be a family ofbounded linear operators on X . Assume that

● F (0) = Id, (consistency)● ∥F (t)∥ ⩽ ewt for some w ∈ R, and all t ⩾ 0, (stability)

● limt→0

F (t)ϕ − ϕt

= Lϕ (consistency)

for all ϕ ∈ D, where D is a core for (L,Dom(L)).

Then it holdsTtϕ = lim

n→∞[F (t/n)]n ϕ, ∀ ϕ ∈ X ,

and the convergence is locally uniform with respect to t ⩾ 0.

Meta-theorem of Numerics: Consistency + stability ⇒ convergence.

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LEGO principle for Chernoff approximation:

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LEGO principle for Chernoff approximation:

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LEGO principle for Chernoff approximation:

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LEGO principle for Chernoff approximation:

Nice processes to start with:

ξt with known P(t, x ,dy);

Feller processes in Rd ;

Brownian motion in a compact Riemannian manifold;

.......

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Page 17: Chernoff approximation of evolution semigroups and beyond · Cherno approximation of evolution semigroups and beyond Yana A. Butko SOTA 2018, 04.10.2018 Yana A. Butko Cherno approximation

LEGO principle for Chernoff approximation:

Nice processes to start with:

ξt with known P(t, x ,dy);

Feller processes in Rd ;

Brownian motion in a compact Riemannian manifold;

.......

Chernoff approximations are known for the following operations:

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Page 18: Chernoff approximation of evolution semigroups and beyond · Cherno approximation of evolution semigroups and beyond Yana A. Butko SOTA 2018, 04.10.2018 Yana A. Butko Cherno approximation

LEGO principle for Chernoff approximation:

Nice processes to start with:

ξt with known P(t, x ,dy);

Feller processes in Rd ;

Brownian motion in a compact Riemannian manifold;

.......

Chernoff approximations are known for the following operations:

operator splitting ⇔ L∗ ∶= L1 + . . . + Lm;

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Page 19: Chernoff approximation of evolution semigroups and beyond · Cherno approximation of evolution semigroups and beyond Yana A. Butko SOTA 2018, 04.10.2018 Yana A. Butko Cherno approximation

LEGO principle for Chernoff approximation:

Nice processes to start with:

ξt with known P(t, x ,dy);

Feller processes in Rd ;

Brownian motion in a compact Riemannian manifold;

.......

Chernoff approximations are known for the following operations:

operator splitting ⇔ L∗ ∶= L1 + . . . + Lm on D:= a core for L∗;

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Page 20: Chernoff approximation of evolution semigroups and beyond · Cherno approximation of evolution semigroups and beyond Yana A. Butko SOTA 2018, 04.10.2018 Yana A. Butko Cherno approximation

LEGO principle for Chernoff approximation:

Nice processes to start with:

ξt with known P(t, x ,dy);

Feller processes in Rd ;

Brownian motion in a compact Riemannian manifold;

.......

Chernoff approximations are known for the following operations:

operator splitting ⇔ L∗ ∶= L1 + . . . + Lm on D:= a core for L∗;

If Fk(t) ∶ Fk(0) = Id, ∥Fk(t)∥ ⩽ etwk , F ′k(0)ϕ = Lkϕ ∀ ϕ ∈ Dthen

F ∗(t) ∶= F1(t) ○ . . . ○ Fm(t) ∼ etL∗

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Page 21: Chernoff approximation of evolution semigroups and beyond · Cherno approximation of evolution semigroups and beyond Yana A. Butko SOTA 2018, 04.10.2018 Yana A. Butko Cherno approximation

LEGO principle for Chernoff approximation:

Nice processes to start with:

ξt with known P(t, x ,dy);

Feller processes in Rd ;

Brownian motion in a compact Riemannian manifold;

.......

