Heterogeneous Diffusion Process

本计画致力于研究具有空间相依不规则扩散系数的重随机粒子运动的范例模型。该模型由随机微分方程松散地确定，在物理文献中称为非均质扩散过程。我们将主要考虑H“older-continuous扩散率的情况，使得原点是扩散退化的唯一不规则点。首先，在物理学中众所周知，随机积分（解释）的选择是物理模型的重要组成部分。因此，我们将考虑上述随机微分方程在所谓的$\lambda$-解释，其中包括It\^o（$\lambda=0$），Stratonovich（$\lambda=\frac12$）和最重要的H\“anggi-Klimontovich（或动力学，$\lambda=1$）情况。申请人最近对Stratonovich方程进行了全面研究。在这个项目中，我们将确定所有弱/强齐次马尔可夫解花费零时间为零的一般$\lambda$-解释。这些解决方案属于一个有物理意义的类永久扩散在一个独特的吸收点，可能包含一个隐藏的接口的非均匀介质。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的非均匀扩散过程（在任何解释中）。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重粒子在随机介质中的物理运动的理想化，该随机介质的原子-分子结构是弱环境噪声的源。我们期望，弱的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物理自然解.噪声的正则化效应在名称选择问题下是已知的。首先考虑Stratonovich解释下的非均匀扩散过程的选择问题，作为我们分析的主要数学工具，我们将使用非正则/奇异随机微分方程、带局部时的随机微分方程、斜Bessel过程和时间反演理论。我们希望通过斜Bessel过程的某些非线性变换得到马尔可夫非均匀扩散方程的显式公式，所得结果将有助于非正则/奇异随机微分方程的一般理论，并将促进我们对物理学、生物学和应用科学中实际随机模型中非线性效应的理解.

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.