# Law (stochastic processes)

Law (stochastic processes)

In mathematics, the law of a stochastic process is the measure that the process induces on the collection of functions from the index set into the state space. The law encodes a lot of information about the process; in the case of a random walk, for example, the law is the probability distribution of the possible trajectories of the walk.

Definition

Let (&Omega;, "F", P) be a probability space, "T" some index set, and ("S", &Sigma;) a measurable space. Let "X" : "T" &times; &Omega; &rarr; "S" be a stochastic process (so the map

:$X_\left\{t\right\} : Omega o S : omega mapsto X \left(t, omega\right)$

is a ("F", &Sigma;)-measurable function for each "t" &isin; "T"). Let "S""T" denote the collection of all functions from "T" into "S" (see remark below). The process "X" induces a function &Phi;"X" : &Omega; &rarr; "S""T", where

:$left\left( Phi_\left\{X\right\} \left(omega\right) ight\right) \left(t\right) := X_\left\{t\right\} \left(omega\right).$

The law of the process "X" is then defined to be the pushforward measure

:$mathcal\left\{L\right\}_\left\{X\right\} := left\left( Phi_\left\{X\right\} ight\right)_\left\{*\right\} \left( mathbf\left\{P\right\} \right)$

on "S""T".

(Cautious readers may wonder for a moment if "S""T" really is a set. Abstractly, a function "T" &rarr; "S" is a certain type of subset of the Cartesian product "T" &times; "S", so the collection of all functions "T" &rarr; "S" is just a collection of certain elements of the power set of "T" &times; "S", and so is definitely a set.)

Example

* The law of standard Brownian motion is classical Wiener measure. (Indeed, many authors define Brownian motion to be a sample continuous process starting at the origin whose law is Wiener measure, and then proceed to derive the independence of increments and other properties from this definition; other authors prefer to work in the opposite direction.)

ee also

* Finite-dimensional distribution

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