Report Title Stochastic Volterra Equations in Banach Spaces and - PowerPoint PPT Presentation

Report Title Stochastic Volterra Equations in Banach Spaces and Stochastic Volterra Equations in SPDEs Banach Spaces and SPDEs Contents ASVE Motivations Xicheng Zhang(HUST, Wuhan) Local Solutions of ASVE Workshop on Stochastic Analysis & Finance Strong Solutions of City University of Hong Kong Semilinear SPDEs Large June 29-July 3 Deviation Principle for ASVE Reference

Contents Stochastic Volterra Equations in 1 ASVE Banach Spaces and SPDEs 2 Motivations Contents ASVE 2 Local Maximal Solutions Motivations Local Solutions of ASVE 3 Strong Solutions of Semilinear SPDEs Strong Solutions of Semilinear 4 Large Deviations for ASVE SPDEs Large Deviation Principle for 5 Reference ASVE Reference

Abstract Stochastic Volterra Equations (ASVE) Stochastic Let X be a 2-smooth and real separable Banach space, i.e., Volterra Equations in there exists a constant C X � 2 such that for all x, y ∈ X Banach Spaces and SPDEs � x + y � 2 X + � x − y � 2 X � 2 � x � 2 X + C X � y � 2 X . Contents Example: Any Hilbert space and any L p -space over measure ASVE space ( E, E , µ ) with p � 2 are 2-smooth. In fact, it follows Motivations from the following two elementary inequalities: for α ∈ [0 , 1] Local and any a, b � 0, Solutions of ASVE 2 α − 1 ( a α + b α ) � ( a + b ) α � a α + b α Strong Solutions of Semilinear SPDEs and for p � 2 and any a, b ∈ R , Large Deviation Principle for 1 2( | a + b | p + | a − b | p ) � | a | p + (2 p − 1 − 1) | b | p . ASVE Reference In this case, C L p = 2(2 p − 1 − 1) 2 /p .

Abstract Stochastic Volterra Equations Let l 2 be the usual sequence Hilbert space and ( e k ) k ∈ N the Stochastic Volterra Equations in canonical basis of l 2 . Let W ( t ) := ( W k ( t )) k ∈ N be an l 2 - Banach Spaces and valued cylindrical Brownian motion on a complete filtered SPDEs probability space (Ω , F , P ; ( F t ) t � 0 ). Contents Definition (See Ondrej´ at(2004)) ASVE A bounded linear operator B : l 2 → X is called radonifying if Motivations Local Solutions of the series Be k · W k (1) converges in L 2 (Ω; X ) . ASVE Strong Solutions of We shall denote by L 2 ( l 2 ; X ) the space of all radonifying op- Semilinear SPDEs erators, and write for B ∈ L 2 ( l 2 ; X ) Large Deviation � 1 / 2 Principle for � � 2 � Be k · W k (1) � � � B � L 2 ( l 2 ; X ) := . ASVE E X Reference L 2 ( l 2 ; X ) together with �·� L 2 ( l 2 ; X ) is a separable Banach space.

Abstract Stochastic Volterra Equations Stochastic We shall consider the following stochastic Volterra integral Volterra Equations in equation in a 2-smooth Banach space X (Berger-Mizel(1980)): Banach Spaces and SPDEs � t � t X t = g ( t ) + 0 A ( t, s, X s )d s + 0 B ( t, s, X s )d W ( s ) , (1) Contents where g ( t ) is an X -valued measurable ( F t )-adapted process, ASVE and Motivations Local Solutions of A : △ × Ω × X → X ∈ M △ × B ( X ) / B ( X ) ASVE B : △ × Ω × X → L 2 ( l 2 ; X ) ∈ M △ × B ( X ) / B ( L 2 ( l 2 ; X )) . Strong Solutions of Semilinear SPDEs Here and below, △ := { ( t, s ) ∈ R 2 + : s � t } , and M △ denotes Large the progressively measurable σ -field on △ × Ω generated by Deviation Principle for the sets E ∈ B ( △ ) × F with properties: 1 E ( t, s, · ) ∈ F s for ASVE all ( t, s ) ∈ △ , and s �→ 1 E ( t, s, ω ) is right continuous for any Reference t ∈ R + and ω ∈ Ω.

