Stochastic evolution equations in Hilbert spaces
Let \(H\) be a separable Hilbert space, \(A\) a single-valued maximal monotone operator on \(H\) and \(W\) a Wiener process in a separable Hilbert space \(K\) with nuclear covariance operator. Consider the stochastic differential equation \[ dx(t) + Ax(t)dt = Bx(t) dw(t), \quad x(0) = x \in H, \tag{1} \] and a time discretization scheme of it, \[ x_k - x_{k - 1} + \Delta t Ax_k = Bx_{k - 1} \bigl( w(k \Delta t) \bigr) - w \bigl( (k - 1) \Delta t \bigr). \tag{2} \] The existence of a solution \(x(t)\) to (1) is established by using the theory of monotone operators on the Hilbert space \(H\). Moreover, the weak convergence of the sequence defined by (2) to \(x(t)\), in an appropriate state space, is proved. Approximation theorems are given. Also, regularity results of solutions of (1) for the operator \(A\) being the subdifferential of a lower semicontinuous convex function on \(H\) are obtained. Stochastic evolution equations under Gelfand triple and hyperbolic stochastic evolution equations are considered. Examples of applications of these results are given.
hyperbolic stochastic evolution equations
scientific article; zbMATH DE number 806343
scientific article