Heterogeneous diffusion process with power-law nonlinearity

TL;DR

This paper investigates solutions to a nonlinear stochastic differential equation modeling heterogeneous diffusion with power-law nonlinearity, revealing they can be represented as transformations of a skew Bessel process, advancing understanding of complex stochastic systems.

Contribution

It introduces a novel representation of solutions to a class of nonlinear diffusion processes as transformations of skew Bessel processes, linking stochastic calculus with special functions.

Findings

Solutions are representable as nonlinear transformations of skew Bessel processes.

The parameters α and λ control the nonlinearity and interpretation of the stochastic integral.

The approach provides new insights into the structure of heterogeneous diffusion processes.

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 $\delta \in \mathbb{R}$.