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

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

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

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

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

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

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

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

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.