Brownian Motion

Brownian motion (Wiener process) models continuous, erratic movement such as that of particles in a fluid or stock price fluctuations. It represents the limiting case of a random walk as the step size approaches zero and the number of steps approaches infinity, connecting back to our first topic.

A standard Brownian motion W(t) has several defining properties: W(0) = 0, it has independent increments, and for any times s < t, the increment W(t) - W(s) follows a normal distribution with mean 0 and variance t-s. Its paths are continuous but nowhere differentiable—mathematically capturing the concept of continuous but extremely erratic motion.

Geometric Brownian Motion (GBM), given by the stochastic differential equation dS(t) = μS(t)dt + σS(t)dW(t), extends this concept to model quantities that can't go negative (like stock prices) and incorporates both drift (μ) and volatility (σ) parameters.

In machine learning, Brownian motion serves as the foundation for models in finance (option pricing), physics-informed neural networks, diffusion models for image generation, and stochastic optimization techniques.

Recent breakthrough generative AI models like DALL-E and Stable Diffusion use diffusion processes—directly inspired by Brownian motion—to generate images by gradually denoising random Gaussian noise. The mathematics of Brownian motion helps these models transform noise into coherent, detailed images through a carefully controlled reverse diffusion process.