Motivations: 1. SPDE driven by space-time white noise Stochastic Let W ( t, x ) be a Brownian sheet on R + × [0 , 1]. Consider the Volterra Equations in following SPDE (Walsh [1986]): Banach Spaces and SPDEs = ∂ 2 u t ( x ) + f ( x, u t ( x )) + σ ( x, u t ( x )) ∂ 2 W  ∂u t ( x ) ∂t∂x,  ∂x 2 ∂t Contents  u ( t, 0) = u ( t, 1) = 0 , u ( t, x ) | t =0 = u 0 ( x ) , ASVE Motivations where f ( x, r ) , σ ( x, r ) : [0 , 1] × R → R are two Borel measur- Local able functions. This equation is understood as Solutions of ASVE � 1 � t � 1 Strong u t ( x ) = G t ( x, y ) u 0 ( y )d y + G t − s ( x, y ) f ( y, u s ( y ))d y d s Solutions of Semilinear 0 0 0 SPDEs � t � 1 Large + G t − s ( x, y ) σ ( y, u s ( y )) W (d s, d y ) , Deviation Principle for 0 0 ASVE where the stochastic integral is the usual Itˆ o’s integral and Reference G t ( x, y ) is the fundamental solutions of homogeneous heat equation.

Motivations: 1. SPDE driven by space-time white noise Stochastic Let ( φ k ) k ∈ N be an orthogonal basis of L 2 (0 , 1). Define Volterra Equations in Banach � 1 Spaces and T t f ( x ) := G t ( x, y ) f ( y )d y, SPDEs 0 � 1 Contents W k ( t ) := φ k ( y ) W ( t, d y ) . ASVE 0 Motivations Then we can write Local Solutions of � t ASVE u t ( x ) = T t u 0 ( x ) + T t − s f ( · , u s ( · ))( x )d s Strong Solutions of 0 Semilinear � t SPDEs T t − s ( σ ( · , u s ( · )) φ k ( · ))( x ) W k (d s ) , + Large 0 Deviation Principle for ASVE which takes form (1). We can take X = L p (0 , 1), p � 2. Reference

Motivations: 2. Stochastic Reaction-Diffusion Equation Stochastic Consider the following stochastic reaction diffusion equation Volterra Equations in in R d : Banach Spaces and � SPDEs d u t ( x ) = [∆ u t ( x ) − u 3 t ( x )]d t + σ k ( x, u t ( x ))d W k ( t ) , (2) u (0 , x ) = u 0 ( x ) . Contents ASVE Define Motivations 1 � R d e − | x − y | 2 T t f ( x ) := f ( y )d y. Local 2 t Solutions of � (2 πt ) d ASVE Strong The mild solution of (2) is given by Solutions of Semilinear � t SPDEs T t − s u 3 u t ( x ) = T t u 0 ( x ) − s ( x )d s Large Deviation 0 Principle for � t ASVE T t − s ( σ k ( · , u s ( · )))( x ) W k (d s ) , + Reference 0 which takes form (1).

Motivations: 3. Stochastic Evolutionary Integral Equation Stochastic Consider the following stochastic evolutionary integral equa- Volterra Equations in tion ( α ∈ [0 , 1)): Banach Spaces and � t � t SPDEs ∆ u s ( x ) u t ( x ) = u 0 ( x ) + ( t − s ) α d s + f ( x, u s ( x ))d s 0 0 Contents � t σ k ( x, u s ( x ))d W k ( s ) . ASVE + (3) Motivations 0 Local Let S t f be the solution of (See Pr¨ uss [1993]) Solutions of ASVE � t ∆ u s ( x ) Strong u t ( x ) = f ( x ) + ( t − s ) α d s. Solutions of Semilinear 0 SPDEs Large By a solution of (3), we mean that Deviation Principle for � t � t ASVE S t − s σ k ( u s ) W k (d s ) , u t = S t u 0 + S t − s f ( u s )d s + Reference 0 0 which still takes form (1).

