Brownian Motion and the Heat Equation
Central Question
Why does the probability distribution of random motion evolve according to a deterministic PDE?
Why This Matters
Diffusion translates random microscopic paths into deterministic macroscopic evolution, joining probability with familiar partial differential equations.
Learning Prompt
I know random walks, basic Brownian motion and undergraduate PDEs. Derive how the discrete evolution equation for a symmetric random walk leads, under diffusive scaling, to the heat equation. Then explain why standard Brownian motion has generator and how a different diffusion coefficient changes the normalization. Connect transition densities, the Gaussian heat kernel and expected values of initial data. Make me distinguish a random sample path from its deterministic density and from the solution operator acting on functions. Explain the forward and backward viewpoints carefully, and identify the assumptions behind the formulas.
Ideas
Examples
Questions
Connections
- Central Limit Theorem
- Random Walks
- From Random Walk to Brownian Motion
- Geometry Meets Physics
- Week overview