diffusion as Brownian motion

Watch a single speck of pollen jiggling in water under a microscope: it stumbles around in an erratic, jittery, unpredictable path — Brownian motion. Now watch a whole cloud of such specks. Each wanders randomly and independently, yet the cloud as a whole spreads out smoothly and predictably, exactly the way heat diffuses. Diffusion as Brownian motion is the discovery that the heat equation is the law obeyed by the density of randomly wandering particles.

Make it precise. Let a particle perform Brownian motion, a random walk whose steps are independent and have no preferred direction, with a spread that grows like the square root of time. Release many such particles and let p(x,t) be their probability density (or, equivalently, the density of a large cloud). Then p satisfies exactly the diffusion equation p_t = D p_xx, with the diffusion coefficient D set by the variance of the random steps. The connection runs both ways. The heat kernel G(x,t), the Gaussian solution from a point source, is precisely the probability density of where a single Brownian particle started at the origin will be at time t — and indeed a Brownian increment over time t is Gaussian with variance proportional to t, which is why the kernel widens like sqrt(t). So solving the heat equation and computing where random walkers go are the same computation.

This probabilistic picture is illuminating and practically powerful. It explains the heat equation's features intuitively: smoothing comes from averaging over all the random paths; infinite propagation speed comes from the (vanishingly unlikely but nonzero) chance a walker has already jumped far; conservation of total heat is conservation of total probability. It also gives a method — the Feynman-Kac formula expresses solutions of heat-type PDE as expectations over Brownian paths, turning PDE problems into Monte Carlo simulations and underlying the Black-Scholes theory of option pricing. Caveat: the clean correspondence is for the linear heat/diffusion equation; adding drift, variable coefficients, or boundaries changes the stochastic process accordingly (drift terms, reflecting or absorbing boundaries), and nonlinear PDE generally do not have such a simple particle picture.