Chernoff approximations are known for the following operations:

operator splitting ⇔ L∗ ∶= L1 + . . . + Lm on D:= a core for L∗;

If Fk(t) ∶ Fk(0) = Id, ∥Fk(t)∥ ⩽ etwk , F ′k(0)ϕ = Lkϕ ∀ ϕ ∈ Dthen

F ∗(t) ∶= F1(t) ○ . . . ○ Fm(t) ∼ etL∗

Corollary (Trotter formula):

etL1 ○ etL2 ∼ et(L1+L2) i.e. et(L1+L2) = limn→∞

[etL1/n ○ etL2/n]n

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Page 22: Chernoff approximation of evolution semigroups and beyond · Cherno approximation of evolution semigroups and beyond Yana A. Butko SOTA 2018, 04.10.2018 Yana A. Butko Cherno approximation

LEGO principle for Chernoff approximation:

Nice processes to start with:

ξt with known P(t, x ,dy);

Feller processes in Rd ;

Brownian motion in a compact Riemannian manifold;

.......

Chernoff approximations are known for the following operations:

operator splitting ⇔ L∗ ∶= L1 + . . . + Lm;

multiplicative perturbations of L ⇔ L∗ ∶= aL ⇔random time change of ξt via an additive functional;

killing of ξt upon leaving a domain G ⊂ Rd ⇔L∗ ∶= L+ Dirichlet boundary / external conditions;

subordination ⇔ L∗ ∶= −f (−L), ξ∗t ∶= ξηt ;“rotation” ⇔ L∗ ∶= iL (F (t) ∼ etL ⇒ F ∗(t) ∶= e i(F(t)−Id) ∼ e itL);

......

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Chernoff approximation for Feller processes

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Chernoff approximation for Feller processes

Feller process ξt ↭ Tt ≡ etL on C∞(Rd) with L ≡ −H:

Hϕ(x) ∶= (2π)−d ∫Rd

∫Rd

e ip⋅(x−q)H(x ,p)ϕ(q)dqdp,

where H(x , ⋅) is given by the Levy–Khintchine formula

H(x ,p) = C(x) + iB(x) ⋅ p + p ⋅A(x)p + ∫y≠0

(1 − e iy ⋅p + iy ⋅ p1 + ∣y ∣2

)N(x ,dy).

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Page 25: Chernoff approximation of evolution semigroups and beyond · Cherno approximation of evolution semigroups and beyond Yana A. Butko SOTA 2018, 04.10.2018 Yana A. Butko Cherno approximation

Chernoff approximation for Feller processes

Feller process ξt ↭ Tt ≡ etL on C∞(Rd) with L ≡ −H:

Hϕ(x) ∶= (2π)−d ∫Rd

∫Rd

e ip⋅(x−q)H(x ,p)ϕ(q)dqdp,

where H(x , ⋅) is given by the Levy–Khintchine formula

H(x ,p) = C(x) + iB(x) ⋅ p + p ⋅A(x)p + ∫y≠0

(1 − e iy ⋅p + iy ⋅ p1 + ∣y ∣2

)N(x ,dy).

Remark: If H = H(x ,p) then

e−tH ≠ e−tH

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Chernoff approximation for Feller processes

Feller process ξt ↭ Tt ≡ etL on C∞(Rd) with L ≡ −H:

Hϕ(x) ∶= (2π)−d ∫Rd

∫Rd

e ip⋅(x−q)H(x ,p)ϕ(q)dqdp,

where H(x , ⋅) is given by the Levy–Khintchine formula

H(x ,p) = C(x) + iB(x) ⋅ p + p ⋅A(x)p + ∫y≠0

(1 − e iy ⋅p + iy ⋅ p1 + ∣y ∣2

)N(x ,dy),

Remark: If H = H(x ,p) then

e−tH ≠ e−tH

Theorem:

F (t) ∶= e−tH ∼ e−tH , i.e. e−tH = limn→∞

[e−tH/n]n

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Page 27: Chernoff approximation of evolution semigroups and beyond · Cherno approximation of evolution semigroups and beyond Yana A. Butko SOTA 2018, 04.10.2018 Yana A. Butko Cherno approximation

Chernoff approximation for Feller processes

Feller process ξt ↭ Tt ≡ etL on C∞(Rd) with L ≡ −H:

Hϕ(x) ∶= (2π)−d ∫Rd

∫Rd

e ip⋅(x−q)H(x ,p)ϕ(q)dqdp,

where H(x , ⋅) is given by the Levy–Khintchine formula

H(x ,p) = C(x) + iB(x) ⋅ p + p ⋅A(x)p + ∫y≠0

(1 − e iy ⋅p + iy ⋅ p1 + ∣y ∣2

)N(x ,dy).

