Uniqueness of Brownian motion

Solution 1:

The question you're asking is interesting in its own right, but is usually not considered a question of importance in probability theory. Generally, uniqueness is important, and a statement such as "is Brownian motion unique?" is asked, but not in the way you describe. Instead, the question of uniqueness of Brownian motion is asked in the following manner:

Consider the measurable space $(C,\mathcal{C})$, where $C$ is the set of continuous functions from $[0,\infty)$ to $\mathbb{R}$ and $\mathcal{C}$ is the $\sigma$-algebra induced by the coordinate projections. If $P$ and $Q$ are two probability measures both satisfying the requirements to be the distribution of a Brownian motion, is $P=Q$?

In this sense, uniqueness is about distributions and not mappings or random variables. Also, the question of uniqueness is rather trivial compared to the question of existence: Since the $P$ and $Q$ measures above have the same finite-dimensional distributions, they are equal on a generating system for $\mathcal{C}$ which is stable under intersections, and therefore standard uniqueness results for probability measures yields $P = Q$.

Also, in probability theory, usually the main result referred to when considering existence of the Brownian motion is the existence of a probability measure on $(C,\mathcal{C})$ satisfying the requirements to be a Brownian motion, and not an actual mapping. If an actual stochastic process is required, one can just use the identity mapping on $(C,\mathcal{C},P)$. These are somewhat abstract considerations, but are useful when making precise what one is constructing. See also the books by Rogers & Williams, in particular Chapter II of Volume I, for more on this.