Remarkable fact about Brownian Motion #1: It exists.

A Brownian Motion in one dimension is a stochastic process $B_t$ adapted to $(\mathcal{F}_t)$ , defined on some probability space, which is almost surely continuous, and has the properties that $B_0=0$ a.s. and for every $0\leq s\leq t, B_t-B_s\sim N(0,t-s)$ and is independent of $\mathcal{F}_s$ . So it has independent increments.

It can be shown that Brownian motion is also surely differentiable nowhere, and is invariant under suitable time-space rescaling. It is therefore not obvious that a probability space with rich enough structure exists to construct such a process.

Theorem (Wiener): There exists a Brownian motion on some probability space.

Proof: We will construct Brownian motion on D, the dyadic rationals in [0,1], then develop the machinery which will enable us to conclude that taking a limit onto the reals retains all the properties we need. The Strong Markov property allows us to extend this to the real line by taking countably many copies, and d independent copies will give BM on $\mathbb{R}^d$ .

We make the following observation. Given independent $X_1,X_2\sim N(0,1)$ , it is a simple check that conditional on $(X_1|X_1+X_2=a)\sim N(\frac{a}{2},\frac{1}{2})$ . From this, it is clear how our construction on D will proceed. Take a family of independent N(0,1) RVs, one for each dyadic rational. Rescale these appropriately to construct the BM on $D_{n+1}$ from the values on $D_n$ . Can check the covariance of the new finer increments, and it is clear that these are independent, provided the original increments were independent.

We need continuity. The precise result needed will be dealt with at the end, but it is easy to check that $B_d$ has dyadic increments bounded as $\mathbb{E}|B_t-B_s|^p=|t-s|^{p/2}\mathbb{E}|N|^p$ . The Kolmogorov criterion thus gives that $D\ni t\mapsto B_t(\omega)$ is Holder continuous for any exponent in (0,1/2). Then define: $B_t=\lim_{D\ni s\downarrow t} B_s, t\in[0,1]$ , which will also have the a.s. Holder property.

Need to check increments property. Given some increments $t_0<t_1<\ldots<t_k$ , approximate from above by dyadic $t_i^n\rightarrow t_i$ . Then $(B_{t_0}^n,\ldots, B_{t_k}^n)\stackrel{\text{a.s.}}{\rightarrow} (B_{t_0},\ldots, B_{t_k})$ , so the joint distributions converge also. Using Levy then gives both the Gaussian property and the independence in one go.

Theorem (Kolmogorov’s Criterion): If there exist $p,\epsilon>0$ such that: $\mathbb{E}|X_t-X_s|^p\leq C|t-s|^{1+\epsilon}\; \forall s,t\in D$ then for every $\alpha\in (0,\frac{\epsilon}{p})$ , X is $\alpha$ -Holder continuous almost surely.

Proof: By Markov and BC, can deduce from the condition the existence of a RV M such that almost surely: $\sup_{n}\max_k 2^{-n\alpha}|X_{k2^{-n}}-X_{(k+1)2^{-n}}|\leq M<\infty$ . Now given dyadic s,t, there is a unique dyadic rational with smallest denominator, say $2^{r}$ , between them. Can then express the difference t-s as a sum of dyadic reciprocals with denominators greater than $2^{(r+1)}$, where each denominator occurs at most twice. So can write: $|X_t-X_s|\leq 2\sum_{n\geq r+1}M2^{-n\alpha}=2M\cdot\frac{2^{-(r+1)\alpha}}{1-2^{-alpha}}$ . Then $M<\infty$ a.s. gives the Holder criterion.

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Remarkable fact about Brownian Motion #1: It exists.

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