A new non-Markovian approach to weak convergence for SPDEs Adam Andersson Joint work with Raphael Kruse and Stig Larsson Mathematical Sciences Chalmers University of Technology G¨ oteborg, Sweden Sixth Workshop on Random Dynamical Systems, Bielefeld 2 Nov, 2013 1 / 22
Outline ◮ Stochastic integration in Hilbert space, 2 / 22
Outline ◮ Stochastic integration in Hilbert space, ◮ Malliavin calculus, 2 / 22
Outline ◮ Stochastic integration in Hilbert space, ◮ Malliavin calculus, ◮ Weak convergence, 2 / 22
Outline ◮ Stochastic integration in Hilbert space, ◮ Malliavin calculus, ◮ Weak convergence, ◮ Strong convergence in a dual Watanabe-Sobolev norm. 2 / 22
Cylindrical Q -Wiener process H , separable Hilbert space ( H = L 2 ( D ), D ⊂ R d ), 3 / 22
Cylindrical Q -Wiener process H , separable Hilbert space ( H = L 2 ( D ), D ⊂ R d ), Q ∈ L ( H ) self-adjoint and positive semi-definite covariance operator, 3 / 22
Cylindrical Q -Wiener process H , separable Hilbert space ( H = L 2 ( D ), D ⊂ R d ), Q ∈ L ( H ) self-adjoint and positive semi-definite covariance operator, 1 2 ( H ), Hilbert space with � u , v � 0 = � Q − 1 2 u , Q − 1 2 v � H , u , v ∈ U 0 , U 0 = Q 3 / 22
Cylindrical Q -Wiener process H , separable Hilbert space ( H = L 2 ( D ), D ⊂ R d ), Q ∈ L ( H ) self-adjoint and positive semi-definite covariance operator, 1 2 ( H ), Hilbert space with � u , v � 0 = � Q − 1 2 u , Q − 1 2 v � H , u , v ∈ U 0 , U 0 = Q An operator I : L 2 ([0 , T ] , U 0 ) → L 2 (Ω) is said to be an isonormal process on a probability space (Ω , F , P ), if 3 / 22
Cylindrical Q -Wiener process H , separable Hilbert space ( H = L 2 ( D ), D ⊂ R d ), Q ∈ L ( H ) self-adjoint and positive semi-definite covariance operator, 1 2 ( H ), Hilbert space with � u , v � 0 = � Q − 1 2 u , Q − 1 2 v � H , u , v ∈ U 0 , U 0 = Q An operator I : L 2 ([0 , T ] , U 0 ) → L 2 (Ω) is said to be an isonormal process on a probability space (Ω , F , P ), if ◮ I ( φ ) ∼ N (0 , � φ � L 2 ([0 , T ] , U 0 ) ) , ∀ φ ∈ L 2 ([0 , T ] , U 0 ), 3 / 22
Cylindrical Q -Wiener process H , separable Hilbert space ( H = L 2 ( D ), D ⊂ R d ), Q ∈ L ( H ) self-adjoint and positive semi-definite covariance operator, 1 2 ( H ), Hilbert space with � u , v � 0 = � Q − 1 2 u , Q − 1 2 v � H , u , v ∈ U 0 , U 0 = Q An operator I : L 2 ([0 , T ] , U 0 ) → L 2 (Ω) is said to be an isonormal process on a probability space (Ω , F , P ), if ◮ I ( φ ) ∼ N (0 , � φ � L 2 ([0 , T ] , U 0 ) ) , ∀ φ ∈ L 2 ([0 , T ] , U 0 ), ◮ E [ I ( φ ) I ( ψ )] = � φ, ψ � L 2 ([0 , T ] , U 0 ) , ∀ φ, ψ ∈ L 2 ([0 , T ] , U 0 ). 3 / 22
