Heterogeneous Diffusion Process

本项目致力于研究具有空间依赖的不规则扩散率的重随机粒子运动的范式模型。该模型由随机微分方程松散确定，在物理文献中称为异质扩散过程。我们将主要考虑Hölder-continuous扩散率的情况，使得原点是扩散退化的唯一不规则点。该项目将处理两个具有明确物理动机的问题。首先，在物理学中，随机积分（解释）的选择是物理模型的重要组成部分，这是众所周知的。因此，我们将在所谓的$\lambda$ -解释中考虑上述随机微分方程，其中包括Itô （$\lambda=0$）， Stratonovich （$\lambda=\frac12$）和最重要的Hänggi- Klimontovich（或动力学，$\lambda=1$）情况。斯特拉诺维奇方程最近被申请人彻底研究了。在这个项目中，我们将确定所有弱/强齐次马尔可夫解在零处花费零时间用于一般$\lambda$ -解释。这些解属于一类具有独特吸收点的非均质介质中具有物理意义的永久扩散，该吸收点可能包含隐藏界面。该项目的第二个目标是研究在存在额外的独立小外部噪声的情况下（在任何解释下）的非均匀扩散过程。这种设置允许考虑由布朗运动$B$驱动的非均匀扩散过程，作为一个理想的重粒子在随机介质中的物理运动，其原子-分子结构是弱环境噪声的来源。我们期望弱外部噪声将使原方程正则化，并在外部噪声的零极限下得到唯一的\emph{物理自然}解。在名称选择问题下，噪声的正则化效应是已知的。选择问题将首先考虑在Stratonovich解释下的非均质扩散过程。作为我们分析的主要数学工具，我们将使用不规则/奇异随机微分方程理论，局部时间随机微分方程，斜贝塞尔过程和时间回归。我们希望从斜贝塞尔过程的某些非线性变换中得到马尔可夫非均相扩散的显式公式。本项目所获得的结果将有助于不规则/奇异随机微分方程的一般理论，并将促进我们对物理、生物和应用科学中实际随机模型的非线性效应的理解。

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.