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

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 $C_c^\infty ({\mathbb{R}^3}\backslash \{ 0\} )$). We will prove that this energy form is a regular Dirichlet form with core $C_c^\infty ({\mathbb{R}^3})$. 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, ∞) and its angular part is a time-changed Brownian motion on the sphere S². The radial part of X is a “reflected” extension of the radial part of X⁰ (the part process of X before hitting {0}). Moreover, X is the unique reflecting extension of X⁰, but X is not a semi-martingale.