Post on 19-Jul-2020
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
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.
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
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.
Assume F0(t)∼ etL0 is constructed. Then
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
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.
Assume F0(t)∼ etL0 is constructed. Then
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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Idea 2. Case (a): (η0t )t≥0 is known explicitly (IG, Gamma, e.t.c.).
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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Idea 2. Case (a): (η0t )t≥0 is known explicitly (IG, Gamma, e.t.c.).
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.
Idea 2. Case (b): (η0t )t≥0 is unknown, but µ is known and bounded.
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Idea 2. Case (a): (η0t )t≥0 is known explicitly (IG, Gamma, e.t.c.).
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.
Idea 2. Case (b): (η0t )t≥0 is unknown, but µ is known and bounded.
Then L0 : L0ϕ=∫∞0+(Tsϕ−ϕ)µ(ds) is bounded =⇒ Id+tL0 ∼ etL0.
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Idea 2. Case (a): (η0t )t≥0 is known explicitly (IG, Gamma, e.t.c.).
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.
Idea 2. Case (b): (η0t )t≥0 is unknown, but µ is known and bounded.
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
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