Cylindrical Q -Wiener process H , separable Hilbert space ( H = L 2 ( D ), D ⊂ R d ), Q ∈ L ( H ) self-adjoint and positive semi-definite covariance operator, 1 2 ( H ), Hilbert space with � u , v � 0 = � Q − 1 2 u , Q − 1 2 v � H , u , v ∈ U 0 , U 0 = Q An operator I : L 2 ([0 , T ] , U 0 ) → L 2 (Ω) is said to be an isonormal process on a probability space (Ω , F , P ), if ◮ I ( φ ) ∼ N (0 , � φ � L 2 ([0 , T ] , U 0 ) ) , ∀ φ ∈ L 2 ([0 , T ] , U 0 ), ◮ E [ I ( φ ) I ( ψ )] = � φ, ψ � L 2 ([0 , T ] , U 0 ) , ∀ φ, ψ ∈ L 2 ([0 , T ] , U 0 ). 3 / 22
Cylindrical Q -Wiener process H , separable Hilbert space ( H = L 2 ( D ), D ⊂ R d ), Q ∈ L ( H ) self-adjoint and positive semi-definite covariance operator, 1 2 ( H ), Hilbert space with � u , v � 0 = � Q − 1 2 u , Q − 1 2 v � H , u , v ∈ U 0 , U 0 = Q An operator I : L 2 ([0 , T ] , U 0 ) → L 2 (Ω) is said to be an isonormal process on a probability space (Ω , F , P ), if ◮ I ( φ ) ∼ N (0 , � φ � L 2 ([0 , T ] , U 0 ) ) , ∀ φ ∈ L 2 ([0 , T ] , U 0 ), ◮ E [ I ( φ ) I ( ψ )] = � φ, ψ � L 2 ([0 , T ] , U 0 ) , ∀ φ, ψ ∈ L 2 ([0 , T ] , U 0 ). W : [0 , T ] × U 0 → L 2 (Ω) cylindrical Q -Wiener process: ∞ � W ( t ) u := I ( χ [0 , t ] ⊗ u ) = � u , u i � 0 β i ( t ) , i =1 where ( u i ) i ∈ N ⊂ U 0 is an ON-basis and ( β i ) i ∈ N are independent standard Brownian motions. 3 / 22
The H -valued Wiener integral Wiener integral for simple integrands: � T χ [ s , t ] ⊗ ( h ⊗ u ) d W = [( W ( t ) − W ( s )) u ] ⊗ h ∈ L 2 (Ω) ⊗ H = L 2 (Ω , H ) 0 Extends directly to linear combinations. 4 / 22
The H -valued Wiener integral Wiener integral for simple integrands: � T χ [ s , t ] ⊗ ( h ⊗ u ) d W = [( W ( t ) − W ( s )) u ] ⊗ h ∈ L 2 (Ω) ⊗ H = L 2 (Ω , H ) 0 Extends directly to linear combinations. Wiener’s isometry: � T � T 2 � � � φ � 2 φ d W H = 2 d t E � � L 0 � � 0 0 4 / 22
The H -valued Wiener integral Wiener integral for simple integrands: � T χ [ s , t ] ⊗ ( h ⊗ u ) d W = [( W ( t ) − W ( s )) u ] ⊗ h ∈ L 2 (Ω) ⊗ H = L 2 (Ω , H ) 0 Extends directly to linear combinations. Wiener’s isometry: � T � T 2 � � � φ � 2 φ d W H = 2 d t E � � L 0 � � 0 0 By density the integral extends to all of L 2 ([0 , T ] , L 0 2 ). For stochastic equations driven by additive noise this definition of the integral suffices. 4 / 22
Malliavin calculus p ( R n ) denote the space of all C ∞ -functions over R n with Let C ∞ polynomial growth. Define S = { X = f ( I ( φ 1 ) , . . . , I ( φ n )): f ∈ C ∞ p ( R n ) , φ 1 , . . . , φ n ∈ L 2 ([0 , T ] , U 0 ) , n ≥ 1 } and n � � � S ( H ) = F = X k ⊗ h k : X 1 , . . . , X n ∈ S , h 1 , . . . , h n ∈ H , n ≥ 1 . k =1 5 / 22
Malliavin calculus p ( R n ) denote the space of all C ∞ -functions over R n with Let C ∞ polynomial growth. Define S = { X = f ( I ( φ 1 ) , . . . , I ( φ n )): f ∈ C ∞ p ( R n ) , φ 1 , . . . , φ n ∈ L 2 ([0 , T ] , U 0 ) , n ≥ 1 } and n � � � S ( H ) = F = X k ⊗ h k : X 1 , . . . , X n ∈ S , h 1 , . . . , h n ∈ H , n ≥ 1 . k =1 We define the Malliavin derivative of F ∈ S ( H ) as the process m n � � D t F = ∂ i f k ( I ( φ 1 ) , . . . , I ( φ n )) ⊗ ( h k ⊗ φ i ( t )) k =1 i =1 and let, for v ∈ U 0 , m n D v � � t F = D t Fv = ∂ i f k ( I ( φ 1 ) , . . . , I ( φ n )) ⊗ � φ i ( t ) , v � 0 ⊗ h k k =1 i =1 5 / 22
