Regular Solutions of Second-Order Stationary Hamilton–Jacobi Equations

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.