On the generator of a killed Feller process Tomasz Luks Paderborn - - PowerPoint PPT Presentation

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On the generator of a killed Feller process

Tomasz Luks

Paderborn University (based on a joint work with B. Baeumer and M. Meerschaert)

Probability and Analysis

Będlewo, May 15, 2017

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Motivation

Consider the following Cauchy problem ∂tu(x, t) = Lu(x, t) ∀ x ∈ Ω, t > 0; u(x, 0) = f (x) ∀ x ∈ Ω; u(x, t) = 0 ∀ x / ∈ Ω, t ≥ 0, where Ω ⊆ Rd is a bounded domain and L is some (pseudo-)differential operator acting on u in x.

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Motivation

Consider the following Cauchy problem ∂tu(x, t) = Lu(x, t) ∀ x ∈ Ω, t > 0; u(x, 0) = f (x) ∀ x ∈ Ω; u(x, t) = 0 ∀ x / ∈ Ω, t ≥ 0, where Ω ⊆ Rd is a bounded domain and L is some (pseudo-)differential operator acting on u in x. Idea: Find a stronlgy continuous semigroup {PΩ

t , t ≥ 0} whose

infinitesimal generator equals L and take u = PΩ

t f .

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Motivation

If {PΩ

t , t ≥ 0} is uniformly bounded, this approach would also

provide a solution to the fractional Cauchy problem ∂β

t v(x, t) = Lv(x, t)

∀ x ∈ Ω, t > 0; v(x, 0) = f (x) ∀ x ∈ Ω; v(x, t) = 0 ∀ x / ∈ Ω, t ≥ 0, where ∂β

t is the Caputo fractional derivative of order 0 < β < 1.

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Motivation

If {PΩ

t , t ≥ 0} is uniformly bounded, this approach would also

provide a solution to the fractional Cauchy problem ∂β

t v(x, t) = Lv(x, t)

∀ x ∈ Ω, t > 0; v(x, 0) = f (x) ∀ x ∈ Ω; v(x, t) = 0 ∀ x / ∈ Ω, t ≥ 0, where ∂β

t is the Caputo fractional derivative of order 0 < β < 1.

Namely, v(x, t) = ∞ gβ(s)PΩ

(t/s)βf (x)ds,

where gβ denotes the probability density function of the standard stable subordinator, with Laplace transform ∞ e−stgβ(t)dt = e−sβ (B.Baeumer, M.Meerschaert, 2001).

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Generator of a Feller process

Definition (Xt)t≥0 is a Feller process on Rd, when Ptf (x) := Exf (Xt) defines a strongly continuous contraction semigroup on C0(Rd).

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Generator of a Feller process

Definition (Xt)t≥0 is a Feller process on Rd, when Ptf (x) := Exf (Xt) defines a strongly continuous contraction semigroup on C0(Rd). L – the infinitesimal generator of {Pt, t ≥ 0} with domain D(L)

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Generator of a Feller process

Definition (Xt)t≥0 is a Feller process on Rd, when Ptf (x) := Exf (Xt) defines a strongly continuous contraction semigroup on C0(Rd). L – the infinitesimal generator of {Pt, t ≥ 0} with domain D(L) L♯f (x) := lim

t→0

Ptf (x) − f (x) t , x ∈ Rd – pointwise formula for L

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Generator of a Feller process

Definition (Xt)t≥0 is a Feller process on Rd, when Ptf (x) := Exf (Xt) defines a strongly continuous contraction semigroup on C0(Rd). L – the infinitesimal generator of {Pt, t ≥ 0} with domain D(L) L♯f (x) := lim

t→0

Ptf (x) − f (x) t , x ∈ Rd – pointwise formula for L Fact: If f ∈ C0(Rd), L♯f (x) exists for all x ∈ Rd and L♯f ∈ C0(Rd), then f ∈ D(L).

