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.
The law of the process X is then defined to be the pushforward measure
- 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.)