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Wiener Process

The formal name, used especially in stochastic calculus, for the same mathematical process known as Brownian motion, emphasizing Norbert Wiener's 1923 rigorous construction and its role as the driving noise term in stochastic differential equations.

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In Plain English

Wiener process and Brownian motion refer to exactly the same object, most authors use the names interchangeably, and this dictionary distinguishes them only by which framing is being emphasized. Brownian motion carries the physical, historical story of jittering pollen grains; Wiener process is the name typically used once the object shows up inside the machinery of stochastic calculus, as the fundamental source of randomness driving a stochastic differential equation, denoted $d W_{t}$ . Norbert Wiener's 1923 paper was the first to construct the process with full mathematical rigor (proving it actually exists as a well-defined probability object, not just an idealized physical description), which is why his name attaches to the formal, calculus-ready version.

Definition

The Wiener process ${W_{t}}_{t \geq 0}$ is defined by the same four properties as standard Brownian motion: $W_{0} = 0$ , independent increments, $W_{t} - W_{s} \sim N (0, t - s)$ , and continuous paths. What differs across the two names is emphasis and context rather than mathematics: 'Wiener process' is the standard term inside stochastic calculus and stochastic differential equations, where its infinitesimal increment $d W_{t}$ (formally, $W_{t + d t} - W_{t}$ , with $E [d W_{t}] = 0$ and $Var (d W_{t}) = d t$ ) serves as the driving noise term, e.g. in $d X_{t} = μ (X_{t}, t) d t + σ (X_{t}, t) d W_{t}$ . The Ito integral $\int_{0}^{t} f (s) d W_{s}$ , defined with respect to the Wiener process, is the foundation on which the entire theory of stochastic integration and Ito's lemma is built.

Properties

The choice between 'Wiener process' and 'Brownian motion' in a given text is almost always a matter of disciplinary convention rather than mathematical substance, physics and applied science papers lean toward 'Brownian motion,' while probability theory and stochastic calculus texts lean toward 'Wiener process.'
Wiener's 1923 construction (via what is now called Wiener measure, a probability measure on the space of continuous paths) resolved a genuine mathematical gap: earlier physical descriptions assumed such a process existed without proving it, Wiener supplied the first fully rigorous existence proof.
Every stochastic differential equation in modern quantitative finance, physics, and engineering is, at its foundation, built on integration against the Wiener process, making it arguably the single most consequential building block in the entire theory of continuous-time stochastic processes.

At a Glance

Relationship to Brownian motionthe identical mathematical object; names are used interchangeably

Rigorous constructionNorbert Wiener, 1923, the first fully rigorous existence proof

Typical usage contextstochastic calculus, SDEs, as the driving noise term

d W_{t}

Foundational rolethe Ito integral and Ito's lemma are defined with respect to it

Last updated September 8, 2026← Back to the Dictionary