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Abstract stochastic integro-differential delay equations. (English) Zbl 1105.60045
The abstract stochastic integro-differential delay equation \[ \begin{aligned} x'(t)+ Ax(t)= f_1(t, x_t,\mu(t)) & + \int^t_0 K_1(t, s) f_2(s, x_s, \mu(s))\,ds\\ & +\int^t_0 K_2(t, s) f_3(s,x_s, \mu(s))\, dW(s),\quad 0\leq t\leq T,\end{aligned}\tag{1} \] \(x(t)= \varphi(t)\), \(-r\leq t\leq 0\), in a real separable Hilbert space \(H\) is considered. Here, \(K\) is a real separable Hilbert space; \(W\) is a given \(K\)-valued Wiener process associated with a positive nuclear covariance operator \(Q\); \(A\) is a linear operator which generates a strongly continuous semigroup \(\{S(t): t\geq 0\}\) on \(H\); \(K_i(t, s)\), \(i= 1,2\), are bounded linear operators on \(H\); \(f_i:[0, T]\times C_r\times{\mathcal P}_{\lambda^2}(H)\to H\), \(i= 1,2\), and \(f_3:[0, T]\times C_r\times{\mathcal P}_{\lambda^2}(H)\to BL(K; H)\), where \(BL(K; H)\) denotes the space of all bounded linear operators from \(K\) into \(H\) and \({\mathcal P}_{\lambda^2}(H)\) denotes a particular subset of probability measures on \(H\), are given mappings; \(\mu(t)\) is the probability law of \(x(t)\) and \(\varphi\in L^2(\Omega; C_r)\) is an \({\mathcal J}_0\)-measurable random variable independent of \(W\) with almost surely continuous paths.
The authors prove the existence and uniqueness of a mild solution of (1), the continuous dependence of mild solutions on the initial data and the boundedness of the \(p\)th moments. They consider the Yosida approximation of (1) \[ \begin{aligned} x_n'(t)+ Ax_n(t) = & nR(n; A)f_1(t, (x_n)_t, \mu_n(t))\\ & +\int^t_0 K_1(t, s)nR(n; A)f_2(s, (x_n)_s, \mu_n(s))\,ds\\ & + \int^t_0 K_2(t, s) nR(n; A)f_3(s, (x_n)_x, \mu_n(s))\,dW(s),\quad 0\leq t\leq T,\end{aligned}\tag{2} \] \(x_n(t)= nR(n; A)\varphi(t)\), \(-r\leq t\leq 0\), where \(\mu(t)\) is the probability law of \(x_n(t)\) and \(R(n; A)= (I- nA)^{-1}\) is the resolvent operator of \(A\), and show that sequence of strong solutions \(x_n(t)\) of (2) converges to unique mild solution of (1). From which they deduce the weak convergence of the measures induced by \(x_n(t)\) to the measure induced by mild solution of (1). The authors also establish the \(p\)th moment and almost sure exponential stability of the mild solution.
These results are applied to generalized stochastic heat equation and a Sobolev-type evolution equation.

MSC:
60H20 Stochastic integral equations
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