We study a second order stationary Hamilton Jacobi equation in infinite dimension. This equation is nonlinear and convex with respect to the first-order term. We use properties of the transition semigroup associated to the linear equation to write the Hamilton Jacobi equation in integral form and we prove existence, uniqueness and regularity of a solution by the theory of maximal monotone operators. We also prove that this solution is the pointwise limit of a uniformly bounded sequence of classical solutions of approximating problems. Finally, the solution is the value function of the associated optimal stochastic control problem. Some examples are given.
The paper is concerned with fully nonlinear second order Hamilton-Jacobi-BellmanIsaacs equations of elliptic type in separable Hilbert spaces which have unbounded first and second order terms. The viscosity solution approach is adapted to the equations under consideration and the existence and uniqueness of viscosity solutions are proved. A stochastic optimal control problem driven by a parabolic stochastic PDE with control of Dirichlet type on the boundary is considered. It is proved that the value function of this problem is the unique viscosity solution of the associated Hamilton-Jacobi-Bellman equation.
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