Stochastic Analysis and its Applications

PI调查随机分析中的几个问题，既有纯粹的，也有应用的。反问题和第三边界问题是从地球物理到医学和物理的应用科学中自然产生的数学模型。PI研究与这些模型相关的基本问题，例如相应方程解的存在和唯一性，并解决每个领域的具体问题。PI研究了反射布朗运动的几个方面，包括同步耦合的合并，反射与惰性漂移，Shy耦合的存在，通过近视条件反射过程的构造，以及Skorohod方程的路径唯一性。其他模型和过程也是独立但相关研究的主题；它们包括斜布朗运动随机流、删失稳定过程和跳跃类型过程。Pi‘s攻击了Neumann特征函数上的一些众所周知的“公开问题”，并反映了布朗运动。该项目致力于在数学层面对涉及随机性的自然、技术、经济和社会现象进行建模。随机分析领域的出现是因为需要了解各种现象，如股票市场、材料的非破坏性分析、医学成像和散热。该项目的数学方面为计算机程序等应用程序提供了基础，这些程序有助于现实生活中的情况。应用科学经常使用随机分析的数学公式。该项目旨在提供有关随机过程的有用信息。

The PIs investigate several problems in stochastic analysis of both a pure and applied nature. Inverse problems and the third boundary problem are mathematical models arising naturally in applied sciences from geophysics to medicine and physics. The PIs study the foundational questions related to these models, such as existence and uniqueness of the solutions to the corresponding equations, and also address specific problems in each of these fields. The PI's investigate several aspects of reflected Brownian motion, including the coalescence of synchronous couplings, reflection with inert drift, existence of shy couplings, construction of reflected processes via myopic conditioning, and the pathwise uniqueness for the Skorohod equation. Other models and processes are the subject of separate but related studies; they include the skew Brownian motion stochastic flow, censored stable processes, and jump-type processes. The PI's attack some well known "open problems" on Neumann eigenfunctions and reflected Brownian motion. The project is devoted to modeling, at the mathematical level, of natural, technical, economic and social phenomena involving randomness. The field of stochastic analysis arose as a need to understand phenomena as diverse as the stock market, non-destructive analysis of materials, medical imaging, and heat dissipation. The mathematical side of the project provides a foundation for the applications such as computer programs that help in real life situations. The applied sciences routinely use mathematical formulas of stochastic analysis. The project aims at providing usable information about stochastic processes.