The Stochastic Reflection Problem in Hilbert Spaces

We construct a Markov process X associated with the stochastic reflection problem on a closed convex subset with non empty interior and smooth boundary in a Hilbert space, as a solution to a random convex control problem. The transition semigroup corresponding to X is exactly that defined by the Kolmogorov equation with Neumann homogeneous boundary conditions (see [3 Barbu , V. , Da Prato , G. , Tubaro , L. ( 2011 ). Kolmogorov equation associated to the stochastic reflection problem on a smooth convex set of a Hilbert space II . Ann. Inst. H. Poincaré 4 : 699 – 724 .[Crossref] , [Google Scholar]]).