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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.

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    $\begingroup$ 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, 2020 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, 2020 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, 2020 at 13:20

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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.

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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.

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