On a randomly accelerated particle

Nuno, Michelle; Jang, Juhi

doi:10.2140/involve.2017.10.399

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On a randomly accelerated particle

Michelle Nuno and Juhi Jang

Vol. 10 (2017), No. 3, 399–415

DOI: 10.2140/involve.2017.10.399

Abstract

The focus of this note is to learn more about the Kolmogorov equation describing the dynamics of a randomly accelerated particle. We first explore some existing results of the Kolmogorov equation from the stochastic and differential equation points of view and discuss its solvability with and without boundary conditions. More specifically, we introduce stochastic processes and Brownian motion and we present a connection between a stochastic process and a differential equation. After looking at stochastic processes, we introduce generalized functions and derive the fundamental solution to the heat equation and to the Fokker–Planck equation. The problem with a reflecting boundary condition is also studied by using various methods such as separation of variables, self-similarity, and the reflection method.

Keywords

Kolmogorov equation, kinetic Fokker–Planck equation, reflection method, specular boundary condition

Mathematical Subject Classification 2010

Primary: 35Q84, 65M80

Milestones

Received: 4 August 2015

Revised: 28 April 2016

Accepted: 13 June 2016

Published: 14 December 2016

Communicated by Kenneth S. Berenhaut

Authors

Michelle Nuno
University of California Irvine, CA 92697 United States

Juhi Jang
Department of Mathematics University of Southern California 3620 S. Vermont Ave. Los Angeles, CA 90089 United States

Vol. 10, No. 3, 2017