Kolmogorov equations and weak order analysis for SPDEs with - PowerPoint PPT Presentation

Kolmogorov equations and weak order analysis for SPDEs with multiplicative noise Charles-Edouard Bréhier Joint work with Arnaud Debussche (ENS Rennes) CNRS & Université Lyon 1, Institut Camille Jordan C-E Bréhier Multiplicative noise SPDEs: weak rates 1 / 21

Plan of the talk Model, assumptions and results 1 SPDEs with non constant diffusion coefficient Numerical scheme Weak convergence result Study of the Kolmogorov equation 2 The Kolmogorov equation Basic regularity estimates Improved estimates: strategy C-E Bréhier Multiplicative noise SPDEs: weak rates 2 / 21

Model and assumptions Parabolic, semilinear, SPDE: t ≥ 0, z ∈ ( 0 , 1 ) � � = ∂ 2 x ( t , z ) f 2 ( x ( t , z )) ∂ x ( t , z ) + ∂ � � � � + f 1 x ( t , z ) + σ x ( t , z ) ξ ( t , z ) ∂ t ∂ z 2 ∂ z driven by space-time white noise ξ ( t , z ) . Boundary conditions: x ( t , 0 ) = x ( t , 1 ) = 0 (Dirichlet). Initial condition: x ( 0 , · ) = x 0 . Coefficients f 1 , f 2 and σ are smooth, bounded and Lipschitz continuous. In the drift: Burgers type nonlinearity. Multiplicative noise: diffusion coefficient is not constant C-E Bréhier Multiplicative noise SPDEs: weak rates 3 / 21

Stochastic evolution equation formulation Formulation: in the Da Prato-Zabczyk framework, X ( t ) = X ( t , · ) ∈ H = L 2 ( 0 , 1 ) dX ( t ) = AX ( t ) dt + G ( X ( t )) dt + σ ( X ( t )) dW ( t ) , with A : D ( A ) → H unbounded, self-adjoint linear operator √ 2 sin ( n π · ) and λ n = − n 2 π 2 ; Ae n = − λ n e n , where e n ( · ) = G = F 1 + BF 2 and σ : H → L ( H ) are Nemytskii coefficients. W is a cylindrical Wiener process: � W ( t ) = β n ( t ) e n . n ∈ N where β n are independent standard Wiener processes, for n ∈ N . C-E Bréhier Multiplicative noise SPDEs: weak rates 4 / 21

Well-posedness The equation dX ( t ) = AX ( t ) dt + G ( X ( t )) dt + σ ( X ( t )) dW ( t ) admits a unique, global, mild solution: for all t ≥ 0 � t � t X ( t ) = e tA x 0 + e ( t − s ) A � e ( t − s ) A σ ( X ( s )) dW ( s ) � F 1 ( X ( s ) + BF 2 ( X ( s )) ds + 0 0 where e tA x = � n ∈ N e − t λ n � x , e n � e n is the semi-group associated with A . C-E Bréhier Multiplicative noise SPDEs: weak rates 5 / 21

Well-posedness The equation dX ( t ) = AX ( t ) dt + G ( X ( t )) dt + σ ( X ( t )) dW ( t ) admits a unique, global, mild solution: for all t ≥ 0 � t � t X ( t ) = e tA x 0 + e ( t − s ) A � e ( t − s ) A σ ( X ( s )) dW ( s ) � F 1 ( X ( s ) + BF 2 ( X ( s )) ds + 0 0 where e tA x = � n ∈ N e − t λ n � x , e n � e n is the semi-group associated with A . Regularization by the semi-group: for every α ∈ [ 0 , 1 ) � � e tA h � H ≤ C α t − α | h | − α . � n = 1 λ − 2 α with | h | 2 − α = � ∞ |� h , e n �| 2 . n C-E Bréhier Multiplicative noise SPDEs: weak rates 5 / 21

