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.
What a learner can do afterwards
- 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
1 · Read
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.
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.
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.
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.
The reading notes, extra questions, worked solutions and flashcards were written for this dataset. Video references use Khan Academy catalog records for normal-distribution background; the linked videos retain their creators' terms.
More to read
Changes and positions are different
A Brownian position B_t includes all the change since time 0. An increment B_t-B_s includes only the change after time s. For standard Brownian motion, that increment has a normal distribution with mean 0 and variance t-s.
Changes on successive time intervals are independent. Sharing the endpoint between two intervals does not create shared randomness in their increments. Positions at two positive times are different: both include the earlier part of the motion. Their covariance is the earlier time, so they are not independent.
For two forward increments, break their intervals into shared and unshared pieces. The unshared pieces are independent. The shared piece contributes its variance, which is its duration. This gives the overlap-length rule for increment covariance.
Independence from the whole observed past also explains the martingale forecast. The future increment has conditional mean 0. Adding it to the observed current position leaves that position as the conditional mean of the future position.
Scaling values and scaling a process
A fixed-time Brownian value has law N(0,t). To express it using a standard normal variable, multiply that variable by sqrt(t). The square root appears because multiplying a random variable by a multiplies its variance by a^2.
The process scaling rule changes every time at once. For c>0, define W_t=B_(ct)/sqrt(c). During a W interval of length h, the original Brownian motion runs for ch time units. Its increment variance is ch. Division of positions by sqrt(c) divides that variance by c, leaving h. The zero starting point, independent increments and continuity are preserved, so W is standard Brownian motion in law.
Matching laws does not identify values path by path. It describes the distribution of the transformed process.
A related diffusion equation uses a different normalization from the unit-diffusivity heat equation. On the real line, the standard Brownian density satisfies p_t=(1/2)p_xx. This is an optional connection, not the fixed-end rod problem from separation of variables.
What converges in the walk limit?
Start with independent fair steps taking values +1 and -1. Their sum after n steps has mean 0 and variance n. Dividing positions by sqrt(n) keeps the variance at 1 after those n steps. Giving each step duration 1/n places that endpoint at time 1.
Join consecutive grid positions with straight segments. Donsker's theorem concerns the distributions of these continuous interpolated paths on a finite time interval. They converge in distribution to Brownian law in continuous-path space equipped with the uniform norm. It does not say that arbitrary individual simulations converge to one particular Brownian path.
The fixed-time central limit theorem explains only part of this result. A theorem about a whole path distribution contains more information than a theorem about one endpoint. The law of large numbers is different again: it concerns averages approaching a population mean.
Although each interpolated walk has slopes inside its grid intervals, Brownian paths are almost surely nowhere differentiable. Taking a path-distribution limit does not permit taking a limit of derivatives. Small Brownian increments have standard deviation sqrt(h), but their difference quotients have standard deviation 1/sqrt(h). That calculation suggests roughness; the nowhere-differentiability conclusion requires a separate theorem.
Explore more
2 · Watch
به انگلیسیTake it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
18 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.