Global existence results for a stochastic differential equation in Hilbert spaces (Q2845228)

The authors consider a class of stochastic differential equations of the form NEWLINE\[NEWLINEd[x(t)-g(t,x(t))]=Ax(t)dt+G(t,x(t))dw(t),\quad t\in\mathbb R,NEWLINE\]NEWLINE where \(A\) is the infinitesimal generator of a hyperbolic \(C_0\)-semigroup of bounded linear operators \(\{T(t)\}_{y\geq0}\) in the Hilbert space \(H\), \(g:\mathbb R\times H\to U\) and \(G:\mathbb R\times H\to L_2^0\) are appropriate functions, and for a symmetric nonnegative operator \(Q\in L_2(K,H)\) with finite trace \(\{w(t): t\in\mathbb R\}\), a \(Q\)-Wiener process is defined on \((\Omega,{\mathcal F}, P)\) with values in \(K\). Using the Banach contraction mapping principle and the fixed point theorem for condensing maps, the existence of a unique mild solution to the considered equation is proved. As an application of the obtained results the authors prove the existence and uniqueness of a global mild solution to the partial neutral stochastic differential equation NEWLINE\[NEWLINEd[x(t,\xi)-\int_{0}^{\pi}b(\eta,\xi)x(t,\eta)d\eta]={\partial^2\over\partial\xi^2}x(t,\xi)dt+a(t,x(t,\xi))dw(t),\,x(t,0)=x(t,\pi)=0,NEWLINE\]NEWLINE for all \((t,\xi)\in\mathbb R\times[0,\pi]\).