# “Universal” properties of Brownian motion

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.

• 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. – Nate Eldredge Jun 14 '20 at 16:08
• 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. – Yfmvch Jun 14 '20 at 18:09
• 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. – 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 ($$(B_{t_{i+1}}-B_{t_i})_{0\leq i are independent for $$t_0<\cdots) 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.

-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)