I am currently reading the book "Brownian Motion, Martingales and Stochastic Calculus" by Jean-Francois Le Gall 2016.
In chapter 02, he constructs the brownian motion through the prebrownian motion and the continuitytheorem of kolmogorov.
I am struggling to understand:
- Why do we need continuity for the paths?
i.e. why isn't $B_0=0$ a.s. and independent, stationary increments $B_t-B_s \sim \mathcal{N}(0,t-s)$ for $t >s$ enough? (not in this book but in general) - What is the difference between the corollary 2.4 (density of a vector of prebrownian motion) and the wiener-measure?
See image for corollary
Thank you in advance.
Best regards.
Best Answer
Generally we can take random process as a random variable $\omega \mapsto X(\omega)$ that takes value in a set of functions, here $X(\omega)\in\mathbb{R}^{[0,\infty)}$ is the path given $\omega$. You can take $X(\omega)$ as an infinitely large vector containing values $X_{t}(\omega)$ for each $t\in[0,\infty)$. Under the condition of independent and normal increment, the best we can do is deriving the joint distribution of any finite collection of points $\bigl(X_{t_{1}}(\omega),X_{t_{2}}(\omega),\ldots,X_{t_{n}}(\omega)\bigr)$, as in Corollary 2.4, but it's still not enough to pin down the entire path $X(\omega)$, since it contains uncountably many points.
To make sure the process is well-defined, we make an additional assumption that $X$ has a continuous path almost surely. Now the space of (a version of) $X$ is restricted to $C_{\mathbb{R}}[0,\infty)$, the set of all $\mathbb{R}$-valued continuous functions. By Corollary 2.4, we can construct a process on positive dyadic rational time points $\mathbb{Q}_{2}\equiv\{k2^{-n}:k,n\in\mathbb{N}\}$, then Kolmogorov-Centsov criterion tells us this process has a unique continuous extension from $\mathbb{Q}_{2}$ to $[0,\infty)$, means the entire path $X(\omega)$ is now pinned down safely.
As for your second question, Wiener measure is defined on $\bigl(C_{\mathbb{R}}[0,\infty),\mathcal{B}({C_{\mathbb{R}[0,\infty)}})\bigr)$, where the distribution in Corollary 2.4 is on $\bigl(\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n})\bigr)$, it's a pushforward measure of Wiener measure under coordinate projection $X(\omega)\mapsto\bigl(X_{t_{1}}(\omega),X_{t_{2}}(\omega),\ldots,X_{t_{n}}(\omega)\bigr)$.