Temporal discretization Discretization of dX ( t ) = AX ( t ) dt + G ( X ( t )) dt + σ ( X ( t )) dW ( t ) : linear-implicit Euler scheme X ∆ t n + 1 = X ∆ t + ∆ tAX ∆ t n + 1 + ∆ tG ( X ∆ t n ) + σ ( X ∆ t n )∆ W ∆ t n n = S ∆ t X ∆ t + ∆ tS ∆ t G ( X ∆ t n ) + S ∆ t σ ( X ∆ t n )∆ W ∆ t n n with S ∆ t = ( I − ∆ tA ) − 1 , T = N ∆ t , X ∆ t = x 0 , 0 ∆ W ∆ t � � = W ( n + 1 )∆ t − W ( n ∆ t ) . n Strong order of convergence is 1 4 : 1 � ≤ C κ ( T , x 0 )∆ t � � X ( N ∆ t ) − X ∆ t � 4 − κ . E N where C κ ( T , x 0 ) ∈ ( 0 , ∞ ) , for all κ ∈ ( 0 , 1 4 ) . C-E Bréhier Multiplicative noise SPDEs: weak rates 6 / 21

Main result: weak order 1 2 Consider real-valued test functions ϕ , with C 3 regularity, and derivatives controlled in the following sense: for some p , q ∈ [ 2 , ∞ ) � ≤ C ( 1 + | x | L p ) K | h 1 | L q . . . | h n | L q . � �� � D n ϕ ( x ) . � h 1 , . . . , h n � 1 Example: ϕ ( x ) = 0 φ ( x ( z )) dz . Theorem (B.-Debussche) For such test functions ϕ , for every T ∈ ( 0 , ∞ ) , and every κ ∈ ( 0 , 1 2 ) , there exists C κ ( T , ϕ ) ∈ ( 0 , ∞ ) such that � �� 1 X ∆ t 2 − κ . � � � X ( N ∆ t ) − E ϕ � ≤ C κ ( T , ϕ ) P ( x 0 )∆ t � E ϕ � � N Preprint 2017 Kolmogorov Equations and Weak Order Analysis for SPDES with Nonlinear Diffusion Coefficient. C-E Bréhier Multiplicative noise SPDEs: weak rates 7 / 21

Comparison with previous results Approaches in the literature (not exhaustive): Kolmogorov equation: SDEs: Talay, Tubaro, Milstein, Bally, Tretyakov, Kloeden, Platen, . . . SPDEs: Debussche, Larsson, Kovacs, Andersson, Printems, Wang, . . . mild Itô calculus: Jentzen, Conus, Kurniawan, Cox, . . . Malliavin calculus: SDEs: Clément, Kohatsu-Higa, Lamberton SPDEs: Andersson, Larsson, Kruse; Lindner C-E Bréhier Multiplicative noise SPDEs: weak rates 8 / 21

Comparison with previous results Approaches in the literature (not exhaustive): Kolmogorov equation: SDEs: Talay, Tubaro, Milstein, Bally, Tretyakov, Kloeden, Platen, . . . SPDEs: Debussche, Larsson, Kovacs, Andersson, Printems, Wang, . . . mild Itô calculus: Jentzen, Conus, Kurniawan, Cox, . . . Malliavin calculus: SDEs: Clément, Kohatsu-Higa, Lamberton SPDEs: Andersson, Larsson, Kruse; Lindner Novelties of our work: 1 extension of the Kolmogorov equation approach when the diffusion coefficient σ is not constant. 2 treatment of Burgers type nonlinearities. C-E Bréhier Multiplicative noise SPDEs: weak rates 8 / 21

Plan of the talk Model, assumptions and results 1 SPDEs with non constant diffusion coefficient Numerical scheme Weak convergence result Study of the Kolmogorov equation 2 The Kolmogorov equation Basic regularity estimates Improved estimates: strategy C-E Bréhier Multiplicative noise SPDEs: weak rates 9 / 21