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Generator of a Feller process

Courrège, von Waldenfels 65’: If C ∞

c (Rd) ⊂ D(L), then for

f ∈ C ∞

c (Rd) we have

Lf (x) = − c(x)f (x) + l(x) · ∇f (x) + ∇(Q(x)∇f (x)) +

• Rd\{0}

(f (x + y) − f (x) − ∇f (x) · y1B(y))N(x, dy) =: PDO[f ](x)

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Generator of a Feller process

Courrège, von Waldenfels 65’: If C ∞

c (Rd) ⊂ D(L), then for

f ∈ C ∞

c (Rd) we have

Lf (x) = − c(x)f (x) + l(x) · ∇f (x) + ∇(Q(x)∇f (x)) +

• Rd\{0}

(f (x + y) − f (x) − ∇f (x) · y1B(y))N(x, dy) =: PDO[f ](x) for some c(x) ≥ 0, l(x) ∈ Rd, Q(x) ∈ Rd×d symmetric and positive semidefinite, B the unit ball, and N(x, ·) a positive measure satisfying

• Rd\{0}

min(|y|2, 1)N(x, dy) < ∞.

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Killed Feller process

Ω ⊆ Rd – bounded domain C0(Ω) := {f : Ω → R : f continuous and f (x) → 0 as x → ∂Ω}

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Killed Feller process

Ω ⊆ Rd – bounded domain C0(Ω) := {f : Ω → R : f continuous and f (x) → 0 as x → ∂Ω} τΩ := inf{t > 0 : Xt / ∈ Ω}

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Killed Feller process

Ω ⊆ Rd – bounded domain C0(Ω) := {f : Ω → R : f continuous and f (x) → 0 as x → ∂Ω} τΩ := inf{t > 0 : Xt / ∈ Ω} Killed Feller process on Ω: X Ω

t :=

• Xt,

t < τΩ ∂, t ≥ τΩ

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Killed Feller process

Ω ⊆ Rd – bounded domain C0(Ω) := {f : Ω → R : f continuous and f (x) → 0 as x → ∂Ω} τΩ := inf{t > 0 : Xt / ∈ Ω} Killed Feller process on Ω: X Ω

t :=

• Xt,

t < τΩ ∂, t ≥ τΩ Definition x ∈ ∂Ω is regular for Ω : ⇐ ⇒ Px(τΩ = 0) = 1. Ω is regular : ⇐ ⇒ all x ∈ ∂Ω are regular.

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Killed Feller process

Definition {Pt, t ≥ 0} is strong Feller : ⇐ ⇒ Ptf ∈ Cb(Rd) for all t > 0 and all f measurable bounded with compact support in Rd. Xt is doubly Feller : ⇐ ⇒ Xt is Feller and {Pt, t ≥ 0} is doubly Feller.

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Killed Feller process

Definition {Pt, t ≥ 0} is strong Feller : ⇐ ⇒ Ptf ∈ Cb(Rd) for all t > 0 and all f measurable bounded with compact support in Rd. Xt is doubly Feller : ⇐ ⇒ Xt is Feller and {Pt, t ≥ 0} is doubly Feller. Chung 86’: Xt doubly Feller, Ω regular = ⇒ PΩ

t f (x) := Exf (X Ω t ),

x ∈ Ω, t ≥ 0 is a Feller semigroup on C0(Ω).

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Killed Feller process

Definition {Pt, t ≥ 0} is strong Feller : ⇐ ⇒ Ptf ∈ Cb(Rd) for all t > 0 and all f measurable bounded with compact support in Rd. Xt is doubly Feller : ⇐ ⇒ Xt is Feller and {Pt, t ≥ 0} is doubly Feller. Chung 86’: Xt doubly Feller, Ω regular = ⇒ PΩ

t f (x) := Exf (X Ω t ),

x ∈ Ω, t ≥ 0 is a Feller semigroup on C0(Ω). LΩ – the infinitesimal generator of {PΩ

t , t ≥ 0} with domain D(LΩ)

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Main results

Theorem (BB,TL,MM) Suppose Xt is doubly Feller on Rd and Ω ⊆ Rd is regular. Then D(LΩ) = {f ∈ C0(Ω) : L♯f ∈ C0(Ω)}.

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Main results

Theorem (BB,TL,MM) Suppose Xt is doubly Feller on Rd and Ω ⊆ Rd is regular. Then D(LΩ) = {f ∈ C0(Ω) : L♯f ∈ C0(Ω)}. Also, for all f ∈ D(LΩ) and x ∈ Ω we have LΩf (x) = L♯f (x) and Ptf − f t → L♯f uniformly on compacta in Ω.

