Stratonovich stochastic differential equation with power non-linearity: (non)-uniqueness and selection problem

I will review results regarding a Stratonovich stochastic differential equation $$ X_t=X_0+\int_0^t |X_s|^\alpha\circ d B_s, $$ which was introduced in the physical literature under the name ``heterogeneous diffusion process''. It turns out that equation has properties quite different from its Ito counterpart. Namely, we show that for $\alpha\in(0,1)$ it has infinitely many strong solutions spending zero time at zero. They are given by $X^\theta = \bigl((1-\alpha)B^\theta+(X_0)^{1-\alpha} \bigr)^{1/(1-\alpha)}$, where for $\theta\in(-1,1)$, $B^\theta$ is the $\theta$-skew Brownian motion, and $(x)^{\gamma} = |x|^\gamma \operatorname{sign} x$. It appears that there are no other homogeneous strong Markov solutions to the equation. To address the non-uniqueness, we consider a perturbation of the equation by a small independent noise. It appears that the solution to such equations converge to the solution of initial equation corresponding to $\theta=0$, i.e. the physically symmetric case.