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

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 1/19

Motivation Consider the following Cauchy problem ∂ t u ( x , t ) = L u ( x , t ) ∀ x ∈ Ω , t > 0 ; u ( x , 0 ) = f ( x ) ∀ x ∈ Ω; u ( x , t ) = 0 ∀ x / ∈ Ω , t ≥ 0 , where Ω ⊆ R d is a bounded domain and L is some (pseudo-)differential operator acting on u in x . 2/19

Motivation Consider the following Cauchy problem ∂ t u ( x , t ) = L u ( x , t ) ∀ x ∈ Ω , t > 0 ; u ( x , 0 ) = f ( x ) ∀ x ∈ Ω; u ( x , t ) = 0 ∀ x / ∈ Ω , t ≥ 0 , where Ω ⊆ R d is a bounded domain and L is some (pseudo-)differential operator acting on u in x . Idea: Find a stronlgy continuous semigroup { P Ω t , t ≥ 0 } whose infinitesimal generator equals L and take u = P Ω t f . 2/19

Motivation If { P Ω t , t ≥ 0 } is uniformly bounded, this approach would also provide a solution to the fractional Cauchy problem ∂ β t v ( x , t ) = L v ( 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. 3/19

Motivation If { P Ω t , t ≥ 0 } is uniformly bounded, this approach would also provide a solution to the fractional Cauchy problem ∂ β t v ( x , t ) = L v ( 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, � ∞ g β ( s ) P Ω v ( x , t ) = ( t / s ) β f ( x ) ds , 0 where g β denotes the probability density function of the standard stable subordinator, with Laplace transform � ∞ e − st g β ( t ) dt = e − s β (B.Baeumer, M.Meerschaert, 2001). 0 3/19

Generator of a Feller process Definition ( X t ) t ≥ 0 is a Feller process on R d , when P t f ( x ) := E x f ( X t ) defines a strongly continuous contraction semigroup on C 0 ( R d ) . 4/19

Generator of a Feller process Definition ( X t ) t ≥ 0 is a Feller process on R d , when P t f ( x ) := E x f ( X t ) defines a strongly continuous contraction semigroup on C 0 ( R d ) . L – the infinitesimal generator of { P t , t ≥ 0 } with domain D ( L ) 4/19

Generator of a Feller process Definition ( X t ) t ≥ 0 is a Feller process on R d , when P t f ( x ) := E x f ( X t ) defines a strongly continuous contraction semigroup on C 0 ( R d ) . L – the infinitesimal generator of { P t , t ≥ 0 } with domain D ( L ) P t f ( x ) − f ( x ) x ∈ R d – pointwise formula for L L ♯ f ( x ) := lim , t t → 0 4/19

Generator of a Feller process Definition ( X t ) t ≥ 0 is a Feller process on R d , when P t f ( x ) := E x f ( X t ) defines a strongly continuous contraction semigroup on C 0 ( R d ) . L – the infinitesimal generator of { P t , t ≥ 0 } with domain D ( L ) P t f ( x ) − f ( x ) x ∈ R d – pointwise formula for L L ♯ f ( x ) := lim , t t → 0 Fact: If f ∈ C 0 ( R d ) , L ♯ f ( x ) exists for all x ∈ R d and L ♯ f ∈ C 0 ( R d ) , then f ∈ D ( L ) . 4/19

Generator of a Feller process Courrège, von Waldenfels 65’: If C ∞ c ( R d ) ⊂ D ( L ) , then for f ∈ C ∞ c ( R d ) we have Lf ( x ) = − c ( x ) f ( x ) + l ( x ) · ∇ f ( x ) + ∇ ( Q ( x ) ∇ f ( x )) � + ( f ( x + y ) − f ( x ) − ∇ f ( x ) · y 1 B ( y )) N ( x , dy ) R d \{ 0 } =: PDO [ f ]( x ) 5/19

