Home Statistical Dictionary Brownian Motion

Brownian Motion

A continuous-time stochastic process with independent, normally distributed increments and continuous (but nowhere smooth) sample paths, originally observed as the erratic jittering of pollen grains suspended in water.

Stochastic ProcessesAdvancedFree

In Plain English

Brownian motion started as a physical observation: in 1827, botanist Robert Brown watched pollen grains suspended in water jitter around erratically under a microscope, buffeted by countless invisible collisions with water molecules. Einstein's 1905 paper explained this physically and, in doing so, derived the exact mathematical process now named for the phenomenon, a path that moves continuously but so erratically that it has no well-defined velocity anywhere, with each increment of movement independent of the last and normally distributed in size. This entry emphasizes that physical motivation and the process's defining probabilistic properties; see Wiener Process for the same mathematical object viewed through the lens of its rigorous construction and use in stochastic calculus.

Definition

A standard Brownian motion is a stochastic process ${W_{t}}_{t \geq 0}$ satisfying: (1) $W_{0} = 0$ ; (2) independent increments, for any $0 \leq t_{1} < t_{2} < \dots < t_{n}$ , the increments $W_{t_{2}} - W_{t_{1}}, \dots, W_{t_{n}} - W_{t_{n - 1}}$ are mutually independent; (3) $W_{t} - W_{s} \sim N (0, t - s)$ for $0 \leq s < t$ (Gaussian, stationary increments); and (4) continuous sample paths, $t \mapsto W_{t}$ is continuous almost surely. A striking consequence of these properties is that Brownian paths, while continuous everywhere, are differentiable nowhere, and have infinite total variation on any interval, a mathematically rigorous version of the physical picture of ceaseless, infinitely-jagged jittering. Brownian motion is simultaneously a Markov process, a martingale, and a Gaussian process, one of very few processes belonging to all three families at once.

Formula

W_{t} - W_{s} \sim N (0, t - s) for 0 \leq s < t

Notation

W_{t}

—Position of the Brownian motion at time $t$

Properties

Brownian motion's nowhere-differentiability is not a mathematical curiosity but the entire reason stochastic calculus (Ito's lemma, stochastic differential equations) had to be invented, ordinary calculus's chain rule and integration theory assume smooth or at least bounded-variation paths, which Brownian motion categorically does not have.
Louis Bachelier's 1900 doctoral thesis modeled Paris stock exchange price fluctuations using what amounts to Brownian motion, five years before Einstein's physics paper, an early and largely unrecognized-at-the-time application that later became foundational to mathematical finance via geometric Brownian motion.
Donsker's invariance principle shows that Brownian motion is the universal scaling limit of a simple random walk (and, more generally, of a broad class of random walks satisfying mild conditions), rescale the walk's step size and time step together and, in the limit, it converges to Brownian motion, explaining why this one process appears so ubiquitously.

At a Glance

Originphysical observation (Brown, 1827); rigorous physical explanation (Einstein, 1905)

Incrementsindependent, Gaussian, with variance equal to the elapsed time

Pathscontinuous everywhere, but differentiable nowhere

Triple identitysimultaneously a Markov process, a martingale, and a Gaussian process

Last updated September 8, 2026← Back to the Dictionary