Stochastic variational inequalities for a wave equation. (Q265176)

The author considers a system of evolution variational inequalities NEWLINE\[NEWLINEu_{tt}-\Delta u-b(\omega,t,x,u)\dot B\geq 0,\quad u_t\geq 0,\quad u_t(u_{tt}-\Delta u-b(\omega,t,x,u)\dot B)=0NEWLINE\]NEWLINE on a smooth bounded domain \(G\) in \(\mathbb R^d\) with the homogeneous Dirichlet boundary condition and initial conditions \(u(0)=u_0\), \(u_t(0)=u_1\) where \(u_1\geq 0\). Here \(B\) is a cylindrical Wiener process and \(b\) either does not depend on \(u\) (i.e. the noise in the equation is additive) or \(b\) is a deterministic and time-homogeneous affine operator applied on \(u\) (i.e. the noise in the equation is multiplicative). By transferring the stochastic problem to a random problem NEWLINE\[NEWLINEv_{tt}-\Delta v-f\geq 0,\quad v_t+M\geq 0,\quad (v_t+M)(v_{tt}-\Delta v-f)=0NEWLINE\]NEWLINE where \(M\) and \(f\) are random processes, the author proves existence, uniqueness and regularity of the solutions to both the stochastic and the random problems in the state space \((H^1_0(G)\cap H^2(G))\times H^1_0(G)\) for the pair \((u,u_t)\). In case of the multiplicative noise, it is additionally assumed that \(d\leq 3\).