We study here the problem of dual representation of the value functions associated to linear-convex stochastic control problems in infinite dimensional Hilbert spaces. Since the dual state runs backwards in time, it turns out that the dual representation has the meaning of a classical (Markov) control problem only if the primal linear state equation is driven by the generator of a group. In the general case, a dual representation of the value function still holds, but such a representation cannot be reduced to solving a dual Hamilton-Jacobi-Bellman equation.
"A dual representation result for value functions in stochastic control of infinite dimensional groups." Topol. Methods Nonlinear Anal. 54 (2B) 907 - 916, 2019. https://doi.org/10.12775/TMNA.2019.054