---
title: "Brownian motion and its defining properties"
description: "Standard Brownian motion starts at zero and has independent mean-zero normal increments with variance equal to elapsed time. Its paths are almost surely continuous and nowhere differentiable. Rescaled"
canonical: https://lightmysky.com/learn/mathematics/brownian-motion-and-its-defining-properties-mt_T5KtWIk07Z
source: https://lightmysky.com/learn/mathematics/brownian-motion-and-its-defining-properties-mt_T5KtWIk07Z.md
retrieved: 2026-11-12
---

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# Brownian motion and its defining properties

Standard Brownian motion starts at zero and has independent mean-zero normal increments with variance equal to elapsed time. Its paths are almost surely continuous and nowhere differentiable. Rescaled fair random walks converge to its path distribution.

Subject: Mathematics · Area: Probability · Ages 23 to 24
Page: https://lightmysky.com/learn/mathematics/brownian-motion-and-its-defining-properties-mt_T5KtWIk07Z

## Ready when they can

- State the defining properties and compute a covariance from them
- Describe the time and space scaling under which interpolated fair random walks converge in distribution to Brownian motion
- Distinguish path continuity from almost-sure nowhere differentiability and interpret the roughness suggested by increment scaling

## Lesson: Brownian increments and continuous paths

Write B_t for the value at time t. N(m,v) denotes a normal law with mean m and variance v. Standard Brownian motion has B_0=0 and continuous paths with probability 1. For 0<=s<t, its increment B_t-B_s has law N(0,t-s). Increments over any finite set of disjoint time intervals are mutually independent. In particular, E[B_t]=0 and Var(B_t)=t.

Your fair-walk lesson used forecasts given the past. Set F_s=sigma(B_u:0<=u<=s), the natural Brownian history sigma-algebra. For 0<=s<t, B_t-B_s is independent of F_s and has mean 0. Thus E[B_t-B_s | F_s]=0 and E[B_t | F_s]=B_s almost surely. Brownian motion is adapted to this filtration and integrable at each finite time, so it is a continuous-time martingale for this filtration.

**Example.** For s<=t, split B_t=B_s+(B_t-B_s). The later increment is independent of B_s, so its covariance with B_s is 0. Hence Cov(B_s,B_t)=Var(B_s)=s, giving Cov(B_s,B_t)=min(s,t). Covariance is linear in each argument: Cov(X-Y,Z)=Cov(X,Z)-Cov(Y,Z). For example, Cov(B_4,B_10)=4.

For two forward Brownian increments, covariance equals the length of their overlapping time intervals. Adjacent increments are independent even when their intervals share an endpoint: an endpoint has zero length. Positions are different. If 0<s<t, B_s and B_t share the earlier increment and have positive covariance s. They cannot be independent, since independent variables with finite second moments have covariance 0.

*(drawing: For X=B_9-B_2 and Y=B_12-B_7, the shared interval is 7 to 9. Thus Cov(X,Y)=2. The line shows time, not a sample path.)*

For finite second moments, E[X^2]=Var(X)+E[X]^2. Adding a constant changes the mean but not the variance. A variable with mean 2 and variance 5 therefore has second moment 9. Also, Var(aX+bY)=a^2 Var(X)+b^2 Var(Y)+2ab Cov(X,Y). The covariance term matters when X and Y are dependent. For example, Var(B_3+B_10)=3+10+2(3)=19.

If Z is standard normal, B_t has the same distribution as sqrt(t)Z. Multiplication by a multiplies variance by a^2; B_16 therefore has the law of 4Z. Scaling also applies to whole processes: for c>0, W_t=B_(ct)/sqrt(c) is standard Brownian motion in law. Its increment over length h has variance ch/c=h. Independence and continuity are preserved. Equal laws do not mean equal values on each sample path.

Let S_k sum k independent fair +1 or -1 steps. Set W_n(k/n)=S_k/sqrt(n), then interpolate linearly. Each step takes time 1/n and has size 1/sqrt(n). Since Var(S_n)=n, Var(W_n(1))=1. Donsker's theorem says these path laws converge to Brownian law on each finite interval, in continuous-path space with the uniform norm. This is convergence in distribution, not a guarantee that every simulated path converges.

Normal increment laws describe distributions at fixed times. Continuity is a separate requirement about whole paths. The Brownian definition requires one probability-one event on which the path is continuous at every time. Specifying only fixed-time increment laws does not require every version of that process to have continuous paths.

A further theorem says that, with probability 1, a Brownian path has no finite derivative anywhere, using the right derivative at time 0. Continuity does not give a velocity. An integral of a smooth finite velocity would be differentiable, so it cannot describe a Brownian path. Interpolated walks have slopes inside grid intervals, but convergence of their path distributions does not imply convergence of derivatives.

For h>0, B_(t+h)-B_t has standard deviation sqrt(h). Dividing this increment by h gives a difference quotient with standard deviation sqrt(h)/h=1/sqrt(h). This spread grows without bound as h tends to 0. It suggests why slopes remain unstable at smaller scales. This calculation alone is not a proof of almost-sure nowhere differentiability.

**Recap.** Brownian increments are centered normal with variance equal to elapsed time. Scaled walk laws converge to Brownian law. Brownian paths are almost surely continuous and nowhere differentiable.

*The lesson, practice questions, worked solutions and flashcards were written for this dataset.*

## Practice

18 questions on this page, each with its working shown.

## Needs first

- [Martingales and Optional Stopping](https://lightmysky.com/learn/mathematics/martingales-and-optional-stopping-mt_198uXX3jRJ)
- [Joint Distributions, Covariance and Independence](https://lightmysky.com/learn/mathematics/joint-distributions-covariance-and-independence-mt_YyTFTAqhSF)

## Optional connections

- [The Law of Large Numbers and the Central Limit Theorem](https://lightmysky.com/learn/mathematics/the-law-of-large-numbers-and-the-central-limit-theorem-mt_NknG3L5rQp)
- [Separation of Variables and the Heat Equation](https://lightmysky.com/learn/mathematics/separation-of-variables-and-the-heat-equation-mt_XnPDL0X1TL)

## Opens up

- [Fluctuations, Linear Response and the Fluctuation-Dissipation Relation](https://lightmysky.com/learn/science/fluctuations-linear-response-and-the-fluctuation-dissipation-relation-mt_Ibsvdyx7uP)