Heterogeneous Diffusion Process

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This project is devoted to a paradigmatic model of motion of a heavy random particle with the space dependent irregular diffusivity. This model is loosely determined by a stochastic differential equation and is known in physical literature as heterogeneous diffusion process. We will mainly consider the cases of H\"older-continuous diffusivity such that the origin is the unique irregular point where the diffusion degenerates.The project will treat 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 are going to consider the above stochastic differential equation in the so-called $\lambda$-interpretation that includes It\^o ($\lambda=0$), Stratonovich ($\lambda=\frac12$) and mostly important H\"anggi--Klimontovich (or kinetic, $\lambda=1$) cases. The Stratonovich equation was completely studied by the applicants recently. In this project, we are going to determine all weak/strong homogeneous Markovian solutions spending zero time at zero for a general $\lambda$-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. The second goal of the project is to study the heterogeneous diffusion process (in any interpretation) in the presence of additional independent small external noise. This setting allows to consider the heterogeneous diffusion process 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. We expect that the weak external noise will regularize the original equation and the unique \emph{physically natural} solution will be obtained in the zero limit of the external noise. The regularization effect of the noise is known under the name selection problem. The selection problem will be first considered for the heterogeneous diffusion process under Stratonovich interpretation.As a main mathematical tool for our analysis, we will use the theory of irregular/singular stochastic differential equations, stochastic differential equations with local time, skew Bessel processes, and time reversion. We hope to get explicit formulae for Markovian heterogeneous diffusions in terms of certain non-linear transformations of skew Bessel processes.The results to be obtained in this project will contribute to the general theory of irregular/singular stochastic differential equations and will advance our understanding of the non-linear effects in realistic stochastic models of physics, biology, and applied sciences.