Introduction Definitions Results Outlook Schauder estimates for non-local operators Franziska K¨ uhn (Institut de Math´ ematiques de Toulouse) Probability and Analysis 2019 May 23, 2019 Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators
Introduction Definitions Results Outlook Outline 1 Introduction 2 Definitions 3 Schauder estimates for L´ evy operators 4 Outlook: Schauder estimates for L´ evy-type operators Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators
Introduction Definitions Results Outlook Introduction Aim: Study pointwise regularity of solutions to the equation Af = g where A is an integro-differential operator ( → infinitesimal generator of a L´ evy(-type) process). Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators
Introduction Definitions Results Outlook Introduction Aim: Study pointwise regularity of solutions to the equation Af = g where A is an integro-differential operator ( → infinitesimal generator of a L´ evy(-type) process). Questions: How regular is f ∈ D ( A ) ? If g has a certain regularity then what additional information do we get on the regularity of f ? Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators
Introduction Definitions Results Outlook Introduction Aim: Study pointwise regularity of solutions to the equation Af = g where A is an integro-differential operator ( → infinitesimal generator of a L´ evy(-type) process). Questions: How regular is f ∈ D ( A ) ? If g has a certain regularity then what additional information do we get on the regularity of f ? Formally, f = A − 1 g i.e. we are interested in the smoothing properties of A − 1 . Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators
Introduction Definitions Results Outlook Example I Laplacian: Af ( x ) = 1 2∆ f ( x ) . . . appears as infinitesimal generator of Brownian motion. Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators
Introduction Definitions Results Outlook Example I Laplacian: Af ( x ) = 1 2∆ f ( x ) . . . appears as infinitesimal generator of Brownian motion. Known: If f ∈ D ( A ) then f is “almost twice differentiable”: ∥ f ∥ C 2 b ( R d ) ≤ M (∥ f ∥ ∞ + ∥ Af ∥ ∞ ) . Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators
Introduction Definitions Results Outlook Example I Laplacian: Af ( x ) = 1 2∆ f ( x ) . . . appears as infinitesimal generator of Brownian motion. Known: If f ∈ D ( A ) then f is “almost twice differentiable”: ∥ f ∥ C 2 b ( R d ) ≤ M (∥ f ∥ ∞ + ∥ Af ∥ ∞ ) . b ( R d ) ⊂ C k b ( R d ) for k ∈ N and C α b ( R d ) = C α b ( R d ) for α ∉ N C k Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators
Introduction Definitions Results Outlook Example I Laplacian: Af ( x ) = 1 2∆ f ( x ) . . . appears as infinitesimal generator of Brownian motion. Known: If f ∈ D ( A ) then f is “almost twice differentiable”: ∥ f ∥ C 2 b ( R d ) ≤ M (∥ f ∥ ∞ + ∥ Af ∥ ∞ ) . If Af = g ∈ C κ b ( R d ) for some κ > 0 then ∥ f ∥ C 2 + κ ( R d ) ≤ M κ (∥ f ∥ ∞ + ∥ Af ∥ C κ b ( R d ) ) . b b ( R d ) ⊂ C k b ( R d ) for k ∈ N and C α b ( R d ) = C α b ( R d ) for α ∉ N C k Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators
Introduction Definitions Results Outlook Example II Fractional Laplacian: Af ( x ) = −(− ∆ ) α / 2 f ( x ) ∶= c ∫ y ≠ 0 ( f ( x + y ) − f ( x ) − ∇ f ( x ) y 1 ( 0 , 1 ) (∣ y ∣)) 1 ∣ y ∣ d + α dy . . . appears as infinitesimal generator of isotropic α -stable L´ evy process. Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators
Introduction Definitions Results Outlook Example II Fractional Laplacian: Af ( x ) = −(− ∆ ) α / 2 f ( x ) ∶= c ∫ y ≠ 0 ( f ( x + y ) − f ( x ) − ∇ f ( x ) y 1 ( 0 , 1 ) (∣ y ∣)) 1 ∣ y ∣ d + α dy . . . appears as infinitesimal generator of isotropic α -stable L´ evy process. Known: If f ∈ D ( A ) then ∥ f ∥ C α b ( R d ) ≤ M (∥ f ∥ ∞ + ∥ Af ∥ ∞ ) . Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators
Introduction Definitions Results Outlook Example II Fractional Laplacian: Af ( x ) = −(− ∆ ) α / 2 f ( x ) ∶= c ∫ y ≠ 0 ( f ( x + y ) − f ( x ) − ∇ f ( x ) y 1 ( 0 , 1 ) (∣ y ∣)) 1 ∣ y ∣ d + α dy . . . appears as infinitesimal generator of isotropic α -stable L´ evy process. Known: If f ∈ D ( A ) then ∥ f ∥ C α b ( R d ) ≤ M (∥ f ∥ ∞ + ∥ Af ∥ ∞ ) . If Af = g ∈ C κ b ( R d ) for some κ > 0 then ∥ f ∥ C α + κ ( R d ) ≤ M κ (∥ f ∥ ∞ + ∥ Af ∥ C κ b ( R d ) ) . b Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators
