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Brownian motion (the Wiener process) is a continuous-time stochastic process W(t) with W(0) = 0, independent increments, stationary Gaussian increments (W(t) - W(s) ~ N(0, t-s)), and almost surely continuous sample paths. It is the canonical building block of stochastic calculus and the continuous-time analogue of a symmetric random walk. The Wiener measure on the space of continuous functions C([0,∞)) provides the rigorous measure-theoretic foundation.
Brownian motion is the foundational object of continuous-time probability. Physically, it models the erratic movement of a pollen grain suspended in water — the phenomenon Robert Brown observed in 1827 and Einstein explained in 1905. Mathematically, it is a stochastic process W(t) defined for t ≥ 0, characterized by four properties: W(0) = 0, independent increments over non-overlapping time intervals, Gaussian increments with W(t) - W(s) ~ N(0, t-s) for t > s, and continuous sample paths almost surely. Norbert Wiener gave the first rigorous construction (1923), which is why the process is also called the Wiener process.
The construction is non-trivial. You need to produce a probability measure on the infinite-dimensional space C([0,∞)) of continuous functions. One approach uses the Kolmogorov extension theorem: the finite-dimensional distributions are multivariate Gaussians (determined by the mean function μ(t) = 0 and covariance function K(s,t) = min(s,t)), and the extension theorem guarantees a consistent probability measure on infinite product spaces. Continuity of paths then follows from the Kolmogorov continuity criterion, using the fact that E[|W(t) - W(s)|⁴] = 3(t-s)² gives sufficient moment control.
From your study of martingales, Brownian motion provides a rich supply of continuous-time martingales. W(t) itself is a martingale (its expected future value given the present is the present value). W(t)² - t is also a martingale — the quadratic variation of Brownian motion grows linearly, and subtracting t compensates for this growth. More generally, exp(θW(t) - θ²t/2) is a martingale for any real θ (the exponential martingale). These martingale properties are the engine behind optional stopping arguments, change-of-measure techniques, and the entire apparatus of stochastic calculus.
The sample paths of Brownian motion are almost surely continuous but almost surely nowhere differentiable. The increments W(t+h) - W(t) have standard deviation √h, which goes to zero (continuity) but not fast enough relative to h for a derivative to exist (the ratio √h/h = 1/√h diverges). This pathological roughness — Brownian paths have Hausdorff dimension 3/2 — is precisely why Brownian motion cannot be analyzed using ordinary calculus and demands the development of Itô's stochastic integral, the next major topic in this course.