# Wiener process

The Wiener process, also called (standard) [Brownian motion](https://www.edgepedia.org/brownian-motion), is a continuous-time stochastic process \( W = \{W(t), t \ge 0\} \) that starts at zero, has stationary and independent increments, and whose increment over an interval of length \( t-s \) is normally distributed with mean zero and variance \( t-s \). It is the mathematical model of a particle's random displacement under molecular bombardment, and it is named after the mathematician [Norbert Wiener](https://www.edgepedia.org/norbert-wiener) (1894–1964), who gave the first rigorous construction of the process.<sup>[1](https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.110/lehre/ws15/Stochastik_II/ln_stochasticsII_chaps_1_to_4.pdf)</sup><sup> • </sup><sup>[2](https://galton.uchicago.edu/~lalley/Courses/383/BrownianMotion.pdf)</sup>

| Key fact | Detail |
|---|---|
| Defining properties | \( W(0) = 0 \) almost surely; stationary, independent increments; \( W(t) - W(s) \sim N(0, t-s) \) for \( 0 \le s < t \)<sup>[1](https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.110/lehre/ws15/Stochastik_II/ln_stochasticsII_chaps_1_to_4.pdf)</sup> |
| Gaussian characterization | A Gaussian process with \( E[W(t)] = 0 \) and \( \mathrm{cov}(W(s), W(t)) = \min\{s, t\} \); the two definitions are equivalent<sup>[1](https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.110/lehre/ws15/Stochastik_II/ln_stochasticsII_chaps_1_to_4.pdf)</sup> |
| Sample paths | Continuous with probability 1, but almost surely nowhere differentiable, with infinite total variation on every interval \( [0, T] \)<sup>[2](https://galton.uchicago.edu/~lalley/Courses/383/BrownianMotion.pdf)</sup><sup> • </sup><sup>[3](https://www.math.kth.se/matstat/gru/sf2940/lectnotemat5.pdf)</sup> |
| Quadratic variation | Over a partition of \( [0, T] \) into \( n \) equal parts, \( \sum_{i=0}^{n-1} (\Delta_i^n W)^2 \) converges in mean square to \( T \)<sup>[3](https://www.math.kth.se/matstat/gru/sf2940/lectnotemat5.pdf)</sup> |
| Random-walk limit | By Donsker's invariance principle (1951), a rescaled random walk converges weakly to the Wiener process in \( C[0,1] \), whatever the step distribution with mean 0 and variance 1<sup>[4](https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.110/mitarbeiter/spodarev/publications/scripts/Stochastics2_2018.pdf)</sup> |
| Rigorous construction | First proved to exist by N. Wiener in about 1920, as a probability measure on continuous functions<sup>[2](https://galton.uchicago.edu/~lalley/Courses/383/BrownianMotion.pdf)</sup><sup> • </sup><sup>[5](https://www.sciencedirect.com/topics/mathematics/brownian-motion)</sup> |
| Physical origin | Einstein's 1905 kinetic theory of Brownian movement, verified experimentally by Jean Perrin, who obtained Avogadro's number between \( 55 \times 10^{22} \) and \( 72 \times 10^{22} \)<sup>[6](https://www.nobelprize.org/prizes/physics/1926/perrin/lecture/)</sup> |

## Definition and equivalent characterizations

One standard definition lists four conditions: \( W(0) = 0 \); the sample paths \( t \mapsto W(t) \) are almost surely continuous; the process has stationary and independent increments; and \( W(t) - W(s) \in N(0, t-s) \) for \( t > s \).<sup>[3](https://www.math.kth.se/matstat/gru/sf2940/lectnotemat5.pdf)</sup> Stationary means the distribution of \( X_{t+s} - X_s \) is the same as that of \( X_t - X_0 \); independent means increments over disjoint intervals are mutually independent random variables.<sup>[2](https://galton.uchicago.edu/~lalley/Courses/383/BrownianMotion.pdf)</sup>

