Diffusion Approximations of Markovian Solutions to Discontinuous ODEs

TL;DR

This paper demonstrates how deterministic and Markov semigroups associated with discontinuous ODEs can be approximated by smooth ODE flows and diffusion processes, respectively, providing a bridge between discontinuous dynamics and smooth approximations.

Contribution

It establishes that all such semigroups can be approximated by smooth ODE flows and diffusion processes, extending the understanding of solutions to discontinuous ODEs.

Findings

Deterministic semigroups are limits of smooth ODE flows.

Markov semigroups are limits of diffusion processes with vanishing noise.

Provides a method to approximate discontinuous ODE solutions with smooth models.

Abstract

In a companion paper, the authors have characterized all deterministic semigroups, and all Markov semigroups, whose trajectories are Carathe'odory solutions to a given ODE x'=f(x), with f possibly discontinuous. The present paper establishes two approximation results. Namely, every deterministic semigroup can be obtained as the pointwise limit of the flows generated by a sequence of ODEs $x'=f_n(x) with smooth right hand sides. Moreover, every Markov semigroup can be obtained as limit of a sequence of diffusion processes with smooth drifts and with diffusion coefficients approaching zero.