Regularization of stochastic problems with respect to variables of different kinds (Q736028)

The author considers the stochastic Cauchy problem \[ X'(t)= AX(t)+ B\mathbb{W}(t),\quad t\in [0,\tau),\;\tau\leq\infty;\;X(0)= \xi,\tag{1} \] where \(A\) is the generator of a regularized semigroup \(S= \{S(t): t\in [0,\tau)\}\) in a Hilbert space \(H\); \(\mathbb{W}= \{\mathbb{W}(t): t\geq 0\}\) is an \(H\)-valued white noise, which is informally defined as a process with independent distributions at different values of \(t\) with zero mean and infinite variation, \(B\in{\mathcal L}(H)\). By regularization of (1), we mean the construction of a new problem (corrected, smoothed) and its solution, including a weak solution of (1). The resulting regularized solutions are not assumed to converge to some exact solution of (1) with respect to a parameter. The author uses regularization (i) with respect to variables of \(H\) and the operator \(A\); (ii) with respect to \(t\); and (iii) with respect to a random variable.