Simplified problem Evolution equation: dY ( t ) = AY ( t ) dt + G ( Y ( t )) dt + σ ( Y ( t )) dW ( t ) , Y ( 0 ) = y . where the coefficients G : H → H and σ : H → L ( H ) are of class C 3 b . Discretization: spectral Galerkin approximation dY N ( t ) = AY N ( t ) dt + P N G ( Y N ( t )) dt + P N σ ( Y N ( t )) dW ( t ) . where P N y = � N n = 1 � y , e n � e n . Test function: ϕ : H → R is of class C 3 b . − 1 � � 2 + κ Weak error estimate: � E ϕ ( Y N ( T )) − E ϕ ( Y ( T )) � ≤ C κ ( ϕ, T ) λ N + 1 . � � C-E Bréhier Multiplicative noise SPDEs: weak rates 10 / 21

The Kolmogorov equation � � � The function ( t , y ) �→ u ( t , y ) = E ϕ ( Y ( t )) � Y ( 0 ) = y is solution of the Kolmogorov equation ∂ u ( t , y ) = L u ( t , y ) ∂ t = � Ay + G ( y ) , Du ( t , y ) � + 1 � � σ ( y ) σ ( y ) ⋆ D 2 u ( t , y ) . 2 Tr C-E Bréhier Multiplicative noise SPDEs: weak rates 11 / 21

The Kolmogorov equation � � � The function ( t , y ) �→ u ( t , y ) = E ϕ ( Y ( t )) � Y ( 0 ) = y is solution of the Kolmogorov equation ∂ u ( t , y ) = L u ( t , y ) ∂ t = � Ay + G ( y ) , Du ( t , y ) � + 1 � � σ ( y ) σ ( y ) ⋆ D 2 u ( t , y ) . 2 Tr Some regularity properties are needed: D 2 u ( t , y ) . ( h , k ) ≤ C β,γ ( t ) | h | − β | k | − γ . Du ( t , y ) . h ≤ C α ( t ) | h | − α , C-E Bréhier Multiplicative noise SPDEs: weak rates 11 / 21

The Kolmogorov equation � � � The function ( t , y ) �→ u ( t , y ) = E ϕ ( Y ( t )) � Y ( 0 ) = y is solution of the Kolmogorov equation ∂ u ( t , y ) = L u ( t , y ) ∂ t = � Ay + G ( y ) , Du ( t , y ) � + 1 � � σ ( y ) σ ( y ) ⋆ D 2 u ( t , y ) . 2 Tr Some regularity properties are needed: D 2 u ( t , y ) . ( h , k ) ≤ C β,γ ( t ) | h | − β | k | − γ . Du ( t , y ) . h ≤ C α ( t ) | h | − α , Basic estimates: α ∈ [ 0 , 1 2 ) and β, γ ∈ [ 0 , 1 2 ) with β + γ < 1 2 . Improved estimates: α ∈ [ 0 , 1 ) and β, γ ∈ [ 0 , 1 2 ) . C-E Bréhier Multiplicative noise SPDEs: weak rates 11 / 21

Decomposition of the weak error E ϕ ( Y ( T )) − E ϕ ( Y N ( T )) = u ( T , y ) − E u ( 0 , Y N ( T )) = u ( T , y ) − u ( T , P N y ) + u ( T , P N y ) − E u ( 0 , Y N ( T )) and using Itô’s formula and the Kolmogorov equation u ( T , P N y ) − E u ( 0 , Y N ( T )) � T L N − ∂ � � � � = E u T − t , Y N ( t ) dt ∂ t 0 � T � � � � = E L N − L T − t , Y N ( t ) u dt 0 C-E Bréhier Multiplicative noise SPDEs: weak rates 12 / 21

Required regularity results for numerical analysis First-order derivative: − α ≤ C α ( T − t ) � ≤ C α ( T − t ) � � � � � � � � ( P N − I ) G ( y ) , Du T − t , Y ( t ) � � ( P N − I ) G ( y ) . � λ α N + 1 So we need α ∈ [ 0 , 1 2 ) . Burgers: G ( y ) = F 1 ( y ) + BF 2 ( y ) : α ∈ [ 0 , 1 ) . C-E Bréhier Multiplicative noise SPDEs: weak rates 13 / 21