Uniqueness of invariant measures for the stochastic Cauchy problem in Banach spaces

2001 ◽  
pp. 491-517 ◽  
Author(s):  
J. M. A. M. Van Neerven
Keyword(s):  
Cauchy Problem ◽  
Banach Spaces ◽  
Invariant Measures ◽  
Stochastic Cauchy Problem

ON THE DOMAIN OF NONSYMMETRIC ORNSTEIN–UHLENBECK OPERATORS IN BANACH SPACES

2008 ◽  
Vol 11 (04) ◽  
pp. 603-626 ◽  
Author(s):  
JAN MAAS ◽  
JAN VAN NEERVEN
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Cauchy Problem ◽  
Brownian Motion ◽  
Invariant Measure ◽  
Sobolev Space ◽  
Banach Spaces ◽  
Bounded Operator ◽  
Frechet Derivatives ◽  
Stochastic Cauchy Problem

We consider the linear stochastic Cauchy problem [Formula: see text] where A generates a C0-semigroup on a Banach space E, WH is a cylindrical Brownian motion over a Hilbert space H, and B: H → E is a bounded operator. Assuming the existence of a unique minimal invariant measure μ∞, let Lp denote the realization of the Ornstein–Uhlenbeck operator associated with this problem in Lp (E, μ∞). Under suitable assumptions concerning the invariance of the range of B under the semigroup generated by A, we prove the following domain inclusions, valid for 1 < p ≤ 2: [Formula: see text] Here [Formula: see text] denotes the kth order Sobolev space of functions with Fréchet derivatives up to order k in the direction of H. No symmetry assumptions are made on Lp.

scholarly journals ON STOCHASTIC GENERALIZED FUNCTIONS

2011 ◽  
Vol 14 (02) ◽  
pp. 237-260 ◽  
Author(s):  
PEDRO CATUOGNO ◽  
CHRISTIAN OLIVERA
Keyword(s):  
Cauchy Problem ◽  
Existence And Uniqueness ◽  
Generalized Functions ◽  
Uniqueness Of The Solution ◽  
Stochastic Cauchy Problem

In this work we introduce a new algebra of stochastic generalized functions. The regular Hida distributions in [Formula: see text] are embedded in this algebra via their chaos expansions. As an application, we prove the existence and uniqueness of the solution of a stochastic Cauchy problem involving singularities.

scholarly journals An Application Of The Theory Of Scale Of Banach Spaces

2015 ◽  
Vol 29 (1) ◽  
pp. 51-59
Author(s):  
Łukasz Dawidowski
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Boundary Conditions ◽  
Heat Equation ◽  
Banach Spaces ◽  
Abstract Cauchy Problem ◽  
Initial Boundary ◽  
Scale Of Banach Spaces ◽  
The Cauchy Problem ◽  
Selection Of

AbstractThe abstract Cauchy problem on scales of Banach space was considered by many authors. The goal of this paper is to show that the choice of the space on scale is significant. We prove a theorem that the selection of the spaces in which the Cauchy problem ut − Δu = u|u|s with initial–boundary conditions is considered has an influence on the selection of index s. For the Cauchy problem connected with the heat equation we will study how the change of the base space influents the regularity of the solutions.

scholarly journals Weak solutions of differential equations in Banach spaces

1986 ◽  
Vol 33 (3) ◽  
pp. 407-418 ◽  
Author(s):  
Nikolaos S. Papageorgiou
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Vector Field ◽  
Banach Spaces ◽  
Existence Result ◽  
Measure Of Noncompactness ◽  
Continuous Vector ◽  
Weakly Continuous ◽  
Continuous Vector Field ◽  
The Cauchy Problem

We consider the Cauchy problem x (t) = f (t,x (t)), x (0) = x0 in a nonreflexive Banach space X and for f: T × X → X a weakly continuous vector field. Using a compactness hypothesis involving a weak measure of noncompactness we prove an existence result that generalizes earlier theorems by Chow-Shur, Kato and Cramer-Lakshmikantham-Mitchell.

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