Controlled stochastic differential equations under constraints in infinite dimensional spaces

In this paper we study the compatibility (or viability) of a given state constraint $K$ with respect to a controlled stochastic evolution equation in a real Hilbert space $H$. We allow the noise to be a cylindrical Wiener process and admit an unbounded linear operator in the state equation. Our assumptions cover, for instance, controlled heat equations with space-time white noise. Our main result is to prove that if $K$ is $\epsilon$-viable, then the square of the distance from $K$: $d_K^2(x)\coloneq\inf_{y\in K}|x-y|^2$ is a viscosity supersolution of a suitable class of fully nonlinear Hamilton-Jacobi-Bellman equations in $H$. This extends already obtained results into the finite-dimensional case. We use the definition of viscosity supersolutions for `unbounded' elliptic equations in infinite variables that have been recently introduced by Świ\c ech and Kelome. We discuss several cases where the above necessary condition is also sufficient.''