On the paradoxical relationship between the heat equation and a traveling-wave solution

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. • Particle motion in traveling waves mixes Brownian motion and drift. • A random-walk model associated with traveling waves is constructed. • The scaling limit yields Brownian motion via weak limit theorems. • Three probabilistic motion types inherent in traveling waves are identified. • Finite-speed motion is reconciled with the heat equation’s infinite propagation.