Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of - PowerPoint PPT Presentation

Presentation Weak solutions Open problems Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces Alessandra Lunardi in collaboration with G. Da Prato Modena, Sept. 9th, 2010 Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Literature Open problems A family of OU operators in infinite dimensions L α ϕ ( x ) = 1 2 Tr [ Q 1 − α D 2 ϕ ( x )] − 1 2 � x , Q − α D ϕ ( x ) � Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Literature Open problems A family of OU operators in infinite dimensions L α ϕ ( x ) = 1 2 Tr [ Q 1 − α D 2 ϕ ( x )] − 1 2 � x , Q − α D ϕ ( x ) � x ∈ H = infinite dimensional separable Hilbert space, with norm | · | ; Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Literature Open problems A family of OU operators in infinite dimensions L α ϕ ( x ) = 1 2 Tr [ Q 1 − α D 2 ϕ ( x )] − 1 2 � x , Q − α D ϕ ( x ) � x ∈ H = infinite dimensional separable Hilbert space, with norm | · | ; Q ∈ L ( H ) self-adjoint positive operator with finite trace; Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Literature Open problems A family of OU operators in infinite dimensions L α ϕ ( x ) = 1 2 Tr [ Q 1 − α D 2 ϕ ( x )] − 1 2 � x , Q − α D ϕ ( x ) � x ∈ H = infinite dimensional separable Hilbert space, with norm | · | ; Q ∈ L ( H ) self-adjoint positive operator with finite trace; 0 ≤ α ≤ 1. Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Literature Open problems A family of OU operators in infinite dimensions L α ϕ ( x ) = 1 2 Tr [ Q 1 − α D 2 ϕ ( x )] − 1 2 � x , Q − α D ϕ ( x ) � x ∈ H = infinite dimensional separable Hilbert space, with norm | · | ; Q ∈ L ( H ) self-adjoint positive operator with finite trace; 0 ≤ α ≤ 1. { e k : k ∈ N } = orthonormal basis in H such that Qe k = λ k e k , D k = derivative in the direction of e k , x k = � x , e k � , Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Literature Open problems A family of OU operators in infinite dimensions L α ϕ ( x ) = 1 2 Tr [ Q 1 − α D 2 ϕ ( x )] − 1 2 � x , Q − α D ϕ ( x ) � x ∈ H = infinite dimensional separable Hilbert space, with norm | · | ; Q ∈ L ( H ) self-adjoint positive operator with finite trace; 0 ≤ α ≤ 1. { e k : k ∈ N } = orthonormal basis in H such that Qe k = λ k e k , D k = derivative in the direction of e k , x k = � x , e k � , ∞ ∞ L α ϕ ( x ) = 1 D kk ϕ ( x ) − 1 λ 1 − α λ − α ∑ ∑ x k D k ϕ ( x ) . k k 2 2 k = 1 k = 1 Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Literature Open problems L α is the Kolmogorov operator of the stochastic problem dX α ( t , x ) = − 1 2 Q − α X α ( t , x ) dt + Q ( 1 − α ) / 2 dW ( t ) , X α ( 0 , x ) = x , where W ( t ) is a standard cylindrical Wiener process in H . Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Literature Open problems L α is the Kolmogorov operator of the stochastic problem dX α ( t , x ) = − 1 2 Q − α X α ( t , x ) dt + Q ( 1 − α ) / 2 dW ( t ) , X α ( 0 , x ) = x , where W ( t ) is a standard cylindrical Wiener process in H . The associated transition Markov semigroups T α ( t ) are the Ornstein-Uhlenbeck semigroups � H ϕ ( y + e − tA α / 2 x ) N Q t ( dy ) , T α ( t ) ϕ ( x ) = E [ ϕ ( X α ( t , x ))] = t > 0 , ϕ ∈ C b ( H ) , with A = Q − 1 , Q t : Q ( I − e − tA α ) , N Q t = the Gaussian measure with mean 0 and covariance Q t . Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Literature Open problems L α is the Kolmogorov operator of the stochastic problem dX α ( t , x ) = − 1 2 Q − α X α ( t , x ) dt + Q ( 1 − α ) / 2 dW ( t ) , X α ( 0 , x ) = x , where W ( t ) is a standard cylindrical Wiener process in H . The associated transition Markov semigroups T α ( t ) are the Ornstein-Uhlenbeck semigroups � H ϕ ( y + e − tA α / 2 x ) N Q t ( dy ) , T α ( t ) ϕ ( x ) = E [ ϕ ( X α ( t , x ))] = t > 0 , ϕ ∈ C b ( H ) , with A = Q − 1 , Q t : Q ( I − e − tA α ) , N Q t = the Gaussian measure with mean 0 and covariance Q t . Important common feature. For 0 ≤ α ≤ 1, T α ( t ) have the same invariant measure µ := N Q . That is, Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Literature Open problems L α is the Kolmogorov operator of the stochastic problem dX α ( t , x ) = − 1 2 Q − α X α ( t , x ) dt + Q ( 1 − α ) / 2 dW ( t ) , X α ( 0 , x ) = x , where W ( t ) is a standard cylindrical Wiener process in H . The associated transition Markov semigroups T α ( t ) are the Ornstein-Uhlenbeck semigroups � H ϕ ( y + e − tA α / 2 x ) N Q t ( dy ) , T α ( t ) ϕ ( x ) = E [ ϕ ( X α ( t , x ))] = t > 0 , ϕ ∈ C b ( H ) , with A = Q − 1 , Q t : Q ( I − e − tA α ) , N Q t = the Gaussian measure with mean 0 and covariance Q t . Important common feature. For 0 ≤ α ≤ 1, T α ( t ) have the same invariant measure µ := N Q . That is, � � T α ( t ) ϕ d µ = H ϕ d µ , t > 0 , ϕ ∈ C b ( H ) . H Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Literature Open problems The Gaussian measure µ = N Q is defined first on the “cylindrical subsets” of H , i.e. subsets of the type n ∈ N , A ∈ B ( R n ) A = { x ∈ H : ( x 1 ,..., x n ) ∈ A } , Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Literature Open problems The Gaussian measure µ = N Q is defined first on the “cylindrical subsets” of H , i.e. subsets of the type n ∈ N , A ∈ B ( R n ) A = { x ∈ H : ( x 1 ,..., x n ) ∈ A } , by � � n x 2 1 � ∑ k µ ( A ) = � − dx , exp 2 λ k ( 2 π ) n λ 1 · ... · λ n A k = 1 Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Literature Open problems The Gaussian measure µ = N Q is defined first on the “cylindrical subsets” of H , i.e. subsets of the type n ∈ N , A ∈ B ( R n ) A = { x ∈ H : ( x 1 ,..., x n ) ∈ A } , by � � n x 2 1 � ∑ k µ ( A ) = � − dx , exp 2 λ k ( 2 π ) n λ 1 · ... · λ n A k = 1 then it is extended to all Borel sets B ( H ) by the Caratheodory teorem. Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Literature Open problems The most popular among the operators L α are L 0 and L 1 : Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Literature Open problems The most popular among the operators L α are L 0 and L 1 : L 0 ϕ ( x ) = 1 2 Tr [ QD 2 ϕ ( x )] − 1 2 � x , D ϕ ( x ) � , is the operator that arises in the Malliavin calculus, Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Literature Open problems The most popular among the operators L α are L 0 and L 1 : L 0 ϕ ( x ) = 1 2 Tr [ QD 2 ϕ ( x )] − 1 2 � x , D ϕ ( x ) � , is the operator that arises in the Malliavin calculus, and � H ϕ ( y + e − t / 2 x ) N ( 1 − e − t ) Q ( dy ) , T 0 ( t ) ϕ ( x ) := t > 0 , Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Literature Open problems The most popular among the operators L α are L 0 and L 1 : L 0 ϕ ( x ) = 1 2 Tr [ QD 2 ϕ ( x )] − 1 2 � x , D ϕ ( x ) � , is the operator that arises in the Malliavin calculus, and � H ϕ ( y + e − t / 2 x ) N ( 1 − e − t ) Q ( dy ) , T 0 ( t ) ϕ ( x ) := t > 0 , while L 1 ϕ ( x ) = 1 2 Tr [ D 2 ϕ ( x )] − 1 2 � x , AD ϕ ( x ) � , (with A = Q − 1 ) is the generator of the Ornstein-Uhlenbeck semigroup T 1 ( t ) with the best smoothing properties (e.g., books of Da Prato-Zabczyk) Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces