Second-order neutral stochastic evolution equations with heredity. (English) Zbl 1077.34062

McKibben, Mark A.

J. Appl. Math. Stochastic Anal. 2004, No. 2, 177-192 (2004).

The author considers an abstract neutral semilinear stochastic evolution equation of the form \[ d(X'(t)-f_1(t,X_t)) =AX(t)dt+f_2(t,X_t)dt+g(t,X_t)dW(t). \] Here, \(A\) generates a strongly continuous cosine family of operators on a Hilbert space, \(W\) is an infinite-dimensional Wiener process and \(X_t\) denotes the solution segment at time \(t\). Various results for this equation are derived. First, existence and uniqueness of a mild solution is established under Lipschitz conditions. Then, continuous dependenc on the initial segment as well as moment estimates are treated. Finally, the existence and uniqueness result is generalized to the case of Carathéodory conditions. The paper closes with an example discussing an initial-boundary value problem.

Reviewer: Markus Reiß (Heidelberg)

MSC:

34F05	Ordinary differential equations and systems with randomness
34K50	Stochastic functional-differential equations
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)

Keywords:

stochastic evolution equation with delay; semilinear equation; strongly continuous cosine family