I've been learning some stochastic calculus (mostly through Oksendal) recently and while I understand the definition of Brownian motion given by Oksendal, I am curious if there are more "categorical" reasons we should be especially interested in this particular stochastic process above others.

Now when I say "categorical," I don't necessarily mean actual category-theoretic statements (although those are welcome), but more broadly properties of Brownian motion that intuitively justify its study to the exclusion of other processes.

As an example, when I first read the martingale representation theorem I thought it was a type of universal statement (until I read more carefully and realized that it of course applies only to martingales with respect to the filtration associated to Brownian motion).

Sorry if this is too vague, but with a background in algebra I get fidgety studying objects that feel somewhat arbitrarily selected.

  • 7
    $\begingroup$ Have you read about Donsker's theorem? It says roughly that Brownian motion is "universal" in the same sense that the central limit theorem says the normal distribution is universal: a large class of random objects converge to it in the scaling limit. $\endgroup$ – Nate Eldredge Jun 14 '20 at 16:08
  • $\begingroup$ Thank you! This is just the kind of statement I was looking for. I had seen it in the particular case of scaled simple random walk, but not in this generality. $\endgroup$ – Yfmvch Jun 14 '20 at 18:09
  • $\begingroup$ Another reason for Brownian motion on $\mathbb{R}^{n}$ being 'canonic' is that its intrinsic metric is the Euclidean distance. A fact for which analogues exist also on manifolds and infinite dimensional Hilbert spaces, keyword here: Varadhan short-time asymptotics. $\endgroup$ – Tobsn Jun 17 '20 at 13:20

The Brownian motion is, up to a scaling factor, the only continuous¹ process in $\mathbb R^d$ whose increments are stationary ($B_{t+dt}-B_t$ has the distribution of $B_{dt}$ for $t,dt\geq 0$), independent ($(B_{t_{i+1}}-B_{t_i})_{0\leq i<k}$ are independent for $t_0<\cdots<t_k$) and rotationally invariant ($RB_t$ has the distribution of $B_t$ for all rotations $R:\mathbb R^d\to\mathbb R^d$).

In fact, a continuous process whose increments are stationary independent has the form $t\mapsto AB_t+tv$, for a well-chosen $v\in\mathbb R^d$ and $A:\mathbb R^d\to\mathbb R^d$ linear. Then it is not difficult to see that the rotational invariance forces $v$ to be zero and $A$ a constant multiple of the identity.

I am not sure this particular result has a name, but it follows directly from the description of the Lévy processes. Similar results in Lie groups are known as Hunt's theorem.

¹ Here, by continuous process, I mean a process whose samples are almost surely continuous.


Donsker's theorem as pointed out in comments is helpful. Here's another helpful characterization.

Brownian motion is the centered continuous Gaussian process satisfying:

-Self similarity. I.e. $B_{ct}=c^H B(t)$ in distribution for some $H$.

-Stationary, independent increments. I.e. $B_t-B_s=B_{t-s}$ in distribution and are independent.

(Also normalized so $E[B_1^2]=1$ but that is for convenience)

You can build the covariance from this characterization.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.