Generator of a Feller process Courrège, von Waldenfels 65’: If C ∞ c ( R d ) ⊂ D ( L ) , then for f ∈ C ∞ c ( R d ) we have Lf ( x ) = − c ( x ) f ( x ) + l ( x ) · ∇ f ( x ) + ∇ ( Q ( x ) ∇ f ( x )) � + ( f ( x + y ) − f ( x ) − ∇ f ( x ) · y 1 B ( y )) N ( x , dy ) R d \{ 0 } =: PDO [ f ]( x ) for some c ( x ) ≥ 0, l ( x ) ∈ R d , Q ( x ) ∈ R d × d symmetric and positive semidefinite, B the unit ball, and N ( x , · ) a positive measure satisfying � min ( | y | 2 , 1 ) N ( x , dy ) < ∞ . R d \{ 0 } 5/19

Killed Feller process Ω ⊆ R d – bounded domain C 0 (Ω) := { f : Ω → R : f continuous and f ( x ) → 0 as x → ∂ Ω } 6/19

Killed Feller process Ω ⊆ R d – bounded domain C 0 (Ω) := { f : Ω → R : f continuous and f ( x ) → 0 as x → ∂ Ω } τ Ω := inf { t > 0 : X t / ∈ Ω } 6/19

Killed Feller process Ω ⊆ R d – bounded domain C 0 (Ω) := { f : Ω → R : f continuous and f ( x ) → 0 as x → ∂ Ω } τ Ω := inf { t > 0 : X t / ∈ Ω } Killed Feller process on Ω : � X t , t < τ Ω X Ω t := ∂, t ≥ τ Ω 6/19

Killed Feller process Ω ⊆ R d – bounded domain C 0 (Ω) := { f : Ω → R : f continuous and f ( x ) → 0 as x → ∂ Ω } τ Ω := inf { t > 0 : X t / ∈ Ω } Killed Feller process on Ω : � X t , t < τ Ω X Ω t := ∂, t ≥ τ Ω Definition ⇒ P x ( τ Ω = 0 ) = 1 . x ∈ ∂ Ω is regular for Ω : ⇐ Ω is regular : ⇐ ⇒ all x ∈ ∂ Ω are regular. 6/19

Killed Feller process Definition ⇒ P t f ∈ C b ( R d ) for all t > 0 and { P t , t ≥ 0 } is strong Feller : ⇐ all f measurable bounded with compact support in R d . X t is doubly Feller : ⇐ ⇒ X t is Feller and { P t , t ≥ 0 } is doubly Feller. 7/19

Killed Feller process Definition ⇒ P t f ∈ C b ( R d ) for all t > 0 and { P t , t ≥ 0 } is strong Feller : ⇐ all f measurable bounded with compact support in R d . X t is doubly Feller : ⇐ ⇒ X t is Feller and { P t , t ≥ 0 } is doubly Feller. Chung 86’: X t doubly Feller, Ω regular = ⇒ P Ω t f ( x ) := E x f ( X Ω t ) , x ∈ Ω , t ≥ 0 is a Feller semigroup on C 0 (Ω) . 7/19

Killed Feller process Definition ⇒ P t f ∈ C b ( R d ) for all t > 0 and { P t , t ≥ 0 } is strong Feller : ⇐ all f measurable bounded with compact support in R d . X t is doubly Feller : ⇐ ⇒ X t is Feller and { P t , t ≥ 0 } is doubly Feller. Chung 86’: X t doubly Feller, Ω regular = ⇒ P Ω t f ( x ) := E x f ( X Ω t ) , x ∈ Ω , t ≥ 0 is a Feller semigroup on C 0 (Ω) . L Ω – the infinitesimal generator of { P Ω t , t ≥ 0 } with domain D ( L Ω ) 7/19

Main results Theorem (BB,TL,MM) Suppose X t is doubly Feller on R d and Ω ⊆ R d is regular. Then D ( L Ω ) = { f ∈ C 0 (Ω) : L ♯ f ∈ C 0 (Ω) } . 8/19

