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 $\mathrm{d}X_t= |X_t|^\alpha\,\mathrm{d}B_t + \alpha\lambda |X_t|^{2\alpha-1}\operatorname{sign}(X_t)\,\mathrm{d}t$, where $\alpha\in (0,1)$ and $\lambda\in[0,1]$. The parameter $\alpha$ controls the nonlinear power-law profile of the diffusion coefficient, while the parameter $\lambda$ specifies the interpretation of the stochastic integral in the pre-equation $\dot X=|X|^\alpha\dot B$. We demonstrate that the solutions of this equation can be represented as nonlinear transformations of a skew Bessel process with dimension $\delta \in \mathbb{R}$.