Theorem:

F (t) ∶= e−tH ∼ e−tH , i.e. e−tH = limn→∞

[e−tH/n]n

Remark: Let Pxt ∶ F [Px

t ] (p) = e−tH(x ,p). Then

F (t)ϕ(x) = (2π)−d ∫Rd

∫Rd

e ip⋅(x−q)e−tH(x ,p)ϕ(q)dqdp = (ϕ ∗ Pxt ) (x).

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Page 28: Chernoff approximation of evolution semigroups and beyond · Cherno approximation of evolution semigroups and beyond Yana A. Butko SOTA 2018, 04.10.2018 Yana A. Butko Cherno approximation

Chernoff approximation for Feller processes

Feller process ξt ↭ Tt ≡ etL on C∞(Rd) with L ≡ −H:

Hϕ(x) ∶= (2π)−d ∫Rd

∫Rd

e ip⋅(x−q)H(x ,p)ϕ(q)dqdp,

where H(x , ⋅) is given by the Levy–Khintchine formula

H(x ,p) = C(x) + iB(x) ⋅ p + p ⋅A(x)p + ∫y≠0

(1 − e iy ⋅p + iy ⋅ p1 + ∣y ∣2

)N(x ,dy).

Theorem:

F (t) ∶= e−tH ∼ e−tH , i.e. e−tH = limn→∞

[e−tH/n]n

Remark: Let µxt ∶ F [µxt ] (p) = e−tH(x ,−p)−ip⋅x . Then

F (t)ϕ(x) = (2π)−d ∫Rd

∫Rd

e ip⋅(x−q)e−tH(x ,p)ϕ(q)dqdp = ∫Rd

ϕ(q)µxt (dq).

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Page 29: Chernoff approximation of evolution semigroups and beyond · Cherno approximation of evolution semigroups and beyond Yana A. Butko SOTA 2018, 04.10.2018 Yana A. Butko Cherno approximation

Chernoff approximation for Feller processes

Feller process ξt ↭ Tt ≡ etL on C∞(Rd) with L ≡ −H,where H(x , ⋅) is given by the Levy–Khintchine formula

H(x ,p) = C(x) + iB(x) ⋅ p + p ⋅A(x)p + ∫y≠0

(1 − e iy ⋅p + iy ⋅ p1 + ∣y ∣2

)N(x ,dy)

Theorem:

F (t) ∶= e−tH ∼ e−tH , i.e. e−tH = limn→∞

[e−tH/n]n

Remark: Let µxt ∶ F [µxt ] (p) = e−tH(x ,−p)−ip⋅x . Then

F (t)ϕ(x) = (2π)−d ∫Rd

∫Rd

e ip⋅(x−q)e−tH(x ,p)ϕ(q)dqdp = ∫Rd

ϕ(q)µxt (dq).

Example: non-degenerate diffusion:

F (t)ϕ(x) = e−tC(x)

√(4πt)d detA(x)

∫Rd

ϕ(q)e−(x−q−tB(x))⋅A−1(x)(x−q−tB(x))

4t dq

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Chernoff approximation for Feller processes

Feller process ξt ↭ Tt ≡ etL on C∞(Rd) with L ≡ −H,Theorem:

F (t) ∶= e−tH ∼ e−tH , i.e. e−tH = limn→∞

[e−tH/n]n

Remark: Let µxt ∶ F [µxt ] (p) = e−tH(x ,−p)−ip⋅x . Then

F (t)ϕ(x) = (2π)−d ∫Rd

∫Rd

e ip⋅(x−q)e−tH(x ,p)ϕ(q)dqdp = ∫Rd

ϕ(q)µxt (dq).

Example: H(x ,p) ∶= a(x)∣p∣ ⇒ L ∶ Lϕ(x) = a(x) (−(−∆)1/2)ϕ(x)

F (t)ϕ(x) = (2π)−d ∫Rd

∫Rd

e ip⋅(x−q)e−ta(x)∣p∣ϕ(q)dqdp

= Γ(d + 1

2)∫Rd

ϕ(q) a(x)t

(π∣x − q∣2 + a2(x)t2)d+1

2

dq.

