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On a randomly accelerated particle

Michelle Nuno and Juhi Jang

Vol. 10 (2017), No. 3, 399–415

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.

Kolmogorov equation, kinetic Fokker–Planck equation, reflection method, specular boundary condition
Mathematical Subject Classification 2010
Primary: 35Q84, 65M80
Received: 4 August 2015
Revised: 28 April 2016
Accepted: 13 June 2016
Published: 14 December 2016

Communicated by Kenneth S. Berenhaut
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