Malliavin calculus: integration by parts For all F ∈ S ( H ) and Φ ∈ L 2 ([0 , T ] , L 0 2 ), � T � � � DF , Φ � L 2 ([0 , T ] × Ω , L 0 2 ) = F , Φ( t ) d W ( t ) L 2 (Ω , H ) . 0 6 / 22
Malliavin calculus: integration by parts For all F ∈ S ( H ) and Φ ∈ L 2 ([0 , T ] , L 0 2 ), � T � � � DF , Φ � L 2 ([0 , T ] × Ω , L 0 2 ) = F , Φ( t ) d W ( t ) L 2 (Ω , H ) . 0 Let D 1 , p ( H ) be the closure of S ( H ) with respect to the norm � � T �� 1 p . � E [ � F � p � D t F � p � F � D 1 , p ( H ) = H ] + E 2 d t L 0 0 6 / 22
Malliavin calculus: integration by parts For all F ∈ S ( H ) and Φ ∈ L 2 ([0 , T ] , L 0 2 ), � T � � � DF , Φ � L 2 ([0 , T ] × Ω , L 0 2 ) = F , Φ( t ) d W ( t ) L 2 (Ω , H ) . 0 Let D 1 , p ( H ) be the closure of S ( H ) with respect to the norm � � T �� 1 p . � E [ � F � p � D t F � p � F � D 1 , p ( H ) = H ] + E 2 d t L 0 0 Let ( δ, D ( δ )) be the adjoint of D : L 2 (Ω , H ) → L 2 ([0 , T ] × Ω , L 0 2 ). � � � DF , Φ � L 2 ([0 , T ] × Ω , L 0 2 ) = F , δ Φ L 2 (Ω , H ) . 6 / 22
Malliavin calculus: integration by parts For all F ∈ S ( H ) and Φ ∈ L 2 ([0 , T ] , L 0 2 ), � T � � � DF , Φ � L 2 ([0 , T ] × Ω , L 0 2 ) = F , Φ( t ) d W ( t ) L 2 (Ω , H ) . 0 Let D 1 , p ( H ) be the closure of S ( H ) with respect to the norm � � T �� 1 p . � E [ � F � p � D t F � p � F � D 1 , p ( H ) = H ] + E 2 d t L 0 0 Let ( δ, D ( δ )) be the adjoint of D : L 2 (Ω , H ) → L 2 ([0 , T ] × Ω , L 0 2 ). � � � DF , Φ � L 2 ([0 , T ] × Ω , L 0 2 ) = F , δ Φ L 2 (Ω , H ) . D ( δ ) ⊂ L 2 ([0 , T ] × Ω , L 0 2 ) is large and contains in particular all � T predictable L 0 2 -valued processes. In this case δ (Φ) = 0 Φ( t ) d W ( t ). 6 / 22
The stochastic equation An easy equation for a difficult problem: d X ( t ) + AX ( t ) d t = F ( X ( t )) d t + d W ( t ) , t ∈ (0 , T ] , X (0) = X 0 . 7 / 22
The stochastic equation An easy equation for a difficult problem: d X ( t ) + AX ( t ) d t = F ( X ( t )) d t + d W ( t ) , t ∈ (0 , T ] , X (0) = X 0 . ◮ H = L 2 ( D ), where D is a bounded, convex and polygonal domain of R d , d = 1 , 2 , 3. 7 / 22
The stochastic equation An easy equation for a difficult problem: d X ( t ) + AX ( t ) d t = F ( X ( t )) d t + d W ( t ) , t ∈ (0 , T ] , X (0) = X 0 . ◮ H = L 2 ( D ), where D is a bounded, convex and polygonal domain of R d , d = 1 , 2 , 3. ◮ ( A , D ( A )) selfadjoint with compact inverse, − A the generator of an analytic semigroup ( S ( t )) t ≥ 0 . 7 / 22
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