Why are Brownian paths continuous but not differentiable?

Over any small interval the increment is of the order of the square root of the interval length, which keeps the path continuous but makes the difference quotients blow up, so no derivative exists at any point.

How is Brownian motion related to the random walk?

Brownian motion is the scaling limit of a random walk: if a simple random walk is sped up in time and shrunk in space at matching rates, its trajectory converges to Brownian motion, as made precise by Donsker's invariance principle.

Brownian Motion

Brownian motion, or the Wiener process, is the continuous-time random walk that arises as the scaling limit of countless small independent steps; its paths are continuous everywhere yet differentiable nowhere.

Definition

Brownian motion is a real-valued stochastic process starting at zero with continuous sample paths whose increments over disjoint intervals are independent and normally distributed with mean zero and variance equal to the length of the interval.

Scope

The topic covers the defining properties of Brownian motion as a process with continuous paths, independent stationary increments, and Gaussian distributions, its existence via Wiener's construction and Donsker's invariance principle, the path properties of continuity, nowhere-differentiability, and quadratic variation, the strong Markov property and reflection principle, the law of the iterated logarithm, and the role of Brownian motion as a continuous martingale and a Gaussian process.

Core questions

What properties uniquely characterize Brownian motion among stochastic processes?
How is the existence of a process with continuous Brownian paths established?
What are the remarkable analytic properties of Brownian paths?
How does the reflection principle yield the distribution of the maximum and of hitting times?

Key concepts

Wiener process
independent Gaussian increments
nowhere-differentiable paths
quadratic variation
reflection principle

Key theories

Donsker's invariance principle: Suitably rescaled random walks converge in distribution to Brownian motion in the space of continuous paths, the functional central limit theorem that explains the universality of Brownian motion as the limit of summed small independent effects.
Path properties and the reflection principle: Brownian paths are almost surely continuous, nowhere differentiable, and of quadratic variation equal to elapsed time, and the reflection principle uses the strong Markov property to give the distributions of the running maximum and of first-passage times in closed form.

Clinical relevance

Brownian motion models the diffusion of particles in physics and chemistry, the noisy evolution of asset prices in the Black-Scholes theory of finance, thermal and electronic noise in engineering, and the random spread of pollutants or genes; it is also the building block from which more general diffusions and stochastic integrals are constructed.

History

Robert Brown observed the erratic motion of pollen grains in 1827, and Einstein and Smoluchowski explained it physically in 1905 and 1906. Norbert Wiener gave the rigorous mathematical construction in 1923, and Levy and others developed the detailed theory of its paths.

Key figures

Robert Brown
Albert Einstein
Norbert Wiener
Paul Levy

Seminal works

karatzas1991

Frequently asked questions

Why are Brownian paths continuous but not differentiable?: Over any small interval the increment is of the order of the square root of the interval length, which keeps the path continuous but makes the difference quotients blow up, so no derivative exists at any point.
How is Brownian motion related to the random walk?: Brownian motion is the scaling limit of a random walk: if a simple random walk is sped up in time and shrunk in space at matching rates, its trajectory converges to Brownian motion, as made precise by Donsker's invariance principle.