Wiener process
A continuous-time stochastic process with independent Gaussian increments.
The Wiener process, often called Brownian motion because of its link to the physical phenomenon, is a stochastic process that takes real values and runs continuously in time. It was named for mathematician Norbert Wiener and is a classic example of a Lévy process—a type of stochastic process with stationary, independent increments and paths that are right-continuous with left limits. This process appears across pure and applied mathematics, economics, quantitative finance, evolutionary biology, and physics.
In pure mathematics, the Wiener process sparked the study of continuous-time martingales and serves as a fundamental building block for describing more complex stochastic processes. It is central to stochastic calculus, diffusion processes, and potential theory, and it drives Schramm–Loewner evolution. In applied mathematics, it represents the integral of Gaussian white noise, making it a model for noise in electronics (Brownian noise), instrument errors in filtering, and disturbances in control theory.
Physicists use the Wiener process to model Brownian motion and other diffusion phenomena, often via the Fokker–Planck and Langevin equations that describe how random motion unfolds over time. It also supports the path integral formulation of quantum mechanics: through the Feynman–Kac formula, solutions to the Schrödinger equation can be expressed in terms of the Wiener process. In physical cosmology, it appears in models of eternal inflation. In finance, it is central to the Black–Scholes option pricing model.
**Definitions**
*Canonical definition* A Wiener process is a real-valued, continuous-time stochastic process with these properties: it starts at zero almost surely; it has independent increments, meaning that for any times \(0 \leq t_1 < t_2 < \dots < t_n\), the future increments are independent of past values; its increments are Gaussian, so for any \(0 \leq s < t\), the increment \(W_t - W_s\) is normally distributed with mean 0 and variance \(t-s\); and its paths are almost surely continuous. The probability law of this process on the space of continuous functions (with the Borel σ-algebra) is called the Wiener measure, and integrals with respect to it are Wiener integrals.
*Lévy characterization* A continuous local martingale \(X_t\) with \(X_0 = 0\) is a Wiener process if and only if its quadratic variation equals \(t\) (meaning \(X_t^2 - t\) is also a local martingale).
*Gaussian process characterization* The Wiener process can be defined as a Gaussian process with continuous paths such that for all \(s,t \geq 0\), the mean is zero and the covariance is \(\min(s,t)\). This uniquely determines its law on the space of continuous functions.
*Wiener representation* Wiener (1923) gave a representation of a Brownian path using a random Fourier series. If \(\xi_n\) are independent standard normal variables, then \[ W_t = \xi_0 t + \sqrt{2} \sum_{n=1}^\infty \xi_n \frac{\sin(n\pi t)}{n\pi} \] and \[ W_t = \sum_{n=0}^\infty \xi_n \frac{2\sqrt{2}}{(2n+1)\pi} \sin\left((2n+1)\pi t\right) \] represent Brownian motion on \([0,1]\). The scaled process \(\sqrt{c}W_{t/c}\) is a Brownian motion on \([0,c]\).
*White noise representation* In physics and engineering, Brownian motion is informally defined as the integral of white noise, where white noise \(\dot{W}_t\) has mean zero and covariance \(\mathbb{E}[\dot{W}_t \dot{W}_s] = \delta(t-s)\). However, Wiener process paths have unbounded variation and are not absolutely continuous, so this representation is not rigorous as an integral over a function. Mathematically, Gaussian white noise is defined as an isometry from square-integrable functions to centered Gaussian random variables, such that for any test function \(f\), the evaluation \(\langle \dot{W}, f \rangle\) is Gaussian with variance \(\int f^2 dt\). Using this, a pre-Brownian motion can be defined as \(B_t = \langle \dot{W}, \mathbf{1}_{[0,t]} \rangle\). Pre-Brownian motions may not have continuous paths, but Kolmogorov's continuity criterion guarantees a modification that does, yielding the Wiener process.
*Donsker's theorem* The Wiener process arises as the scaling limit of a random walk or other discrete-time processes with stationary independent increments—a result known as Donsker's theorem. Like a random walk, the Wiener process is recurrent in one or two dimensions (it returns almost surely to any fixed neighborhood of the origin infinitely often) but not in three or higher dimensions (where a multidimensional Wiener process has independent coordinates). For i.i.d. random variables \(\xi_i\) with mean 0 and variance 1, the step function \(S_n(t) = \frac{1}{\sqrt{n}} \sum_{i=1}^{\lfloor nt \rfloor} \xi_i\) converges in distribution to a Wiener process as \(n\) increases.
- field
- Mathematics
- known_for
- Wiener process (Brownian motion)
Lore & Background
The Wiener process is characterized by several properties: it starts at zero almost surely, has independent increments, and its increments are normally distributed with mean zero and variance equal to the time step. Its paths are almost surely continuous. The Wiener measure is the probability law of the Wiener process on the space of continuous functions with g(0)=0. An alternative characterization, the Lévy characterization, states that a continuous local martingale W with W0=0 is a Wiener process if and only if its quadratic variation is [W,W]t = t. The Wiener process can also be defined as a Gaussian process with zero mean and covariance equal to the minimum of the two times.