Main results Theorem (BB,TL,MM) Suppose X t is doubly Feller on R d and Ω ⊆ R d is regular. Then D ( L Ω ) = { f ∈ C 0 (Ω) : L ♯ f ∈ C 0 (Ω) } . Also, for all f ∈ D ( L Ω ) and x ∈ Ω we have L Ω f ( x ) = L ♯ f ( x ) and P t f − f → L ♯ f uniformly on compacta in Ω . t 8/19

Main results Theorem (BB,TL,MM) Suppose X t is doubly Feller on R d and Ω ⊆ R d is regular. Then D ( L Ω ) = { f ∈ C 0 (Ω) : L ♯ f ∈ C 0 (Ω) } . Also, for all f ∈ D ( L Ω ) and x ∈ Ω we have L Ω f ( x ) = L ♯ f ( x ) and P t f − f → L ♯ f uniformly on compacta in Ω . t Examples: If X t is a Brownian motion in R d and Ω ⊆ R d is regular, then c (Ω) ⊆ D ( L Ω ) and L Ω f = PDO [ f ] = 1 C 2 2 ∆ f for f ∈ C 2 c (Ω) . 8/19

Main results Theorem (BB,TL,MM) Suppose X t is doubly Feller on R d and Ω ⊆ R d is regular. Then D ( L Ω ) = { f ∈ C 0 (Ω) : L ♯ f ∈ C 0 (Ω) } . Also, for all f ∈ D ( L Ω ) and x ∈ Ω we have L Ω f ( x ) = L ♯ f ( x ) and P t f − f → L ♯ f uniformly on compacta in Ω . t Examples: If X t is a Brownian motion in R d and Ω ⊆ R d is regular, then c (Ω) ⊆ D ( L Ω ) and L Ω f = PDO [ f ] = 1 C 2 2 ∆ f for f ∈ C 2 c (Ω) . If X t is a rotationally invariant α -stable Lévy process in R d with 0 < α < 2, then Lf = PDO [ f ] = − ( − ∆) α/ 2 f for f ∈ C 2 0 ( R d ) , but C ∞ c (Ω) �⊂ D ( L Ω ) . 8/19

Main results Remark: D ( L Ω ) typically contains functions whose zero extensions are not elements of D ( L ) . 9/19

Main results Remark: D ( L Ω ) typically contains functions whose zero extensions are not elements of D ( L ) . Theorem (BB,TL,MM) Suppose X t is doubly Feller on R d and Ω ⊆ R d is regular. Then � D ( L Ω ) = f ∈ C 0 (Ω) : ∃ g ∈ C 0 (Ω) , ( f n ) ⊆ D ( L ) such that � f n → f in C 0 ( R d ) and Lf n → g unif. on compacta in Ω , and for f , g as above we have L Ω f = g. 9/19

Main results C 2 0 (Ω) := C 0 (Ω) ∩ C 2 (Ω) Theorem (BB,TL,MM) Suppose X t is doubly Feller on R d , Ω ⊆ R d is regular and C ∞ c ( R d ) ⊂ D ( L ) . Then: For every f ∈ D ( L Ω ) there exists ( f n ) ⊆ C 2 0 (Ω) such that f n → f uniformly in C 0 (Ω) and PDO [ f n ] → L Ω f uniformly on compacta in Ω . 10/19

Main results C 2 0 (Ω) := C 0 (Ω) ∩ C 2 (Ω) Theorem (BB,TL,MM) Suppose X t is doubly Feller on R d , Ω ⊆ R d is regular and C ∞ c ( R d ) ⊂ D ( L ) . Then: For every f ∈ D ( L Ω ) there exists ( f n ) ⊆ C 2 0 (Ω) such that f n → f uniformly in C 0 (Ω) and PDO [ f n ] → L Ω f uniformly on compacta in Ω . If f n ⊆ C 2 0 (Ω) is such that f n → f ∈ C 0 (Ω) uniformly and PDO [ f n ] → g ∈ C 0 (Ω) uniformly on compacta in Ω , then f ∈ D ( L Ω ) and L Ω f = g. 10/19