# HJB equations in infinite dimension and optimal control of stochastic evolution equations via generalized Fukushima decomposition

## TL;DR

This paper develops a novel approach to stochastic optimal control in infinite-dimensional spaces by leveraging the generalized Fukushima decomposition, leading to milder assumptions and new verification techniques for HJB equations.

## Contribution

It introduces a new method using weak Dirichlet processes to analyze HJB equations in infinite dimensions, relaxing previous regularity assumptions.

## Key findings

- Value process is a real weak Dirichlet process.
- Verification theorem established under milder assumptions.
- Enhanced understanding of HJB equations in infinite-dimensional stochastic control.

## Abstract

A stochastic optimal control problem driven by an abstract evolution equation in a separable Hilbert space is considered. Thanks to the identification of the mild solution of the state equation as $\nu$-weak Dirichlet process, the value processes is proved to be a real weak Dirichlet process. The uniqueness of the corresponding decomposition is used to prove a verification theorem. Through that technique several of the required assumptions are milder than those employed in previous contributions about non-regular solutions of Hamilton-Jacobi-Bellman equations.