# Stochastic Optimal Control as Non-equilibrium Statistical Mechanics: Calculus of Variations over Density and Current

arXiv: https://arxiv.org/abs/1306.6572 · HTML: https://mlsanity.com/paper/1306.6572/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Jun 2013.

## Built on by (most cited first, 3 of 3)

- [Reinforcement learning of rare diffusive dynamics](https://mlsanity.com/paper/2105.04321/) (2021)
- [Generative Stochastic Optimal Transport: Guided Harmonic Path-Integral Diffusion](https://mlsanity.com/paper/2512.11859/) (2025)
- [Analytic Bridge Diffusions for Controlled Path Generation](https://mlsanity.com/paper/2605.02961/) (2026)

## Abstract (arXiv)

In Stochastic Optimal Control (SOC) one minimizes the average cost-to-go, that consists of the cost-of-control (amount of efforts), cost-of-space (where one wants the system to be) and the target cost (where one wants the system to arrive), for a system participating in forced and controlled Langevin dynamics. We extend the SOC problem by introducing an additional cost-of-dynamics, characterized by a vector potential. We propose derivation of the generalized gauge-invariant Hamilton-Jacobi-Bellman equation as a variation over density and current, suggest hydrodynamic interpretation and discuss examples, e.g., ergodic control of a particle-within-a-circle, illustrating non-equilibrium space-time complexity.