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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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