Boundary classification of diffusion processes via local time

Stroock和Williams热方程的初边值问题包含一个非Feller类型的边界条件，当参数(对应于速度度量)为负时，该边界条件不是Feller类型的。因此，Feller的半群方法不能通过取由初始数据组成的扩散过程的期望值来构造解。Peskir最近的一篇论文提出了一种完全不同的方法来概率地解决Stroock和Williams的初边值问题，该方法适用于初始数据在无穷远处消失的情况。这种解是通过取扩散过程的期望值及其由初始数据形成的确定性泛函组成的局部时间来构造的。引用扩散过程的定律和它的局部时间，这也给出了初边值问题的解的封闭公式。导数同时适用于参数的所有值，对其符号没有限制，扩散过程(及其本地时间)在此上下文中扮演基本解的角色(所有其他解的构建块)。该项目的目的是在现有的初边值问题及其解的例子的基础上，使用所建议的方法研究更多的例子，并对扩散过程的边界分类建立一个更统一的观点，该观点超越了Feller的经典观点。还将研究衍生结果的应用(例如在金融数学中)。

The initial boundary value problem for the heat equation of Stroock and Williams includes a boundary condition which is not of Feller type when a parameter (corresponding to a speed measure) is negative. Consequently, Feller's semigroup approach cannot be used to construct a solution by taking the expected value of a diffusion process composed with the initial data. A recent paper by Peskir develops an entirely different approach to solving the initial boundary value problem of Stroock and Williams probabilistically that applies to initial data vanishing at infinity. Such a solution is constructed by taking the expected value of a diffusion process and its local time composed with a deterministic functional formed by the initial data. Invoking the law of the diffusion process and its local time this also yields a closed formula for the solution to the initial boundary value problem. The derivation applies simultaneously to all values of the parameter with no restriction on its sign, and the diffusion process (with its local time) plays the role of a fundamental solution in this context (a building block for all other solutions). The aim of the project is to build on the existing example of the initial boundary value problem and its solution, study further examples using the suggested methodology, and build a more unifying view towards boundary classification of diffusion processes that goes beyond the classic one due to Feller. Applications of the derived results (e.g. in financial mathematics) will also be studied.