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=|X|α∘dB, α∈(0,1). These solutions have the form Xθt=F(Bθt), where F(x)=11−α|x|1/(1−α)signx and Bθ is the skew Brownian motion with skewness parameter θ∈[−1,1] starting at F−1(X0). In this paper we show how an addition of small external additive noise εW restores uniqueness. In the limit as ε→0, we recover heterogeneous diffusion corresponding to the physically symmetric case θ=0.