Entropy dissipation via Information Gamma calculus: Non-reversible stochastic differential equations

TL;DR

This paper develops explicit bounds for exponential convergence of certain non-gradient stochastic differential equations to their invariant distributions using an extended Gamma calculus approach in Wasserstein space.

Contribution

It introduces a novel Fisher information induced Gamma calculus for non-gradient drifts and derives explicit dissipation bounds and non-reversible Poincaré inequalities.

Findings

Derived explicit exponential dissipation bounds.

Extended Gamma calculus to non-gradient SDEs.

Provided an analytical example with non-reversible Langevin dynamics.

Abstract

We formulate explicit bounds to guarantee the exponential dissipation for some non-gradient stochastic differential equations towards their invariant distributions. Our method extends the connection between Gamma calculus and Hessian operators in $L^2$--Wasserstein space. In details, we apply Lyapunov methods in the space of probabilities, where the Lyapunov functional is chosen as the relative Fisher information. We derive the Fisher information induced Gamma calculus to handle non-gradient drift vector fields. We obtain the explicit dissipation bound in terms of $L_1$ distance and formulate the non-reversible Poincar{\'e} inequality. An analytical example is provided for a non-reversible Langevin dynamic.