# Wiener process

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2020 Mathematics Subject Classification: Primary: 60J65 [MSN][ZBL]

A homogeneous Gaussian process $X( t)$ with independent increments. A Wiener process serves as one of the models of Brownian motion. A simple transformation will convert a Wiener process into the "standard" Wiener process $X( t)$, $t \geq 0$, for which

$$X ( 0) = 0,\ \ {\mathsf E} ( X ( t) - X ( s)) = 0,$$

$${\mathsf D} [ X ( t) - X ( s)] = t - s,\ s \leq t.$$

For these average values and incremental variances, this is the only almost-surely continuous process with independent increments. In what follows, the Wiener process will be understood to be this process.

The Wiener process $X( t)$, $0 \leq t \leq 1$, can also be defined as the Gaussian process with zero expectation and covariance function

$$B ( s, t) = \min ( s, t).$$

The Wiener process $X= X( t)$, $t \geq 0$, may also be defined as the homogeneous Markov process with transition function

$$P ( t, x, \Gamma ) = \int\limits _ \Gamma p ( t, x, y) dy,$$

where the transition density $p( t, x, y)$ is the fundamental solution of the parabolic differential equation

$$\frac{\partial p }{\partial t } = \ { \frac{1}{2} } \frac{\partial ^ {2} p }{\partial x ^ {2} } ,$$

given by the formula

$$p ( t, x, y) = \ \frac{1}{\sqrt {2 \pi t } } e ^ {-( y- x) ^ {2} / 2t } .$$

The transition function $P( t, x, \Gamma )$ is invariant with respect to translations in the phase space:

$$P ( t, x + y, \Gamma ) = P ( t, x, \Gamma - y),$$

where $\Gamma - y$ denotes the set $\{ {z } : {z + y \in \Gamma } \}$.

The Wiener process is the continuous analogue of the random walk of a particle which, at discrete moments of time $t= k \Delta t$( multiples of $\Delta t$), is randomly displaced by a quantity $\Delta X( t)$, independent of the past ( ${\mathsf E} \Delta X( t) = 0$, ${\mathsf D} \Delta X( t) = \Delta t$); more precisely, if

$$X ( t) = \sum _{k = 0 } ^ {m-1} \Delta X \left ( { \frac{k}{n} } \right ) + ( nt - m) \Delta X \left ( { \frac{m}{n} } \right ) ,\ \ 0 \leq t \leq 1,$$

is the random trajectory of the motion of such a particle on the interval $[ 0, 1]$( where $m= [ nt]$ is the integer part of $nt$, $X( t) = nt \Delta X ( 0)$ if $0 \leq t < 1/n$ and ${\mathsf P} _ {n}$ is the corresponding probability distribution in the space of continuous functions $x= x( t)$, $0 \leq t \leq 1$), then the probability distribution ${\mathsf P}$ of the trajectory of the Wiener process $X( t)$, $0 \leq t \leq 1$, is the limit (in the sense of weak convergence) of the distributions ${\mathsf P} _ {n}$ as $n \rightarrow \infty$.

As a function with values in the Hilbert space $L _ {2} ( \Omega )$ of all random variables $X$ with ${\mathsf E} X ^ {2} < \infty$, in which the scalar product is defined by the formula

$$\langle X _ {1} , X _ {2} \rangle = {\mathsf E} X _ {1} X _ {2} ,$$

the Wiener process $X = X( t)$, $0 \leq t \leq 1$, may be canonically represented as follows:

$$X ( t) = \sum _ {k = 0 } ^ \infty z _ {k} \phi _ {k} ( t),$$

where $z _ {k}$ are independent Gaussian variables:

$${\mathsf E} z _ {k} = 0,\ \ {\mathsf D} z _ {k} = \frac{1}{\left [ { \frac \pi {2} } ( 2k + 1) \right ] ^ {2} } ,$$

and

$$\phi _ {k} ( t) = \ \sin \left [ { \frac \pi {2} } ( 2k + 1) t \right ] ,\ \ k = 0, 1 \dots$$

are the eigenfunctions of the operator $B$ defined by the formula

$$B \phi ( t) = \int\limits _ { 0 } ^ { 1 } B ( s, t) \phi ( s) ds$$

in the Hilbert space $L _ {2} [ 0, 1]$ of all square-integrable (with respect to Lebesgue measure) functions $\phi$ on $[ 0, 1]$.

