# 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 (Ω, *F*, **P**) be a probability space, *T* some index set, and (*S*, Σ) a measurable space. Let *X* : *T* × Ω → *S* be a stochastic process (so the map

- [math]\displaystyle{ X_{t} : \Omega \to S : \omega \mapsto X (t, \omega) }[/math]

is an (*S*, Σ)-measurable function for each *t* ∈ *T*). Let *S*^{T} denote the collection of all functions from *T* into *S*. The process *X* (by way of currying) induces a function Φ_{X} : Ω → *S*^{T}, where

- [math]\displaystyle{ \left( \Phi_{X} (\omega) \right) (t) := X_{t} (\omega). }[/math]

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

- [math]\displaystyle{ \mathcal{L}_{X} := \left( \Phi_{X} \right)_{*} ( \mathbf{P} ) = \mathbf P(\Phi_X^{-1}[\cdot]) }[/math]

on *S*^{T}.

## 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.)

## See also

Original source: https://en.wikipedia.org/wiki/Law (stochastic processes).
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