##  [Wiener Measure](/wiener-measure-0) 

 Definition

A probability measure on the space of continuous paths (usually starting at a specified point) under which the coordinate maps have Gaussian finite-dimensional marginals with independent increments; it is the canonical law of continuous stochastic motion with stationary, normally distributed increments.

 

 

 

 

 

 





## Principle

Principle

Constructed as the weak limit of scaled random-walk path measures or via projective consistency of finite-dimensional normal distributions; it endows path space with a probability that encodes continuous, nowhere-differentiable sample trajectories almost surely.

 

 

 

 

 





## Demonstration

Demonstration

On C([0,T], R^n) the Wiener measure assigns to cylinder sets the multivariate normal probabilities determined by covariance min(s,t) so that coordinate evaluation yields the classical continuous random motion process.

 

 

 

 

## Misapplication

Misapplication

Using Wiener measure to model processes that admit jumps or discontinuities; the measure concentrates on continuous paths and thus cannot represent jump dynamics.

 

 

 

 

 





## Consequence

Consequence

Wiener measure provides the rigorous probabilistic foundation for stochastic integration and differential equations driven by continuous random motion; many functional limit theorems use it as a universal scaling limit.

 

 

 

 

## Reversal

Reversal

Replacing the continuous-increment law by a measure with jump structure (for example, a compound Poisson law) yields path distributions with discontinuities and fundamentally different analytic properties.

 

 

 

 

 





## Boundary

Boundary

Defined on spaces of continuous functions with a chosen topology (uniform on compacts); it presupposes the Gaussian increment structure and does not apply when increments are non-Gaussian or heavy-tailed.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension appears between Wiener measure as a true probability measure on function space and heuristic path-integral objects that attempt a 'flat' uniform measure on infinite-dimensional spaces, which typically do not exist.

 

 

 

 

 





## Synthesis

Synthesis

Wiener measure is the canonical probability law on continuous path space that captures continuous random motion with independent, stationary normal increments and underpins stochastic calculus and scaling limits.