Boundary classification of diffusion processes via local time

点击翻译按钮获取中文摘要

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.