Stochastic Control as a Non-Equilibrium Statistical Physics: Gauge Invariant Bellman Equation

In Stochastic Control (SC) one minimizes average cost-to-go, consisting of the cost-of-control (amount of efforts), the cost-of-space (where one wants the system to be) and the target cost (where one wants the system to finish), for the system obeying a forced and controlled Langevien dynamics. We generalize the SC problem adding to the cost-to-go a term accounting for the cost-of dynamics, characterized by a vector potential. We provide variational derivation of the generalized gauge-invariant Bellman-Hamilton-Jacobi equation for the optimal average cost-to-go, where the control is expressed in terms of current and density functionals, and discuss examples, e.g.ergodic control of particle-on-a-circle illustrating non-equilibrium space-time complexity over current/flux. The talk is based on a joint work with M. Chertkov, J. Bierkens and H.J. Kappen.