On the Dirichlet form of three-dimensional Brownian motion conditioned to hit the origin

TL;DR

This paper studies a special diffusion process in three dimensions conditioned to hit the origin, revealing its structure, properties, and how it differs from standard Brownian motion.

Contribution

It characterizes the Dirichlet form and the diffusion process associated with a self-adjoint extension of the Laplacian, highlighting its unique behavior near the origin.

Findings

The energy form is a regular Dirichlet form with core $C_c^ ablafty(\

The diffusion behaves like Brownian motion with a radial drift towards zero.

The process is the unique reflecting extension of the Brownian motion before hitting zero.

Abstract

Our concern in this paper is the energy form induced by an eigenfunction of a self-adjoint extension of the restriction of the Laplace operator to $C_c^\infty(\mathbf{R}^3\setminus \{0\})$. We will prove that this energy form is a regular Dirichlet form with core $C_c^\infty(\mathbf{R}^3)$. The associated diffusion $X$ behaves like a $3$-dimensional Brownian motion with a mild radial drift when far from $0$, subject to an ever-stronger push toward $0$ near that point. In particular $\{0\}$ is not a polar set with respect to $X$. The diffusion $X$ is rotation invariant, and admits a skew-product representation before hitting $\{0\}$: its radial part is a diffusion on $(0,\infty)$ and its angular part is a time-changed Brownian motion on the sphere $S^2$. The radial part of $X$ is a "reflected" extension of the radial part of $X^0$ (the part process of $X$ before hitting $\{0\}$). Moreover, $X$ is the unique reflecting extension of $X^0$, but $X$ is not a semi-martingale.