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Main results

Theorem (BB,TL,MM) Suppose Xt is doubly Feller on Rd and Ω ⊆ Rd is regular. Then D(LΩ) = {f ∈ C0(Ω) : L♯f ∈ C0(Ω)}. Also, for all f ∈ D(LΩ) and x ∈ Ω we have LΩf (x) = L♯f (x) and Ptf − f t → L♯f uniformly on compacta in Ω. Examples: If Xt is a Brownian motion in Rd and Ω ⊆ Rd is regular, then C 2

c (Ω) ⊆ D(LΩ) and LΩf = PDO[f ] = 1 2∆f for f ∈ C 2 c (Ω).

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Main results

Theorem (BB,TL,MM) Suppose Xt is doubly Feller on Rd and Ω ⊆ Rd is regular. Then D(LΩ) = {f ∈ C0(Ω) : L♯f ∈ C0(Ω)}. Also, for all f ∈ D(LΩ) and x ∈ Ω we have LΩf (x) = L♯f (x) and Ptf − f t → L♯f uniformly on compacta in Ω. Examples: If Xt is a Brownian motion in Rd and Ω ⊆ Rd is regular, then C 2

c (Ω) ⊆ D(LΩ) and LΩf = PDO[f ] = 1 2∆f for f ∈ C 2 c (Ω).

If Xt is a rotationally invariant α-stable Lévy process in Rd with 0 < α < 2, then Lf = PDO[f ] = −(−∆)α/2f for f ∈ C 2

0 (Rd), but C ∞ c (Ω) ⊂ D(LΩ).

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Main results

Remark: D(LΩ) typically contains functions whose zero extensions are not elements of D(L).

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Main results

Remark: D(LΩ) typically contains functions whose zero extensions are not elements of D(L). Theorem (BB,TL,MM) Suppose Xt is doubly Feller on Rd and Ω ⊆ Rd is regular. Then D(LΩ) =

• f ∈ C0(Ω) : ∃g ∈ C0(Ω), (fn) ⊆ D(L) such that

fn → f in C0(Rd) and Lfn → g unif. on compacta in Ω

• ,

and for f , g as above we have LΩf = g.

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Main results

C 2

0 (Ω) := C0(Ω) ∩ C 2(Ω)

Theorem (BB,TL,MM) Suppose Xt is doubly Feller on Rd, Ω ⊆ Rd is regular and C ∞

c (Rd) ⊂ D(L). Then:

For every f ∈ D(LΩ) there exists (fn) ⊆ C 2

0 (Ω) such that

fn → f uniformly in C0(Ω) and PDO[fn] → LΩf uniformly on compacta in Ω.

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Main results

C 2

0 (Ω) := C0(Ω) ∩ C 2(Ω)

Theorem (BB,TL,MM) Suppose Xt is doubly Feller on Rd, Ω ⊆ Rd is regular and C ∞

c (Rd) ⊂ D(L). Then:

For every f ∈ D(LΩ) there exists (fn) ⊆ C 2

0 (Ω) such that

fn → f uniformly in C0(Ω) and PDO[fn] → LΩf uniformly on compacta in Ω. If fn ⊆ C 2

0 (Ω) is such that fn → f ∈ C0(Ω) uniformly and

PDO[fn] → g ∈ C0(Ω) uniformly on compacta in Ω, then f ∈ D(LΩ) and LΩf = g.

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Main results

C 2

0 (Ω) := C0(Ω) ∩ C 2(Ω)

Theorem (BB,TL,MM) Suppose Xt is doubly Feller on Rd, Ω ⊆ Rd is regular and C ∞

c (Rd) ⊂ D(L). Then:

For every f ∈ D(LΩ) there exists (fn) ⊆ C 2

0 (Ω) such that

fn → f uniformly in C0(Ω) and PDO[fn] → LΩf uniformly on compacta in Ω. If fn ⊆ C 2

0 (Ω) is such that fn → f ∈ C0(Ω) uniformly and

PDO[fn] → g ∈ C0(Ω) uniformly on compacta in Ω, then f ∈ D(LΩ) and LΩf = g. In particular, if f ∈ C 2

0 (Ω) and PDO[f ] ∈ C0(Ω), then f ∈ D(LΩ)

and LΩf (x) = PDO[f ](x) for every x ∈ Ω.

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Proofs

For f ∈ C0(Ω) we consider Ptf (x) − f (x) t − PΩ

t f (x) − f (x)

t =Ex[f (Xt)1{τΩ<t}] t .

