Heterogeneous diffusion process with power-law nonlinearity

In this paper, we study solutions of the heterogeneous diffusion process with power-law nonlinearity governed by the stochastic differential equation $\mathrm{d}{X}_t=\mathrm{\mid }{X}_t{\mathrm{\mid }}^{\alpha }\mathrm{d}{B}_t+\alpha \lambda \mathrm{\mid }{X}_t{\mathrm{\mid }}^{2\alpha -1}\mathrm{sign}(X_t)\mathrm{d}t$, 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 $\dot{X}=\mathrm{\mid }X{\mathrm{\mid }}^{\alpha }\dot{B}$. We demonstrate that the solutions of this equation can be represented as nonlinear transformations of a skew Bessel process with dimension $\delta \in \mathbb{R}$.