Heterogeneous Diffusion Process
Final Report Abstract
This project was devoted to a paradigmatic model of motion of a heavy diffusive particle with the space dependent irregular diffusivity. This model is loosely determined by the stochastic differential equation X “ |X|α B and is known in physical literature as a heterogeneous 9 9 ¨ diffusion process. We considered the case of Holder-continuous diffusivity with α P p0, 1q such that the origin is the unique irregular point where the diffusion degenerates. The project will treated two questions that have clear physical motivation. First, it is well known in Physics that the choice of the stochastic integral (interpretation) is the essential part of a physical model. Hence we considered a singular drift version X “ |X|α B ` αλ|X|2α´1 sgn X of the stochastic differential equation. The parameter λ P 9 9 ˆ r0, 1s determines the interpretation of the stochastic integral. The Ito case Stratonovich and Hanggi–Klimontovich SDEs correspond to the values λ “ 0, λ “ 1 and λ “ 1 respectively. ¨ 2 In this project, we determined all weak time-autonomous strongly Markovian solutions spending zero time at zero for a general λ-interpretation. These solutions belong to a physically meaningful class of perpetual diffusions in a heterogeneous medium with a unique absorbing point that might contain a hidden interface. These solutions are obtained by means of a non-linear transformation of a skew Bessel process of dimension δ “ p1 ´ 2αpλ ´ 1qq{p1 ´ αq. In the second part of the project we studied the heterogeneous diffusion process in the Stratonovich interpetation in the presence of additional independent small external noise. This setting allows us to consider the heterogeneous diffusion driven by the Brownian motion B as an idealization of a physical motion of a heavy particle in a random medium whose atomic-molecular structure is a source of weak ambient noise. The weak external noise regularizes the original equation and the unique physically natural solution is obtained in the zero limit of the external noise. This regularization effect of the noise is known as selection problem. Our result is a manifestation of the selection procedure in the case of degenerate stochastic differential equations. The results obtained in this project contribute to the general theory of irregular/singular stochastic differential equations and advance our understanding of the non-linear effects in realistic stochastic models of physics, biology, and applied sciences.
Publications
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Stochastic selection problem for a Stratonovich SDE with power non-linearity. Bernoulli, 31(2).
Pavlyukevich, Ilya & Shevchenko, Georgiy
