Stochastic selection problem for a Stratonovich SDE with power non-linearity

TL;DR

This paper investigates how adding small external noise to a specific Stratonovich SDE with power non-linearity restores solution uniqueness, connecting the limit to symmetric heterogeneous diffusion.

Contribution

It demonstrates that external additive noise restores uniqueness in a non-unique solution regime of the Stratonovich SDE, linking the limit to symmetric diffusion.

Findings

Adding small noise restores solution uniqueness.

Limit as noise vanishes yields symmetric heterogeneous diffusion.

Solutions are related to skew Brownian motion with variable skewness.

Abstract

In our paper [Bernoulli 26(2), 2020, 1381-1409], we found all strong Markov solutions that spend zero time at $0$ of the Stratonovich stochastic differential equation $d X=|X|^{\alpha}\circ dB$, $\alpha\in (0,1)$. These solutions have the form $X_t^\theta=F(B^\theta_t)$, where $F(x)=\frac{1}{1-\alpha}|x|^{1/(1-\alpha)}\text{sign}\, x$ and $B^\theta$ is the skew Brownian motion with skewness parameter $\theta\in [-1,1]$ starting at $F^{-1}(X_0)$. In this paper we show how an addition of small external additive noise $\varepsilon W$ restores uniqueness. In the limit as $\varepsilon\to 0$, we recover heterogeneous diffusion corresponding to the physically symmetric case $\theta=0$.