The first passage time density of Brownian motion and the heat equation with Dirichlet boundary condition in time dependent domains

2018/11/13 by Jang‐Ming Lee, Lee, JM
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.1811.05069

openalex publication_date 2018/11/13 · openalex created_date 2022/08/02 · openalex updated_date 2026/07/28

Abstract

In [4], it is proved that we can have a continuous first-passage-time density function of one dimensional standard Brownian motion when the boundary is Hölder continuous with exponent greater than 1/2. For the purpose of extending [4] into multidimensional domains, we show that there exists a continuous first-passage-time density function of standard d-dimensional Brownian motion in moving boundaries in ℝ^d, d≥ 2, under a C³-diffeomorphism. Similarly as in [4], by using a property of local time of standard d-dimensional Brownian motion and the heat equation with Dirichlet boundary condition, we find a sufficient condition for the existence of the continuous density function.

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