What is the typical probability space when we talk about a Brownian motion?

Question

What is the typical probability space $(\Omega, \mathcal{F}, P)$ when we talk about a Brownian motion $(B_t)$?

I found some papers used $\Omega = C[0,\infty), \mathcal{F}=\mathcal{B}(C[0,\infty))$, and $P$ being the Wiener measure. The process is then defined by the coordinate mapping process $B(t,\omega) := \omega(t).$

However, in another post, the $\sigma$- algebra $\mathcal{F}$ is taken to be the smallest one for which the projection $\omega \mapsto \omega(t)$ is measurable for each $t$. Moreover, rather than having one probability measue on $(\Omega,F)$ we now have a family $(P^x)$ of probability measures indexed by $x \in \mathbb{R}$. The probability measure is the distribution of $x+B(.),$ where $B$ is standard Brownian motion.

Which one is more typical to use? Are they essentially the same thing?

Answer 1

After some careful reading of Brownian Motion and Stochastic Calculus, I think I have found a comprehensive answer to my question. The standard protocol is:

(1) We first construct a standard Brownian motion on the "canonical'' probability space $(C[0,\infty),\mathcal{B}(C[0,\infty),P_*)$, where $P_*$ is the Wiener measure, using the coordinating mapping process $B_t(\omega):=\omega(t)$ and its natual filtration $\{F^B_t\}$. See Chap. 2.4 of the book.

(2) Then, we can extend the construction by stating that any process, on any probability space, which has state space $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ and the same finite-dimensional distribution as $B$, is a standard Brownian motion (see pg. 3 of the book). We can also extend the filtration to any $\{F_t\}$ that is strictly larger than $\{F^B_t\}$ and satsifies $B_t-B_s \perp \mathcal{F}_s$, e.g., the augmented filtration and the universal filtration (see pg. 48, pg. 92 of the book).

(3) Finally, to construct a Brownian motion with initial distribution $\mu$, i.e., we now require $B_0 \sim \mu$ instead of $B_0=0$ a.s., we can use the protocol stated in this post. This is detailed in pg. 72 of the book.