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Chernoff approximation for Killed Feller processes

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Chernoff approximation for Killed Feller processes

Feller process ξt ↭ Tt ≡ etL on C∞(Rd) with L ≡ −H

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Chernoff approximation for Killed Feller processes

Feller process ξt ↭ Tt ≡ etL on C∞(Rd) with L ≡ −H

Consider G ⊂ Rd bdd, regular, τG ∶= inf {t > 0 ∶ ξt ∉ G} and

ξot ∶= { ξt , t < τG∂, t ⩾ τG .

Hence

T ot ∶ T o

t ϕ(x) = E [ϕ(ξot ) ∣ ξo0 = x]

is a Feller SG on C∞(G); T ot ≡ etL

o,

Dom(Lo) ∶= {ϕ ∈ C∞(G) ∶ Lϕo ∈ C∞(G)} , Loϕ(x) ∶= Lϕo(x), x ∈ G

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Chernoff approximation for Killed Feller processes

Feller process ξt ↭ Tt ≡ etL on C∞(Rd) with L ≡ −H

Consider G ⊂ Rd bdd, regular, τG ∶= inf {t > 0 ∶ ξt ∉ G} and

ξot ∶= { ξt , t < τG∂, t ⩾ τG .

Hence

T ot ∶ T o

t ϕ(x) = E [ϕ(ξot ) ∣ ξo0 = x]

is a Feller SG on C∞(G); T ot ≡ etL

o,

Dom(Lo) ∶= {ϕ ∈ C∞(G) ∶ Lϕo ∈ C∞(G)} , Loϕ(x) ∶= Lϕo(x), x ∈ G

Aim: F (t) ∼ Tt ≡ etL on C∞(Rd) ⇒ F o(t) ∼ T ot ≡ etL

oon C∞(G)

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Aim: F (t) ∼ Tt ≡ etL on C∞(Rd) ⇒ F o(t) ∼ T ot ≡ etL

oon C∞(G)

Yana A. Butko Chernoff approximation SOTA 2018, 04.10.2018 35 / 56

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Aim: F (t) ∼ Tt ≡ etL on C∞(Rd) ⇒ F o(t) ∼ T ot ≡ etL

oon C∞(G)

Wish:F o(t) ∶ F o(t)ϕ(x) = 1G(x) (F (t)ϕo) (x)

Yana A. Butko Chernoff approximation SOTA 2018, 04.10.2018 36 / 56

Page 37: Chernoff approximation of evolution semigroups and beyond · Cherno approximation of evolution semigroups and beyond Yana A. Butko SOTA 2018, 04.10.2018 Yana A. Butko Cherno approximation

Aim: F (t) ∼ Tt ≡ etL on C∞(Rd) ⇒ F o(t) ∼ T ot ≡ etL

oon C∞(G)

Wish:F o(t) ∶ F o(t)ϕ(x) = 1G(x) (F (t)ϕo) (x)

Problems:

F o(t) ∶ C∞(G)→ C∞(G)

Yana A. Butko Chernoff approximation SOTA 2018, 04.10.2018 37 / 56

Page 38: Chernoff approximation of evolution semigroups and beyond · Cherno approximation of evolution semigroups and beyond Yana A. Butko SOTA 2018, 04.10.2018 Yana A. Butko Cherno approximation

Aim: F (t) ∼ Tt ≡ etL on C∞(Rd) ⇒ F o(t) ∼ T ot ≡ etL

oon C∞(G)

Wish:F o(t) ∶ F o(t)ϕ(x) = 1G(x) (F (t)ϕo) (x)

Problems:

F o(t) ∶ C∞(G)→ C∞(G)Hence

F o(t) ∶ F o(t)ϕ(x) = χt(x) (F (t)ϕo) (x)

Yana A. Butko Chernoff approximation SOTA 2018, 04.10.2018 38 / 56

Page 39: Chernoff approximation of evolution semigroups and beyond · Cherno approximation of evolution semigroups and beyond Yana A. Butko SOTA 2018, 04.10.2018 Yana A. Butko Cherno approximation