Reader's Guide
The Wiener process plays an important role in both pure and applied mathematics. In pure mathematics, it gave rise to the study of continuous time martingales and is a key process for describing more complicated stochastic processes. It is vital in stochastic calculus, diffusion processes, and potential theory, and is the driving process of Schramm–Loewner evolution. In applied mathematics, it represents the integral of a white noise Gaussian process and is useful as a model of noise in electronics engineering, instrument errors in filtering theory, and disturbances in control theory. In physics, it models Brownian motion and other diffusion through the Fokker–Planck and Langevin equations, and underpins the path integral formulation of quantum mechanics via the Feynman–Kac formula. It also appears in physical cosmology and is prominent in the mathematical theory of finance, particularly the Black–Scholes option pricing model.
Did You Know?
- The Wiener process is also called Brownian motion due to its historical connection with the physical process of the same name.
- It is one of the best known Lévy processes, which are càdlàg stochastic processes with stationary independent increments.
- The Wiener process has Gaussian increments: a time step u results in an increment normally distributed with mean 0 and variance u.
- Norbert Wiener gave a representation of a Brownian path in terms of a random Fourier series.
Origins and Independent Discovery
The Wiener process, also called the Brownian motion process, occupies a position of singular importance in probability theory, standing alongside the Poisson process as one of the two most fundamental stochastic models. What makes its intellectual history especially striking is that no single person can claim sole credit for its discovery. Louis Bachelier was the first to apply it to modeling price changes on the Paris Bourse, yet the underlying mathematics was rediscovered multiple times by different mathematicians, both preceding and following Bachelier's contribution, in varied contexts and across multiple nations. The Poisson process, which A. K. Erlang employed to describe the count of telephone calls arriving within a fixed interval, shares this identical pattern of repeated independent invention. Both processes are regarded as foundational pillars of the broader stochastic process theory. Their recurring independent emergence in different countries and eras suggests that the mathematics of continuous random motion and discrete random counting are not arbitrary constructions but natural, almost inevitable, descriptions of how randomness manifests in the physical and social world.
Mathematical Architecture and Classification
Formally, a stochastic process is a family of random variables in a probability space, with each member of the family uniquely associated with an element of an index set. Historically that index set has been a subset of the real line, lending the variables a temporal reading, while their values are drawn from a common state space—whether the integers, the real line, or n-dimensional Euclidean space. The Wiener process is one member of a broad taxonomy. Based on their mathematical properties, stochastic processes are sorted into families such as random walks, martingales, Markov processes, Lévy processes, Gaussian processes, random fields, renewal processes, and branching processes. A key dividing line is the cardinality of the index set: a finite or countable index set yields a discrete-time process, whereas an interval of the real line produces a continuous-time one. The latter class is notably harder to analyze because the index set is uncountable, demanding tools from real analysis, measure theory, Fourier analysis, and functional analysis in addition to probability, calculus, linear algebra, set theory, and topology. This breadth of required techniques makes the field one of the most mathematically demanding in modern research.
Reach Across the Sciences and Industry
The Wiener process and the wider family of stochastic processes have penetrated an extraordinary range of disciplines. In the life and physical sciences they describe how bacterial colonies expand, how thermal noise causes electrical currents to jitter, and how individual gas molecules drift unpredictably. In engineering and applied mathematics they underpin work in image processing, signal processing, control theory, and information theory. The digital and communication domains—computer science and telecommunications—rely on stochastic modeling as a core tool. In the social and economic sciences, the apparently erratic swings observed in financial markets have driven the heavy adoption of stochastic modeling, with Bachelier's original Bourse application serving as the historical seed. The theory of stochastic processes is regarded as a major contribution to mathematics and remains an active area of research, motivated by both pure theoretical questions and the constant stream of new real-world phenomena that call for fresh probabilistic models. In this sense, the Wiener process is less a static theorem than a living framework that keeps expanding as new applications demand it.
Etymology and the Language of Randomness
The vocabulary surrounding the Wiener process has a layered linguistic history. The adjective stochastic entered English with the meaning pertaining to conjecturing, rooted in a Greek word that conveys the idea of aiming at a target or making a guess. The Oxford English Dictionary traces its first English usage to 1662. In his 1713 Latin treatise on probability, Ars Conjectandi, Jakob Bernoulli employed the phrase Ars Conjectandi sive Stochastice, rendered in English as the art of conjecturing or stochastics. Decades later, Ladislaus Bortkiewicz picked up the term and, in a 1917 German-language work, used the word stochastik to mean random. The specific compound stochastic process entered English mathematical vocabulary through a 1934 publication by Joseph Doob. Related terminology further distinguishes contexts: a stochastic process may also be called a random function, reflecting its interpretation as a random element in a function space, while collections indexed by the Cartesian plane or higher-dimensional Euclidean spaces are typically termed random fields rather than processes. These naming conventions encode the geometry of the underlying index space.
More in Probability And Stochastic Processes 1-21
Spotted an error? Know more?
This is a living reference — every entry is fact-audited, and reader corrections feed straight into our audit queue. Suggest an edit · See this site's audit record