Let
- $(\Omega,\mathcal{A})$ be a measurable space
- $E$ be a Polish space and $\mathcal{E}$ be the Borel-$\sigma$-algebra on $E$
- $I\subseteq\mathbb{R}$
- $X_t$ be a random variable on $(\Omega,\mathcal{A})$ with values in $(E,\mathcal{E})$
Then, $X:=(X_t,t\in I)$ is called a stochastic process. I've seen that people talk about the distribution $\mathcal{L}[X]$ of a stochastic process $X$, but how is $\mathcal{L}[X]$ defined?
Let $E^I$ denote the set of all functions from $I$ to $E$.
Note that $X$ can be viewed as a mapping from $\Omega$ into $E^I$, with $[X(\omega)](t) = X_t(\omega)$.
Also, $X$ is measurable with respect to the $\sigma$-algebra $\mathcal{F}$ on $E^I$ which is generated by all of the projections $\pi_t: E^I\to E$, given by $\pi_t(f) = f(t)$.
Hence there is a pushforward measure $X_*P$ on $(E^I,\mathcal{F})$ defined by $(X_*P)(A) = P(X \in A)$, and this pushforward measure is usually called the $law$ (or distribution) of $X$. My best guess is that $\mathcal{L}[X]$ refers to this pushforward measure.
It can be proved that if $Y=(Y_t)_{t \in I}$ is another stochastic process on $E$ such that the finite-dimensional distributions of $Y$ agree with the finite-dimensional distributions of $X$, then $\mathcal{L}[X]=\mathcal{L}[Y]$.
http://en.wikipedia.org/wiki/Law_%28stochastic_processes%29