An equivalent characterization treats the process as a whole. A [Gaussian process](https://www.edgepedia.org/gaussian-process) with \( E[W(t)] = 0 \) and covariance function \( \mathrm{cov}(W(s), W(t)) = \min\{s, t\} \) is a Wiener process, and the two definitions imply each other. From the covariance form, \( \mathrm{Var}(W(t) - W(s)) = t - 2\min\{s,t\} + s = t - s \), and the increments come out uncorrelated, hence independent, because uncorrelated coordinates of a multivariate Gaussian vector are independent.<sup>[1](https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.110/lehre/ws15/Stochastik_II/ln_stochasticsII_chaps_1_to_4.pdf)</sup>

Existence itself requires proof. Kolmogorov's theorem guarantees a Gaussian process for any positive semidefinite covariance function, and \( C(s,t) = \min\{s,t\} \) is positive semidefinite; a version with continuous trajectories then exists by a path-regularity theorem, so continuity of the paths is usually assumed outright.<sup>[1](https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.110/lehre/ws15/Stochastik_II/ln_stochasticsII_chaps_1_to_4.pdf)</sup> Wiener's own theorem states that on any probability space supporting an infinite sequence of independent standard normal random variables, a standard Brownian motion exists.<sup>[2](https://galton.uchicago.edu/~lalley/Courses/383/BrownianMotion.pdf)</sup>

The definition extends to several dimensions: a \( d \)-dimensional Brownian motion is \( W(t) = (W_1(t), \ldots, W_d(t)) \) where the \( W_i \) are independent one-dimensional Wiener processes; the increment over an interval of length \( t \) then has the \( d \)-dimensional normal distribution with mean vector 0 and covariance matrix \( tI \).<sup>[1](https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.110/lehre/ws15/Stochastik_II/ln_stochasticsII_chaps_1_to_4.pdf)</sup><sup> • </sup><sup>[2](https://galton.uchicago.edu/~lalley/Courses/383/BrownianMotion.pdf)</sup>

## Sample paths

The paths are the source of the process's distinctive mathematics. Although \( t \mapsto W(t) \) is continuous with probability 1, it is almost surely non-differentiable at every \( t \ge 0 \).<sup>[3](https://www.math.kth.se/matstat/gru/sf2940/lectnotemat5.pdf)</sup> The normal distribution supplies exactly the degree of fluctuation that permits continuity; other increment distributions would give discontinuous paths or infinite variance.<sup>[7](https://kfoster.ccny.cuny.edu/classes/spring2010/eco275/lecturenotes7.html)</sup>

**Quadratic variation.** For the partition of \( [0,T] \) into \( n \) equal parts with increments \( \Delta_i^n W = W(t_{i+1}^{(n)}) - W(t_i^{(n)}) \), the sum of squared increments converges in mean square to \( T \). This nonzero quadratic variation is what forces the paths to be rough: the same argument shows the total variation of the paths on any \( [0,T] \) is infinite, since \( \sum |\Delta_i^n W| \ge (\sum (\Delta_i^n W)^2) / \max_i |\Delta_i^n W| \), and the denominator tends to zero while the numerator tends to \( T \).<sup>[3](https://www.math.kth.se/matstat/gru/sf2940/lectnotemat5.pdf)</sup> Infinite path length on every finite interval is the geometric counterpart of self-similarity.<sup>[3](https://www.math.kth.se/matstat/gru/sf2940/lectnotemat5.pdf)</sup>

**Scaling invariances.** If \( W \) is a standard Brownian motion, then so are each of \( \{-W(t)\} \), \( \{W(t+s) - W(s)\} \) for fixed \( s \), \( \{aW(t/a^2)\} \), and \( \{tW(1/t)\} \). The third of these expresses self-similarity: magnifying the process in time by a factor \( a^2 \) and in space by \( a \) reproduces the same process, which is why a zoomed-in Brownian path looks statistically identical to the original.<sup>[2](https://galton.uchicago.edu/~lalley/Courses/383/BrownianMotion.pdf)</sup><sup> • </sup><sup>[3](https://www.math.kth.se/matstat/gru/sf2940/lectnotemat5.pdf)</sup> The second property says the process continually starts afresh: for each fixed \( s \), the shifted process \( b(t+s) - b(s) \) has the same joint distributions as \( b \) and is independent of \( b(s) \).<sup>[8](https://www.sciencedirect.com/topics/mathematics/wiener-measure)</sup>

