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

• 来源期刊: 中国科学：数学（英文版）
• 年，卷（期）: 2019,（8）
• 页码范围: 1477-1492
• 页数: 16页
• 语种: 英文

摘要

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 Cc∞(R3 \ {0}).We will prove that this energy form is a regular Dirichlet form with core Cc∞ (R3).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.