Aim: F (t) ∼ Tt ≡ etL on C∞(Rd) ⇒ F o(t) ∼ T ot ≡ etL

oon C∞(G)

Wish:F o(t) ∶ F o(t)ϕ(x) = 1G(x) (F (t)ϕo) (x)

Problems:

F o(t) ∶ C∞(G)→ C∞(G)Dom(Lo)↪ Dom(L) or core(Lo)↪ core(L)

Yana A. Butko Chernoff approximation SOTA 2018, 04.10.2018 39 / 56

Page 40: Chernoff approximation of evolution semigroups and beyond · Cherno approximation of evolution semigroups and beyond Yana A. Butko SOTA 2018, 04.10.2018 Yana A. Butko Cherno approximation

Aim: F (t) ∼ Tt ≡ etL on C∞(Rd) ⇒ F o(t) ∼ T ot ≡ etL

oon C∞(G)

Wish:F o(t) ∶ F o(t)ϕ(x) = 1G(x) (F (t)ϕo) (x)

Problems:

F o(t) ∶ C∞(G)→ C∞(G)Dom(Lo)↪ Dom(L) or core(Lo)↪ core(L)

HenceF o(t) ∶ F o(t)ϕ(x) = χt(x) (F (t)E(ϕ)) (x)

Yana A. Butko Chernoff approximation SOTA 2018, 04.10.2018 40 / 56

Page 41: Chernoff approximation of evolution semigroups and beyond · Cherno approximation of evolution semigroups and beyond Yana A. Butko SOTA 2018, 04.10.2018 Yana A. Butko Cherno approximation

Aim: F (t) ∼ Tt ≡ etL on C∞(Rd) ⇒ F o(t) ∼ T ot ≡ etL

oon C∞(G)

Wish:F o(t) ∶ F o(t)ϕ(x) = 1G(x) (F (t)ϕo) (x)

Problems:

F o(t) ∶ C∞(G)→ C∞(G)Dom(Lo)↪ Dom(L) or core(Lo)↪ core(L)

HenceF o(t) ∶ F o(t)ϕ(x) = χt(x) (F (t)E(ϕ)) (x)

Hence additional assumptions on ξt .

Yana A. Butko Chernoff approximation SOTA 2018, 04.10.2018 41 / 56

Page 42: Chernoff approximation of evolution semigroups and beyond · Cherno approximation of evolution semigroups and beyond Yana A. Butko SOTA 2018, 04.10.2018 Yana A. Butko Cherno approximation

Aim: F (t) ∼ Tt ≡ etL on C∞(Rd) ⇒ F o(t) ∼ T ot ≡ etL

oon C∞(G)

Wish:F o(t) ∶ F o(t)ϕ(x) = 1G(x) (F (t)ϕo) (x)

Problems:

F o(t) ∶ C∞(G)→ C∞(G)Dom(Lo)↪ Dom(L) or core(Lo)↪ core(L)

HenceF o(t) ∶ F o(t)ϕ(x) = χt(x) (F (t)E(ϕ)) (x)

Hence additional assumptions on ξt .

Theorem:

F o(t) ∼ T ot , i.e. T o

t ϕ = limn→∞

[F o(t/n)]n

Yana A. Butko Chernoff approximation SOTA 2018, 04.10.2018 42 / 56

Page 43: Chernoff approximation of evolution semigroups and beyond · Cherno approximation of evolution semigroups and beyond Yana A. Butko SOTA 2018, 04.10.2018 Yana A. Butko Cherno approximation

Aim: F (t) ∼ Tt ≡ etL on C∞(Rd) ⇒ F o(t) ∼ T ot ≡ etL

oon C∞(G)

Wish:F o(t) ∶ F o(t)ϕ(x) = 1G(x) (F (t)ϕo) (x)

Problems:

F o(t) ∶ C∞(G)→ C∞(G)Dom(Lo)↪ Dom(L) or core(Lo)↪ core(L)

HenceF o(t) ∶ F o(t)ϕ(x) = χt(x) (F (t)E(ϕ)) (x)

Hence additional assumptions on ξt .