## Random-walk limit and construction

The Wiener process arises as the limit of a symmetric random walk observed at ever finer scales. Taking steps of size \( \Delta x = \sigma\sqrt{\Delta t} \) every \( \Delta t \) time units, left or right with equal probability, gives \( \mathrm{Var}[X(t)] = \sigma^2 t \) and \( E[X(t)] = 0 \); as \( \Delta t \to 0 \), the central limit theorem makes \( X(t) \) normal with mean 0 and variance \( \sigma^2 t \), and the random-walk independence carries over as independent, stationary increments.<sup>[9](https://www.biostat.umn.edu/~baolin/teaching/probmods/ipm-ch10.html)</sup>

Donsker's invariance principle (1951) makes this precise: the distributions of the interpolated, rescaled random walks \( \widetilde{W}^{(n)} \) converge weakly to the distribution of \( W \) in the space \( C[0,1] \) of continuous functions, and the limit does not depend on the step distribution beyond mean 0 and variance 1.<sup>[4](https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.110/mitarbeiter/spodarev/publications/scripts/Stochastics2_2018.pdf)</sup> The limiting probability measure on path space is called the Wiener measure.<sup>[10](https://www-personal.umd.umich.edu/~tiananw/426Notes.pdf)</sup>

## Historical setting

The physical phenomenon came first. Einstein founded a kinetic theory of Brownian movement in 1905, deriving the diffusion coefficient \( D = (RT/N)(6\pi a z)^{-1} \) for spheres of radius \( a \) in a liquid of viscosity \( z \), and [Jean Baptiste Perrin](https://www.edgepedia.org/jean-baptiste-perrin)'s measurements of suspended grains confirmed the theory and yielded determinations of Avogadro's number; Perrin received the 1926 [Nobel Prize in Physics](https://www.edgepedia.org/nobel-prize-in-physics) for this work.<sup>[6](https://www.nobelprize.org/prizes/physics/1926/perrin/lecture/)</sup><sup> • </sup><sup>[11](https://www.nobelprize.org/prizes/themes/the-nobel-prize-in-physics-1901-2000/)</sup> Einstein and Smoluchowski had treated what happens to particles at given times or in aggregate, but not the mathematical properties of the curve traced by a single particle; Wiener, to whom the topic was proposed by [Bertrand Russell](https://www.edgepedia.org/bertrand-russell), constructed a probability distribution on a suitable sigma-field of continuous functions, and his Brownian motion theory was not widely appreciated at the time.<sup>[5](https://www.sciencedirect.com/topics/mathematics/brownian-motion)</sup><sup> • </sup><sup>[12](https://nap.nationalacademies.org/skim.php?chap=388-437&record_id=2037)</sup>

## Place among stochastic processes and applications

The Wiener process is the archetype of Gaussian processes, of continuous-time martingales, and of Markov processes.<sup>[13](https://www.cambridge.org/core/books/abs/stochastic-processes/brownian-motion/38BC96068F991DC120982463317B48C7)</sup> It is also a [Lévy process](https://www.edgepedia.org/levy-process), meaning a process with stationary, independent increments, and right-continuous paths, and a theorem of Lévy states that, up to rescaling and addition of linear drift, it is the only Lévy process with continuous sample paths.<sup>[2](https://galton.uchicago.edu/~lalley/Courses/383/BrownianMotion.pdf)</sup>

