An abstract stochastic Cauchy problem (Q2713899)

The article is a brief review of earlier author's results devoted to the Cauchy problem for the first order operator-differential equations NEWLINE\[NEWLINE u'(t) =A u(t) +B\chi(t)\quad (t\in [0,T),\;T\leq \infty),\qquad u(0)=\xi \tag{1} NEWLINE\]NEWLINE and related questions. Here \(A\) is a generator of a semigroup in a given Hilbert space \(H\), \(B\) is a bounded operator defined on a Hilbert space \(U\) with values in \(H\), \(\xi=\xi(\omega)\) is a random variable on a probabilistic space \((\Omega,{\mathcal F},P)\), and \(\chi(t)=\chi(t,\omega)\) is some random process. The equation (1) is also rewritten as NEWLINE\[NEWLINEdu(t)=Au(t) dt+ B dW(t)\quad \Biggl(W(t)=\int_0^t\chi(s) ds \Biggr)NEWLINE\]NEWLINE with a Wiener process \(W\). The questions of existence and constructing weak or generalized solutions to problem (1) are discussed. In particular, a solution is sought in the distribution space or in the space of stochastic distributions. Some results concerning with the stochastic Cauchy problem for the differential inclusions NEWLINE\[NEWLINE u'(t)\in Au(t), \quad t\geq 0,\qquad u(0)=\xi NEWLINE\]NEWLINE are also presented.NEWLINENEWLINEFor the entire collection see [Zbl 0956.00039].