On the paradoxical relationship between the heat equation and a traveling-wave solution
Abstract
We investigate the relationship between Brownian motion, which has infinite propagation speed, and a traveling-wave solution with finite propagation speed in the heat equation. First, we define a random walk associated with the Goldstein-Taylor model, a finite-speed approximation of the heat equation. We then introduce a random walk associated with the traveling-wave solutions and analyze its scaling limits, yielding three distinct stochastic processes. Finally, we conclude that the particle motion in a traveling-wave solution includes a mixture of Brownian motion and drift.
Fields of Study & Subjects
Stochastic processes and statistical mechanics
Nonlinear Partial Differential Equations
Geometric Analysis and Curvature Flows
Publication details
- Journal / source
- Physica A Statistical Mechanics and its Applications
- Volume
- 685
- Pages
- 131296–131296
- Publication date
- 13 January 2026
- Language
- EN