Its transition density is the Gauss kernel,

\[ P(W_{t+s} \in dy \mid W_s = x) = p_t(x,y)\,dy = \frac{1}{\sqrt{2\pi t}} \exp\{-(y-x)^2/2t\}\,dy, \]

which ties Brownian motion to the heat and diffusion equations; the Black–Scholes partial differential equation of mathematical finance is the multiplicative version of the heat equation.<sup>[2](https://galton.uchicago.edu/~lalley/Courses/383/BrownianMotion.pdf)</sup><sup> • </sup><sup>[14](https://www.fwf.ac.at/en/research-radar/10.55776/P25815)</sup> Because ordinary calculus fails on nowhere-differentiable paths, integration against the Wiener process requires the Itô stochastic integral, whose chain rule carries a correction term coming from the nonzero quadratic variation.<sup>[15](https://www.statslab.cam.ac.uk/~mike/AFM/notes.pdf)</sup>

The process also serves as the base model that generalizations modify. The Wiener process, geometric Brownian motion, fractional Brownian motion, and the [Ornstein–Uhlenbeck process](https://www.edgepedia.org/ornstein-uhlenbeck-process) are commonly used to characterize randomness, diffusion, and mean-reverting behavior in finance, physics, biology, mathematics, and engineering.<sup>[16](https://www.frontiersin.org/journals/computer-science/articles/10.3389/fcomp.2025.1649256/full)</sup>

## References

1. [Stochastics II, Chapter 3: Wiener Process, University of Ulm lecture notes](https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.110/lehre/ws15/Stochastik_II/ln_stochasticsII_chaps_1_to_4.pdf)
2. [S. R. S. Varadhan-style course notes: Introduction to Brownian Motion, University of Chicago (Steven Lalley)](https://galton.uchicago.edu/~lalley/Courses/383/BrownianMotion.pdf)
3. [Lecture notes on the Wiener Process, KTH Royal Institute of Technology (sf2940)](https://www.math.kth.se/matstat/gru/sf2940/lectnotemat5.pdf)
4. [Stochastics II (2018 edition), University of Ulm, Spodarev](https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.110/mitarbeiter/spodarev/publications/scripts/Stochastics2_2018.pdf)
5. [Brownian Motion – an overview, ScienceDirect Topics](https://www.sciencedirect.com/topics/mathematics/brownian-motion)
6. [Jean Baptiste Perrin – Nobel Lecture, NobelPrize.org](https://www.nobelprize.org/prizes/physics/1926/perrin/lecture/)
7. [Lecture Notes 7, Wiener Processes and Itô's Lemma, CCNY (K. Foster)](https://kfoster.ccny.cuny.edu/classes/spring2010/eco275/lecturenotes7.html)
8. [Wiener Measure – an overview, ScienceDirect Topics](https://www.sciencedirect.com/topics/mathematics/wiener-measure)
9. [Gaussian process and Brownian motion, lecture notes, University of Minnesota (B. Laolin)](https://www.biostat.umn.edu/~baolin/teaching/probmods/ipm-ch10.html)
10. [Stochastic Processes 7, lecture notes, University of Michigan–Dearborn](https://www-personal.umd.umich.edu/~tiananw/426Notes.pdf)
11. [The Nobel Prize in Physics 1901–2000, NobelPrize.org](https://www.nobelprize.org/prizes/themes/the-nobel-prize-in-physics-1901-2000/)
12. [Biographical memoir of Norbert Wiener, National Academies Press](https://nap.nationalacademies.org/skim.php?chap=388-437&record_id=2037)
13. [Brownian motion, Chapter 2 of Stochastic Processes, Cambridge Core](https://www.cambridge.org/core/books/abs/stochastic-processes/brownian-motion/38BC96068F991DC120982463317B48C7)
14. [Research project description, Austrian Science Fund (FWF), P25815](https://www.fwf.ac.at/en/research-radar/10.55776/P25815)
15. [Itô's processes and quadratic variation, lecture notes, University of Cambridge Statistical Laboratory (M. Rogers)](https://www.statslab.cam.ac.uk/~mike/AFM/notes.pdf)
16. [Brownian motion models: cryptographic applications, capabilities, and limitations, Frontiers in Computer Science (2025)](https://www.frontiersin.org/journals/computer-science/articles/10.3389/fcomp.2025.1649256/full)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Stochastic processes › Continuous-time and continuous-state processes › Gaussian and Wiener processes › Wiener process and Brownian motion*

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