Wiener Process

A Wiener process is a real-valued, continuous-time stochastic process that starts at zero, has continuous sample paths, and has independent, normally distributed increments whose variance equals the elapsed time. Also called standard Brownian motion, it supplies a precise mathematical model for irregular motion and a fundamental source of randomness in stochastic analysis. Its name honors Norbert Wiener, who established a rigorous mathematical foundation for the process in the 1920s. The mathematical object is distinguished from the physical phenomenon it idealizes. (stat.uchicago.edu)

Definition and distributions

On a probability space $(\Omega,\mathcal F,\mathbb P)$, a standard Wiener process is a family of random variables $(W_t)_{t\geq0}$ satisfying:

1. $W_0=0$ with probability one.

2. Its sample paths $t\mapsto W_t(\omega)$ are continuous almost surely.

3. Increments over disjoint time intervals exhibit statistical independence.

4. For $0\leq s<t$,

   $$W_t-W_s\sim\mathcal N(0,t-s).$$

Here $\mathcal N(m,v)$ denotes a normal distribution with mean $m$ and variance $v$. Thus, the increment distribution depends only on the interval’s length, not its starting time: the increments are stationary. Continuity is a separate requirement; specifying Gaussian distributions at individual times alone does not define a Wiener process. (stat.uchicago.edu)

At each time $t$, the expected value is zero and the variance is $t$. Its covariance function is

More generally, every finite collection of process values has a multivariate normal distribution. Consequently, the Wiener process is a Gaussian process completely characterized by its zero mean and covariance function, together with the continuous-path requirement. Values at different times are generally dependent, despite the independence of disjoint increments. (stat238.berkeley.edu)

Scaling, memory, and conditioning

For every $c>0$, the rescaled process

has the same law as $W_t$. This Brownian scaling expresses the square-root relationship between elapsed time and typical displacement. Reflection also preserves the law: $-W_t$ is a standard Wiener process. For a fixed $s$, the shifted process $W_{s+t}-W_s$ is another standard Wiener process, independent of the history through time $s$. (stat.uchicago.edu)

The natural filtration $\mathcal F_t=\sigma(W_r:0\leq r\leq t)$ records that history. Relative to it, the process has the Markov property: conditional future behavior depends on the present value rather than the full past. It is also a martingale, satisfying

This identity uses conditional expectation to express the absence of predictable drift. With the usual augmentation of its natural filtration, Brownian motion also has the strong Markov property, extending the restart principle to suitable random stopping times. (stat.uchicago.edu)

Sample-path irregularity

Although each typical path is a continuous function, it is nowhere differentiable and has infinite total variation on every nondegenerate bounded interval. Ordinary velocity therefore does not exist for a Wiener path. On compact intervals, paths are almost surely Hölder continuous of every exponent strictly below $1/2$, providing a quantitative description of their roughness. (wias-berlin.de)

Their quadratic variation, however, is finite. For deterministic partitions $0=t_0<\cdots<t_n=t$ whose maximum interval length tends to zero,

in mean square, and hence in probability. Along dyadic partitions, convergence also holds almost surely. This is conventionally written $[W]_t=t$. Nonzero quadratic variation distinguishes Wiener paths from continuously differentiable paths and is central to stochastic integration. (ocw.mit.edu)

Construction and random-walk limits

One construction specifies consistent finite-dimensional Gaussian distributions and then establishes the existence of a continuous version. The resulting probability law on continuous paths starting at zero is called Wiener measure. It describes probabilities of entire trajectories, rather than merely positions at selected times. (wias-berlin.de)

The process also arises as a scaling limit of a random walk. If independent, identically distributed steps have mean zero and finite positive variance $\sigma^2$, form their partial sums $S_k$. Linearly interpolate the values $S_k/(\sigma\sqrt n)$ at times $k/n$. Donsker’s theorem states that these random paths converge in distribution to a standard Wiener process. This functional extension of the central limit theorem concerns whole paths, not just a single normalized sum. (math.cmu.edu)

Stochastic calculus and diffusion

Because Wiener paths have infinite variation, integration against them requires a framework beyond ordinary pathwise integration. The Itô integral $\int_0^t H_s\,dW_s$ is constructed for suitable nonanticipating integrands. For a twice continuously differentiable function $f$, Itô’s formula gives

The second-order term reflects quadratic variation and modifies the ordinary chain rule. Wiener processes consequently serve as drivers of stochastic differential equations, separating deterministic drift from random fluctuations. (live.ocw.mit.edu)

Their connection to diffusion is expressed by the transition density

It satisfies the heat equation $\partial_t p=\tfrac12\partial_{yy}p$. Thus probabilistic evolution under Brownian motion corresponds to deterministic evolution of a density under a differential equation. (stat.uchicago.edu)

For numerical simulation on a finite grid, successive values can be generated using

where the $Z_k$ are independent standard normal variables. These grid values have the exact Wiener finite-dimensional distribution; linear interpolation between them is only an approximation to the continuous random path. (stat238.berkeley.edu)

References

1. Brownian Motionstat.uchicago.edu
2. Wiener’s Construction of the Brownian Motionlink.springer.com
3. STAT 238 — Bayesian Statistics Lecture Twenty Onestat238.berkeley.edu
4. Stochastic Calculusmath.cmu.edu
5. Brownian Motion and Itô Calculuswias-berlin.de
6. Lecture 8: Quadratic Variationocw.mit.edu
7. Lecture 17: Ito Process and Formulalive.ocw.mit.edu