Theorem:

F o(t) ∼ T ot , i.e. T o

t ϕ = limn→∞

[F o(t/n)]n

Remark: If H ∈ C(R2d) then loc. unif. w.r.t. x ∈ G and t > 0

T ot ϕ(x) = lim

n→∞[F o(t/n)]n ϕ(x) with F o(t)ϕ(x) ∶= 1G(x) (F (t)ϕo) (x)

= limn→∞∫G

. . .∫G

∫G

ϕ(xn)µxn−1

t/n(dxn)µxn−2

t/n(dxn−1)⋯µxt/n(dx1)

Yana A. Butko Chernoff approximation SOTA 2018, 04.10.2018 43 / 56

Page 44: Chernoff approximation of evolution semigroups and beyond · Cherno approximation of evolution semigroups and beyond Yana A. Butko SOTA 2018, 04.10.2018 Yana A. Butko Cherno approximation

Aim: F (t) ∼ Tt ≡ etL on C∞(Rd) ⇒ F o(t) ∼ T ot ≡ etL

oon C∞(G)

Theorem:

F o(t) ∼ T ot , i.e. T o

t ϕ = limn→∞

[F o(t/n)]n ,

F o(t) ∶ F o(t)ϕ(x) = χt(x) (F (t)E(ϕ)) (x)

Remark: If H ∈ C(R2d) then loc. unif. w.r.t. x ∈ G and t > 0

T ot ϕ(x) = lim

n→∞[F o(t/n)]n ϕ(x) with F o(t)ϕ(x) ∶= 1G(x) (F (t)ϕo) (x)

= limn→∞∫G

. . .∫G

∫G

ϕ(xn)µxn−1

t/n(dxn)µxn−2

t/n(dxn−1)⋯µxt/n(dx1)

Example: non-deg. diffusion:

F o(t)ϕ(x) = e−tC(x)

√(4πt)d detA(x)

∫G

ϕ(q)e−(x−q−tB(x))⋅A−1(x)(x−q−tB(x))

4t dq

Yana A. Butko Chernoff approximation SOTA 2018, 04.10.2018 44 / 56

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Approximation of non-Markovian evolution

Yana A. Butko Chernoff approximation SOTA 2018, 04.10.2018 45 / 56

Page 46: Chernoff approximation of evolution semigroups and beyond · Cherno approximation of evolution semigroups and beyond Yana A. Butko SOTA 2018, 04.10.2018 Yana A. Butko Cherno approximation

Approximation of non-Markovian evolution

Some physical, chemical, biological phenomena can be modelled viaevolution equations of the form

∂βt f = Lf , β ∈ (0,1),

where ∂βt is the Caputo derivative of order β:

∂βt u(t) ∶=1

Γ(1 − β)d

dt ∫t

0

u(r)(t − r)β

dr − t−β

Γ(1 − β)u(0+),

and (L,Dom(L)) generates a unif. bdd C0-SG (Tt)t⩾0 (on a BS X ).

Yana A. Butko Chernoff approximation SOTA 2018, 04.10.2018 46 / 56

Page 47: Chernoff approximation of evolution semigroups and beyond · Cherno approximation of evolution semigroups and beyond Yana A. Butko SOTA 2018, 04.10.2018 Yana A. Butko Cherno approximation

Approximation of non-Markovian evolution

Some physical, chemical, biological phenomena can be modelled viaevolution equations of the form

∂βt f = Lf , β ∈ (0,1),

where ∂βt is the Caputo derivative of order β:

∂βt u(t) ∶=1

Γ(1 − β)d

dt ∫t

0

u(r)(t − r)β

dr − t−β

Γ(1 − β)u(0+),

and (L,Dom(L)) generates a unif. bdd C0-SG (Tt)t⩾0 (on a BS X ).

More general: We consider equations of the form:

Dµt f (t) = Lf (t),

where Dµt is the Caputo derivative of distributed order given by a finiteBorel measure µ on (0,1):

Dµt u(t) ∶= ∫1

0∂βt u(t)µ(dβ).