Introduction Definitions Results Outlook Infinitesimal generator . . . appears in the study of Markov processes. Idea: P t f − f Af = d dt P t f ∣ = lim t t → 0 t = 0 where P t f ( x ) ∶= E x f ( X t ) = E 0 f ( x + X t ) . L´ evy is the semigroup. Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators
Introduction Definitions Results Outlook Infinitesimal generator . . . appears in the study of Markov processes. Idea: P t f − f Af = d dt P t f ∣ = lim t t → 0 t = 0 where P t f ( x ) ∶= E x f ( X t ) = E 0 f ( x + X t ) . L´ evy is the semigroup. Hence, E x f ( X t ) ≈ f ( x ) + tAf ( x ) for small t i.e. A describes small-time asymptotics. Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators
Introduction Definitions Results Outlook Generator of a L´ evy process Theorem Let ( X t ) t ≥ 0 be a L´ evy process. If f ∈ C ∞ c ( R d ) then Af ( x ) = b ⋅ ∇ f ( x ) + 1 2 tr ( Q ⋅ ∇ 2 f ( x )) + ∫ y ≠ 0 ( f ( x + y ) − f ( x ) − ∇ f ( x ) ⋅ y 1 ( 0 , 1 ) (∣ y ∣)) ν ( dy ) Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators
Introduction Definitions Results Outlook Generator of a L´ evy process Theorem Let ( X t ) t ≥ 0 be a L´ evy process. If f ∈ C ∞ c ( R d ) then Af ( x ) = b ⋅ ∇ f ( x ) + 1 2 tr ( Q ⋅ ∇ 2 f ( x )) + ∫ y ≠ 0 ( f ( x + y ) − f ( x ) − ∇ f ( x ) ⋅ y 1 ( 0 , 1 ) (∣ y ∣)) ν ( dy ) evy triplet ( b , Q ,ν ) consists of where the L´ b ∈ R d (drift vector), Q ∈ R d × d symmetric and positive semidefinite (diffusion matrix) a measure ν with ∫ y ≠ 0 min { 1 , ∣ y ∣ 2 } ν ( dy ) < ∞ (L´ evy measure) Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators
Introduction Definitions Results Outlook Generator of a L´ evy process Theorem Let ( X t ) t ≥ 0 be a L´ evy process. If f ∈ C ∞ c ( R d ) then Af ( x ) = b ⋅ ∇ f ( x ) + 1 2 tr ( Q ⋅ ∇ 2 f ( x )) + ∫ y ≠ 0 ( f ( x + y ) − f ( x ) − ∇ f ( x ) ⋅ y 1 ( 0 , 1 ) (∣ y ∣)) ν ( dy ) evy triplet ( b , Q ,ν ) consists of where the L´ b ∈ R d (drift vector), Q ∈ R d × d symmetric and positive semidefinite (diffusion matrix) a measure ν with ∫ y ≠ 0 min { 1 , ∣ y ∣ 2 } ν ( dy ) < ∞ (L´ evy measure) Equivalent characterization via the characteristic exponent ψ ( ξ ) = − ib ⋅ ξ + 1 2 ξ ⋅ Q ξ + ∫ y ≠ 0 ( 1 − e iy ⋅ ξ + iy ⋅ ξ 1 ( 0 , 1 ) (∣ y ∣)) ν ( dy ) Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators
Introduction Definitions Results Outlook H¨ older–Zygmund space ⎧ ⎫ ⎪ ⎪ ∣ ∆ k h f ( x )∣ ⎪ ⎪ b ( R d ) ∶= ⎨ f ∈ C b ( R d ) ; ∥ f ∥ C α b ( R d ) ∶= ∥ f ∥ ∞ + sup < ∞ ⎬ C α ⎪ ⎪ x ∈ R d sup ∣ h ∣ α ⎪ ⎪ ⎩ ⎭ 0 < ∣ h ∣ ≤ 1 where k = ⌊ α ⌋ + 1 and ∆ h ∶= f ( x + h ) − f ( x ) h f ( x ) ∶= ∆ h ( ∆ k − 1 f )( x ) . ∆ k h Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators
Introduction Definitions Results Outlook H¨ older–Zygmund space ⎧ ⎫ ⎪ ⎪ ∣ ∆ k h f ( x )∣ ⎪ ⎪ b ( R d ) ∶= ⎨ f ∈ C b ( R d ) ; ∥ f ∥ C α b ( R d ) ∶= ∥ f ∥ ∞ + sup < ∞ ⎬ C α ⎪ ⎪ x ∈ R d sup ∣ h ∣ α ⎪ ⎪ ⎩ ⎭ 0 < ∣ h ∣ ≤ 1 where k = ⌊ α ⌋ + 1 and ∆ h ∶= f ( x + h ) − f ( x ) h f ( x ) ∶= ∆ h ( ∆ k − 1 f )( x ) . ∆ k h Theorem b ( R d ) ⊂ C k b ( R d ) for all k ∈ N , 1 C k b ( R d ) = C α b ( R d ) for α ∈ ( 0 , ∞)/ N . 2 C α Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators
Introduction Definitions Results Outlook Schauder estimates for L´ evy generators How to obtain Schauder estimates for solutions to Af = g where A is the generator of a L´ evy process? Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators
Introduction Definitions Results Outlook Schauder estimates for L´ evy generators How to obtain Schauder estimates for solutions to Af = g where A is the generator of a L´ evy process? Known results: Bass ’09, Ros-Oton & Serra ’16: stable operators Bae & Kassmann ’15: ν ( dy ) = 1 /(∣ y ∣ d ϕ ( y )) classical theory for pseudo-differential operators: ∫ ∣ y ∣ > 1 ∣ y ∣ n ν ( dy ) < ∞ for n ≫ 1 Franziska K¨ uhn (IMT Toulouse) Schauder estimates for non-local operators
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