Reflection principle

Abstract

The reflection principle, also known as the method of images, is a fundamental technique in probability theory and mathematical statistics with origins in classical physics, specifically geometrical optics, heat conduction, and electrostatics. In stochastic processes, the principle facilitates the calculation of probability distributions for random walks constrained by reflecting or absorbing barriers. For a symmetric random walk on a straight line, the probability of reaching a specific position relative to a reflecting barrier is determined by summing the unrestricted probabilities of the target point and its mirror image. In the case of absorbing barriers, the solution is derived from the difference between these probabilities. These methods extend to complex scenarios involving multiple barriers through periodic summation and are applicable to higher-dimensional spaces and non-symmetric walks. The mathematical treatment of these stochastic problems is directly analogous to the physical superposition of light sources or electrical charges. Beyond discrete random walks, the principle remains a vital tool for solving problems in Brownian motion, ballot theory, order statistics, and sequential analysis. – AI-generated abstract.