Peano phenomenon for stochastic equations with local time

TL;DR

This paper studies the weak convergence of measures from solutions of stochastic equations involving local time as the diffusion term vanishes, especially when the associated ODE has multiple solutions, and derives formulas for the limiting measure's weights.

Contribution

It provides a novel analysis of the limit behavior of stochastic equations with local time, including explicit formulas for weights on extreme solutions when multiple solutions exist.

Findings

Limit measures concentrate on extreme solutions with specific weights.

Formulas for weights in the limit measure are derived.

The results extend understanding of stochastic equations with local time and multiple ODE solutions.

Abstract

We investigate weak convergence of measures generated by solutions of stochastic equations with local time and small diffusion while the last one tends to zero. In case the correspondent ordinary differential equation has infinitely many solutions we prove that limit measure concentrated with some weights on its extreme solutions. Formulae for weights are obtained.