A stochastic version and a Liouville theorem for hamiltonian inclusions with convex dissipation

TL;DR

This paper introduces a stochastic extension of Hamiltonian systems with convex dissipation and establishes a Liouville theorem based on a minimal dissipation cost functional, advancing the understanding of their statistical properties.

Contribution

It presents a novel stochastic formulation of Hamiltonian systems with convex dissipation and proves a Liouville theorem in this context, filling a gap in the statistical theory.

Findings

Developed a stochastic version of the SBEN principle.

Proved a Liouville type theorem for dissipative Hamiltonian systems.

Connected dissipation cost functional with statistical properties.

Abstract

The statistical counterpart of the formalism of hamiltonian systems with convex dissipation arXiv:0810.1419 , arXiv:1408.3102 is a completely open subject. Here are described a stochastic version of the SBEN principle and a Liouville type theorem which uses a minimal dissipation cost functional.