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Diffusion Processes: from the Heat Equation to Brownian Motion

Rev. 2 · Journal of Applied Analysis

We study the evolution of a temperature field in a homogeneous medium and show that the macroscopic flow of heat admits a simple microscopic description. The central object is the parabolic equation

∂u/∂t = α ∇²u

where α is the thermal diffusivity of the medium. Its fundamental solution is a Gaussian kernel whose variance grows linearly in time — the signature of a random walk.

∂u/∂t = α ∇²u

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00:14 / 01:52

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