Heterogeneous diffusion process with power-law nonlinearity

Abstract

In this paper, we study solutions of the heterogeneous diffusion process with power-law nonlinearity governed by the stochastic differential equation $\mathrm{d}{X}_t=|X_t{|}^{\hat{I}}\pm \mathrm{d}{B}_t+\hat{I}\pm \hat{I}»|X_t{|}^{2\hat{I}\pm -1}\mathrm{sign}(X_t)\mathrm{d}t$, where $\hat{I}\pm \in (0,1)$ and $\hat{I}»\in [0,1]$. The parameter $\hat{I}\pm$ controls the nonlinear power-law profile of the diffusion coefficient, while the parameter $\hat{I}»$ specifies the interpretation of the stochastic integral in the pre-equation $\dot{X}=|X{|}^{\hat{I}}\pm \dot{B}$. We demonstrate that the solutions of this equation can be represented as nonlinear transformations of a skew Bessel process with dimension $\hat{I}´\in \mathbb{R}$.

23 pages, 1 figure