Second-order neutral stochastic evolution equations with heredity
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.
stochastic evolution equation with delay
scientific article; zbMATH DE number 2166664