Brownian Motion

A clear, accessible development of Brownian motion starts from the historic observation of jittery pollen particles and traces how statistical reasoning and early 20th-century physics transformed that mystery into a precise stochastic model. The exposition recounts Brown’s observation, Einstein and Smoluchowski’s molecular bombardment explanation, Perrin’s experimental confirmation, and Wiener's mathematical formalization, then sets the goal of reconstructing the marginal distribution of a particle’s position at a fixed time by building up from a simple discrete-time model.

The concrete argument models motion as a random walk with independent coin flips determining ±u steps, represents all possible sample paths with a binomial tree, and counts endpoints by the number of heads k in n trials. Combinatorics yields the binomial coefficient n choose k and the exact probability P(K_n = k) = (n choose k) p^k q^{n-k}. Expressing K_n as a sum of (Z_i+1)/2 gives E[K_n] = np and Var[K_n] = npq. Those discrete probabilities and moment calculations form the backbone for taking a continuous-time limit, which produces the Gaussian marginal distribution that characterizes Brownian motion.