Yana A. Butko Chernoff approximation SOTA 2018, 04.10.2018 47 / 56

Page 48: Chernoff approximation of evolution semigroups and beyond · Cherno approximation of evolution semigroups and beyond Yana A. Butko SOTA 2018, 04.10.2018 Yana A. Butko Cherno approximation

Approximation of non-Markovian evolution

Consider ”inverse subordinator” (Eµt )t⩾0 (inverse to (ξµt )t⩾0):

Eµt ∶= inf {τ ⩾ 0 ∶ ξµτ > t} ,

where ξµt is a mixture of β−stable subordinators given by the measure µ.

Yana A. Butko Chernoff approximation SOTA 2018, 04.10.2018 48 / 56

Page 49: Chernoff approximation of evolution semigroups and beyond · Cherno approximation of evolution semigroups and beyond Yana A. Butko SOTA 2018, 04.10.2018 Yana A. Butko Cherno approximation

Approximation of non-Markovian evolution

Consider ”inverse subordinator” (Eµt )t⩾0 (inverse to (ξµt )t⩾0):

Eµt ∶= inf {τ ⩾ 0 ∶ ξµτ > t} ,

where ξµt is a mixture of β−stable subordinators given by the measure µ.

Then Eµt

has a.s. non-decreasing paths;

is non-Markovian!!!;

has a smooth PDF pµ(t, x).

Yana A. Butko Chernoff approximation SOTA 2018, 04.10.2018 49 / 56

Page 50: Chernoff approximation of evolution semigroups and beyond · Cherno approximation of evolution semigroups and beyond Yana A. Butko SOTA 2018, 04.10.2018 Yana A. Butko Cherno approximation

Approximation of non-Markovian evolution

We consider equations of the form:

Dµt f (t) = Lf (t),

where (L,Dom(L)) generates a unif. bdd C0-SG (Tt)t⩾0 (on a BS X ).

Yana A. Butko Chernoff approximation SOTA 2018, 04.10.2018 50 / 56

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Approximation of non-Markovian evolution

We consider equations of the form:

Dµt f (t) = Lf (t),

where (L,Dom(L)) generates a unif. bdd C0-SG (Tt)t⩾0 (on a BS X ).

Theorem (Hahn, Kobayashi, Umarov, Mijena, Nane, 2012-2014):Consider the family (Tt)t⩾0 of linear operators on X such that

Ttϕ ∶= ∫∞

0Tsϕpµ(t, s)ds, ϕ ∈ X .

Then (Tt)t⩾0 is a strongly continuous family (not a SG!!!) and, for∀ f0 ∈ Dom(L), the function

f (t) ∶= Tt f0

solves the Cauchy problem

Dµt f (t) = Lf (t), f (0) = f0.

Yana A. Butko Chernoff approximation SOTA 2018, 04.10.2018 51 / 56

Page 52: Chernoff approximation of evolution semigroups and beyond · Cherno approximation of evolution semigroups and beyond Yana A. Butko SOTA 2018, 04.10.2018 Yana A. Butko Cherno approximation

Approximation of non-Markovian evolution

We consider equations of the form:

Dµt f (t) = Lf (t),

where (L,Dom(L)) generates a unif. bdd C0-SG (Tt)t⩾0 (on a BS X ).

Theorem (Hahn, Kobayashi, Umarov, Mijena, Nane, 2012-2014):Consider the family (Tt)t⩾0 of linear operators on X such that

Ttϕ ∶= ∫∞

0Tsϕpµ(t, s)ds, ϕ ∈ X .

Then (Tt)t⩾0 is a strongly continuous family (not a SG!!!) and, for∀ f0 ∈ Dom(L), the function

f (t) ∶= Tt f0 = E [f0(ξEµt) ∣ ξEµ

0= x] , ξt ↭ L

solves the Cauchy problem

Dµt f (t) = Lf (t), f (0) = f0.

Yana A. Butko Chernoff approximation SOTA 2018, 04.10.2018 52 / 56

Page 53: Chernoff approximation of evolution semigroups and beyond · Cherno approximation of evolution semigroups and beyond Yana A. Butko SOTA 2018, 04.10.2018 Yana A. Butko Cherno approximation

Approximation of non-Markovian evolution

Theorem (Hahn, Kobayashi, Umarov, Mijena, Nane, 2012-2014):Consider the family (Tt)t⩾0 of linear operators on X such that

Ttϕ ∶= ∫∞

0Tsϕpµ(t, s)ds, ϕ ∈ X .

