Chapter

From Brownian Motion to Schrödinger’s Equation

Volume 312 of the series Grundlehren der mathematischen Wissenschaften pp 31-60

Killed Brownian Motion

  • Kai Lai ChungAffiliated withDepartment of Mathematics, Stanford University
  • , Zhongxin ZhaoAffiliated withDepartment of Mathematics, Stanford University

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Abstract

Let X t be the d-dimensional Brownian motion and D a domain in ℝ d (d≥ 1). Adjoin an extra point ∂ to D and set
$$ X_t^D = \left\{ {\begin{array}{*{20}c} {X_t on (t < \tau _D )} \\ {\partial on (t \geqslant \tau _D ).} \\ \end{array} } \right. $$