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

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 $dX=\mathrm{\mid }X{\mathrm{\mid }}^{\alpha }\circ dB$, $\alpha \in (0,1)$. These solutions have the form $X_t^{\theta }=F(B_t^{\theta })$, where $F(x)=\frac{1}{1-\alpha }\mathrm{\mid }x{\mathrm{\mid }}^{1\mathrm{/}(1-\alpha )}\text{sign}x$ 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 $\epsilon W$ restores uniqueness. In the limit as $\epsilon \to 0$, we recover heterogeneous diffusion corresponding to the physically symmetric case $\theta =0$.