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Proofs

For f ∈ C0(Ω) we consider Ptf (x) − f (x) t − PΩ

t f (x) − f (x)

t =Ex[f (Xt)1{τΩ<t}] t . Strong continuity and strong Markov property of Pt = ⇒ |Ex[f (Xt)1{τΩ<t}]| t ≤ εPx(τΩ < t) t for sufficiently small t > 0 and all x ∈ Ω.

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Proofs

For f ∈ C0(Ω) we consider Ptf (x) − f (x) t − PΩ

t f (x) − f (x)

t =Ex[f (Xt)1{τΩ<t}] t . Strong continuity and strong Markov property of Pt = ⇒ |Ex[f (Xt)1{τΩ<t}]| t ≤ εPx(τΩ < t) t for sufficiently small t > 0 and all x ∈ Ω. Let U ⊂⊂ Ω and choose r > 0 such that B(x, r) ⊂ Ω ∀x ∈ U. R.Schilling, J.Wang, 2012: ∃M > 0: Px(τB(x,r) < t) t ≤ M ∀x ∈ U, t > 0.

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Proofs

Suppose f ∈ C0(Ω) and that Ptf (x) − f (x) t → g(x) as t → 0 for some g ∈ C0(Ω), for all x ∈ Ω.

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Proofs

Suppose f ∈ C0(Ω) and that Ptf (x) − f (x) t → g(x) as t → 0 for some g ∈ C0(Ω), for all x ∈ Ω. The resolvent (λ − LΩ)−1 exists for all λ > 0, and maps C0(Ω)

• nto D(LΩ) =

⇒ ∃h ∈ D(LΩ) such that (I − LΩ)h = f − g.

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Proofs

Suppose f ∈ C0(Ω) and that Ptf (x) − f (x) t → g(x) as t → 0 for some g ∈ C0(Ω), for all x ∈ Ω. The resolvent (λ − LΩ)−1 exists for all λ > 0, and maps C0(Ω)

• nto D(LΩ) =

⇒ ∃h ∈ D(LΩ) such that (I − LΩ)h = f − g. By previous calculation applied to h, LΩh(x) − g(x) = lim

t→0

Pth(x) − h(x) − (Ptf (x) − f (x)) t , x ∈ Ω.

13/19

Proofs

Hence, for u = h − f we get u(x) = lim

t→0

Ptu(x) − u(x) t , x ∈ Ω.

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Proofs

Hence, for u = h − f we get u(x) = lim

t→0

Ptu(x) − u(x) t , x ∈ Ω. Without loss of generality let x0 ∈ Ω be such that u = sup

x∈Ω

|u(x)| = u(x0) > 0.

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Proofs

Hence, for u = h − f we get u(x) = lim

t→0

Ptu(x) − u(x) t , x ∈ Ω. Without loss of generality let x0 ∈ Ω be such that u = sup

x∈Ω

|u(x)| = u(x0) > 0. Since Pt is a contraction, Ptu(x0) ≤ Ptu ≤ u = u(x0) and therefore 0 ≥ Ptu(x0) − u(x0) t → u(x0) > 0 as t → 0, which is a contradiction. Hence supx∈Ω |u(x)| = 0 and therefore h = f .

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Proofs

Next we show that f ∈ D(LΩ) can be approximated locally in the graph norm by functions in D(L), namely by the functions fλ = (λ − L)−1λf .

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Proofs

Next we show that f ∈ D(LΩ) can be approximated locally in the graph norm by functions in D(L), namely by the functions fλ = (λ − L)−1λf . As Ptf is continuous in t and Ptf ≤ f , it is not hard to check that fλ = λ ∞

0 e−λtPtf dt and

lim

λ→∞ fλ = P0f = f

in C0(Rd).

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Proofs

Next we show that f ∈ D(LΩ) can be approximated locally in the graph norm by functions in D(L), namely by the functions fλ = (λ − L)−1λf . As Ptf is continuous in t and Ptf ≤ f , it is not hard to check that fλ = λ ∞

0 e−λtPtf dt and

lim

λ→∞ fλ = P0f = f

in C0(Rd). Furthermore, fλ ∈ D(L) and by definition, Lfλ = λfλ − λf .

