Heterogeneous diffusion process with power-law nonlinearity
Jorge E. Cardona, Ilya Pavlyukevich
TL;DR
This work analyzes a one-dimensional heterogeneous diffusion with a power-law diffusion coefficient, interpreting multiplicative noise via a parameter λ and showing that solutions can be obtained by nonlinear transformations of skew Bessel processes of dimension δ = δ_{α,λ}. By time-changing Brownian motion and applying nonlinear maps, the authors construct weak solutions and classify the behavior at the origin across δ ranges, linking singular SDEs to classical diffusions. The main contribution is a complete characterization: for δ ∈ (0,2) the solution is a transformed skew Bessel process with sign changes at 0, while for δ ∈ [2,∞) the process either never hits zero or is trapped, with strong solutions in many cases. This framework unifies Itô/Stratonovich/Hängi–Klimontovich interpretations within a single diffusion-driven construction and provides rigorous insight into diffusion with power-law nonlinearities.
Abstract
In this paper, we study solutions of the heterogeneous diffusion process with power-law nonlinearity governed by the stochastic differential equation $\mathrm{d}X_t= |X_t|^α\,\mathrm{d}B_t + αλ|X_t|^{2α-1}\operatorname{sign}(X_t)\,\mathrm{d}t$, where $α\in (0,1)$ and $λ\in[0,1]$. The parameter $α$ controls the nonlinear power-law profile of the diffusion coefficient, while the parameter $λ$ specifies the interpretation of the stochastic integral in the pre-equation $\dot X=|X|^α\dot B$. We demonstrate that the solutions of this equation can be represented as nonlinear transformations of a skew Bessel process with dimension $δ\in \mathbb{R}$.