Then (Tt)t⩾0 is a strongly continuous family (not a SG!!!) and, for∀ f0 ∈ Dom(L), the function f (t) ∶= Tt f0 solves the Cauchy problem

Dµt f (t) = Lf (t), f (0) = f0.

Yana A. Butko Chernoff approximation SOTA 2018, 04.10.2018 53 / 56

Page 54: Chernoff approximation of evolution semigroups and beyond · Cherno approximation of evolution semigroups and beyond Yana A. Butko SOTA 2018, 04.10.2018 Yana A. Butko Cherno approximation

Approximation of non-Markovian evolution

Theorem (Hahn, Kobayashi, Umarov, Mijena, Nane, 2012-2014):Consider the family (Tt)t⩾0 of linear operators on X such that

Ttϕ ∶= ∫∞

0Tsϕpµ(t, s)ds, ϕ ∈ X .

Then (Tt)t⩾0 is a strongly continuous family (not a SG!!!) and, for∀ f0 ∈ Dom(L), the function f (t) ∶= Tt f0 solves the Cauchy problem

Dµt f (t) = Lf (t), f (0) = f0.

Theorem: Let F (t) ∼ Tt . Consider fn ∶ [0,∞)→ X such that

fn(t) ∶= ∫∞

0F n(t/n)f0 pµ(t, s)ds.

Then, for all ∀ f0 ∈ Dom(L),

∥fn(t) − f (t)∥X → 0, n →∞,

locally uniformly w.r.t. t ⩾ 0.Yana A. Butko Chernoff approximation SOTA 2018, 04.10.2018 54 / 56

Page 55: Chernoff approximation of evolution semigroups and beyond · Cherno approximation of evolution semigroups and beyond Yana A. Butko SOTA 2018, 04.10.2018 Yana A. Butko Cherno approximation

Approximation of non-Markovian evolution

Example: time-space-fractional diffusion with β = 12 , α = 1

2 , x ∈ Rd :

∂1/2t f (t, x) = a(x) (−(−∆)1/2) f (t, x), f (0, x) = f0(x).

Then

p1/2(t, τ) = 1√πt

e−τ2

4t ,

and F (t) ∼ Tt ≡ et[a(−(−∆)1/2)] where

F (t)ϕ(x) ∶= Γ(d + 1

2)∫Rd

ϕ(q) a(x)t

(π∣x − q∣2 + a2(x)t2)d+1

2

dq.

Hence for any x0 ∈ Rd ∶

f (t, x0) = limn→∞

Γn (d + 1

2)×

×∞

∫0

∫Rnd

⎡⎢⎢⎢⎢⎣

n

∏k=1

a(xk−1)τ/n

(π∣xk − xk−1∣2 + (a(xk−1)τ/n)2)d+1

2

⎤⎥⎥⎥⎥⎦

f0(xn)√πt

e−τ2

4t dx1⋯dxndτ.

Yana A. Butko Chernoff approximation SOTA 2018, 04.10.2018 55 / 56

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References

Yana A. Butko (2018)

Chernoff approximation of subordinate semigroups

Stoch. Dyn., 18 (3), 1850021, 19 pp.

Yana A. Butko (2017)

Chernoff approximation for semigroups generated by killed Feller processes andFeynman formulae for time-fractional Fokker-Planck-Kolmogorov equations

ArXiv-preprint. https://arxiv.org/pdf/1708.02503.pdf

Yana A. Butko , Martin Grothaus, Oleg G. Smolyanov (2016)

Feynman formulae and phase space Feynman path integrals for tau-quantization ofsome Lvy-Khintchine type Hamilton functions,

J. Math. Phys., 57, 023508, 23 pp.

Yana A. Butko, Rene L. Schilling, Oleg G. Smolyanov (2012)

Lagrangian and Hamiltonian Feynman formulae for some Feller semigroups andtheir perturbations

Inf. Dim. Anal. Quant. Probab. Rel. Top. 15 (3), 26 pp.

Yana A. Butko Chernoff approximation SOTA 2018, 04.10.2018 56 / 56