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Proofs

Previous theorem implies that Ptf (x) − f (x) t → LΩf (x) uniformly in x ∈ U ⊂⊂ Ω,

15/19

Proofs

Previous theorem implies that Ptf (x) − f (x) t → LΩf (x) uniformly in x ∈ U ⊂⊂ Ω, and we get lim

λ→∞ Lfλ(x) = lim λ→∞ λ2

∞ e−λtPtf (x) dt − λf (x) = lim

λ→∞

∞ ue−u P(u/λ)f (x) − f (x) (u/λ) du =LΩf (x) uniformly in x ∈ U.

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Proofs

Reverse set inclusion: suppose f ∈ C0(Ω) and for some (fn) ⊆ D(L) we have fn → f in C0(Rd) and Lfn(x) → g(x) uniformly in x ∈ U ⊂⊂ Ω for some g ∈ C0(Ω).

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Proofs

Reverse set inclusion: suppose f ∈ C0(Ω) and for some (fn) ⊆ D(L) we have fn → f in C0(Rd) and Lfn(x) → g(x) uniformly in x ∈ U ⊂⊂ Ω for some g ∈ C0(Ω). (Uniqueness of g: tricky, apply PMP in an appropriate way)

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Proofs

Reverse set inclusion: suppose f ∈ C0(Ω) and for some (fn) ⊆ D(L) we have fn → f in C0(Rd) and Lfn(x) → g(x) uniformly in x ∈ U ⊂⊂ Ω for some g ∈ C0(Ω). (Uniqueness of g: tricky, apply PMP in an appropriate way) Let h = (I − LΩ)−1(f − g) so that h − f = LΩh − g.

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Proofs

Reverse set inclusion: suppose f ∈ C0(Ω) and for some (fn) ⊆ D(L) we have fn → f in C0(Rd) and Lfn(x) → g(x) uniformly in x ∈ U ⊂⊂ Ω for some g ∈ C0(Ω). (Uniqueness of g: tricky, apply PMP in an appropriate way) Let h = (I − LΩ)−1(f − g) so that h − f = LΩh − g. The resolvent maps C0(Ω) onto D(LΩ) = ⇒ ∃hn ∈ D(L) : hn → h in C0(Rd) and Lhn(x) → LΩh(x) ∀x ∈ Ω, uniformly on compacta, by what we have already proven.

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Proofs

Let u = h − f and assume u(x0) = u > ǫ for some ǫ > 0. Let un = hn − fn so that Lun(x) → LΩh(x) − g(x) = u(x) uniformly in x ∈ U ⊂⊂ Ω.

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Proofs

Let u = h − f and assume u(x0) = u > ǫ for some ǫ > 0. Let un = hn − fn so that Lun(x) → LΩh(x) − g(x) = u(x) uniformly in x ∈ U ⊂⊂ Ω. un converges uniformly to u = ⇒ ∃N > 0, U ⊂⊂ Ω such that {xn : un(xn) = un} ⊂ U ∀n > N.

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Proofs

Let u = h − f and assume u(x0) = u > ǫ for some ǫ > 0. Let un = hn − fn so that Lun(x) → LΩh(x) − g(x) = u(x) uniformly in x ∈ U ⊂⊂ Ω. un converges uniformly to u = ⇒ ∃N > 0, U ⊂⊂ Ω such that {xn : un(xn) = un} ⊂ U ∀n > N. un(xn) > ǫ/2 for large n and Lun(xn) ≤ 0 by PMP = ⇒ un(x) − Lun(x) cannot converge uniformly on U to 0. This is a contradiction, and hence u ≡ 0 = ⇒ h = f ∈ D(LΩ).

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Example: Application to stable processes

The positive and negative Riemann-Liouville fractional derivatives are defined by Dα

[L,x]f (x) =

1 Γ(n − α) dn dxn x

L

f (y)(x − y)n−α−1dy, Dα

[x,R]f (x) =

(−1)n Γ(n − α) dn dxn R

x

f (y)(y − x)n−α−1dy for any non-integer α > 0 and any −∞ ≤ L < x < R ≤ ∞, where n − 1 < α < n. Theorem (BB,TL,MM) Let Xt be any stable Lévy process with index 1 < α < 2 and let Ω = (L, R). Then for all x ∈ Ω and any f ∈ C 2

0 (Ω) such that

PDO[f ] ∈ C0(Ω) we have LΩf (x) = −af ′(x) + b Dα

[L,x]f (x) + c Dα [x,R]f (x).

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THANK YOU FOR YOUR ATTENTION !