Motivations: 4. Stochastic Navier-Stokes Equation Stochastic Consider the following d -dimensional stochastic Navier-Stokes Volterra Equations in equation in R d : Banach Spaces and SPDEs � �  d u t = ∆ u t + ( u t · ∇ ) u t + ∇ π ( t ) d t   + f ( t, u t )d t + σ k ( t, u t )d W k ( t ) (4) Contents  div u t = 0 , u | t =0 = u 0 , ASVE  Motivations Let ( T t ) t � 0 be the Gaussian heat semigroup and P the or- Local Solutions of thogonal projection from L 2 ( R d ; R d ) to the divergence free ASVE subspace L 2 σ ( R d ; R d ). The solution of (4) can be written as Strong Solutions of Semilinear � t SPDEs u t = T t u 0 + T t − s P [( u s · ∇ ) u s ]d s Large Deviation 0 Principle for � t � t ASVE T t − s P σ k ( s, u s ) W k (d s ) , + T t − s P f ( s, u s )d s + Reference 0 0 which also takes form (1).

Motivations: 5. SPDEs driven by additive fractional Brownian motions Stochastic For H ∈ (0 , 1), let Volterra Equations in � c H ( t − s ) H − 1 2 + s H − 1 � Banach 2 F ( t/s ) K H ( t, s ) := 1 { s<t } , s, t ∈ [0 , 1] , Spaces and SPDEs � 1 / 2 � 2 H Γ(3 / 2 − H ) where c H := , Γ denotes the usual Gamma Γ( H +1 / 2)Γ(2 − 2 H ) Contents function, and ASVE � u F ( u ) := c H (1 Motivations ( r − 1) H − 3 2 (1 − r H − 1 2 )d r. 2 − H ) Local 1 Solutions of ASVE The sequence of independent fractional Brownian motions Strong with Hurst parameter H ∈ (0 , 1) may be defined by (See Solutions of Decreusefond and ¨ Semilinear Ust¨ unel [1999]) SPDEs � t Large W k K H ( t, s )d W k ( s ) , Deviation H ( t ) := k = 1 , 2 , · · · , Principle for 0 ASVE which has the covariance function Reference H ( s )) = 1 2( s 2 H + t 2 H − | t − s | 2 H ) . R H ( t, s ) = E ( W k H ( t ) W k

Motivations: 5. SPDEs driven by additive fractional Brownian motions Stochastic Consider the following SPDE driven by { W k H , k ∈ N } Volterra Equations in Banach � d u t ( x ) = [∆ u t ( x ) + f ( t, ω, x, u t ( x ))]d t + σ k ( t, x )d W k H ( t ) , Spaces and SPDEs u ( t, x ) | t =0 = u 0 ( x ) , where σ k ( t, x ) is a deterministic function. As above, we con- Contents sider the mild solution: ASVE � t � t Motivations T t − s σ k ( s )d W k u t = T t u 0 + T t − s f ( s, u s )d s + H ( s ) . Local Solutions of 0 0 ASVE By the integration by parts formula, the stochastic integral Strong Solutions of can be formally written as Semilinear SPDEs � t � t � Large T t − s σ k ( s )d W k H ( s ) = σ k ( t ) K H ( t, s ) Deviation Principle for 0 0 ASVE � t � Reference d W k ( s ) . � � + K H ( u, s ) ∆ T t − u σ k ( u ) − T t − s ˙ σ k ( s ) d u s