Almost-all trajectories of the Wiener process have the following properties:

$${\lim\limits \sup } _ {h \rightarrow \infty } \ \frac{X ( h) }{\sqrt {2 h \mathop{\rm ln} \mathop{\rm ln} { \frac{1}{h} } } } = 1,\ \ X ( 0) = 0,$$

which is the law of the iterated logarithm;

$$\lim\limits _ {h \rightarrow + 0 } \ \sup _ {0 \leq t \leq \delta - h } \ \frac{| X ( t+ h) - X ( t) | }{\sqrt {2 h \mathop{\rm ln} { \frac \delta {h} } } } = 1,$$

characterizing the modulus of continuity on $[ 0, h]$; and

$$\lim\limits _ {h \rightarrow \infty } \sum_{k = 0 } ^ {n-1} | \Delta X ( kh) | ^ {2} = \delta ,$$

$$h = \frac \delta {n} ,\ \Delta X ( t) = X ( t+ h) - X ( t).$$

When applied to the Wiener process $X _ {1} ( t) = tX( 1/t)$, $0 \leq t < \infty$, the law of the iterated logarithm reads:

$${\lim\limits \sup } _ {t \rightarrow \infty } \frac{X ( t) }{\sqrt {2 t \mathop{\rm ln} \mathop{\rm ln} t } } = 1.$$

The distributions of the maximum $\max _ {0 \leq s \leq t } X( s)$, of the time $r _ {x}$ at which the Brownian particle first reaches a fixed point $x > 0$ and of the location $\tau$ of the maximum $\max _ {0 \leq s \leq t } X( s)$ give insight in the nature of the movement of a Brownian particle; these distributions are given by the following formulas:

$${\mathsf P} \left \{ \max _ {0 \leq s \leq t } X( s) \right \} = \frac{2}{\sqrt {2 \pi t } } \int\limits _ { x } ^ \infty e ^ {- u ^ {2} /2t } du ,$$

$$P \{ \tau _ {x} \geq t \} = P \left \{ \max _ {0 \leq s \leq t } X ( s) \leq x \right \} =$$

$$= \ \sqrt { \frac{2} \pi } \int\limits _ { 0 } ^ { {x/ } \sqrt t } e ^ {- u ^ {2} /2 } d u ,$$

while the simultaneous density of the maximum $\max _ {0 \leq s \leq t } X( s)$ and its location $\tau$ is given by:

$${\mathsf P} \{ \tau _ {x} \geq t \} = \ \sqrt { \frac{2} \pi } \int\limits _ { 0 } ^ { {x } / \sqrt t } e ^ {- u ^ {2} /2t } d u ,$$

$$0 < s < t ,\ 0 \leq x < \infty .$$

(The laws of the Wiener process remain unchanged on transforming the phase space $x \rightarrow - x$.) The joint distribution of the maximum point $\tau$, $0 \leq \tau \leq t$, and of the maximum $\max _ {0\leq s \leq t } X( s)$ itself has the probability density

$$p ( s, x) = \ { \frac{1}{\pi \sqrt {s ( t- s) } } } { \frac{x}{s} } e ^ {- x ^ {2} / 2s } ,$$

$$0 \leq s \leq t,\ 0 \leq x < \infty ,$$

while the point $\tau$ by itself (with probability one there is only one maximum on the interval $0 \leq s \leq t$) is distributed according to the arcsine law:

$${\mathsf P} \{ \tau \leq s \} = \ { \frac{2} \pi } \mathop{\rm arc} \sin \sqrt { \frac{s}{t} } ,\ \ 0 \leq s \leq t,$$

with the probability density:

$$p ( s) = { \frac{1}{\pi \sqrt s( t- s) } } ,\ \ 0 \leq s \leq t.$$

The following properties of the Wiener process are readily deduced from the formulas given above. The Brownian trajectory is nowhere differentiable; on starting from any point $x$ this trajectory crosses the "level" $x$( returns to its initial point) infinitely many times with probability one, however short the time $\delta$; the Brownian trajectory passes through all points $x$( more precisely, $\tau _ {x} < \infty$) with probability one (the most probable value of $\tau _ {x}$ is of the order $x ^ {2}$ for large $x$); this trajectory, if considered on a fixed interval $[ 0, t]$, tends to attain the extremal values near the end-points $s= 0$ and $s= t$.

Since a Wiener process is a homogeneous Markov process, there exists an invariant measure $Q( d x)$ for it, namely:

$$Q ( A) \equiv \int\limits Q ( dx) P ( t, x, A),$$

which, since the transition function $P( t, x, A)$ has been seen to be invariant, coincides with the Lebesgue measure on the real line: $Q( d x) = d x$. The time $T( A)$ which a Brownian particle spends in $A$ between the times 0 and $T$ is such that

$$\frac{T ( A _ {1} ) }{T ( A _ {2} ) } \rightarrow \ \frac{Q ( A _ {1} ) }{Q ( A _ {2} ) }$$

as $T \rightarrow \infty$, with probability one for any bounded Borel sets $A _ {1}$ and $A _ {2}$.

Wiener random fields, introduced by P. Lévy [L], are analogues of the Wiener process $X = X( t)$ for a vector parameter $t = ( t _ {1} \dots t _ {n} )$.

#### References

 [IM] K. Itô, H.P. McKean jr., "Diffusion processes and their sample paths" , Springer (1974) MR0345224 Zbl 0285.60063 [PR] Yu.V. Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian) MR0251754 [L] P. Lévy, "Processus stochastiques et mouvement Brownien" , Gauthier-Villars (1965) MR0190953 Zbl 0137.11602 [P] V.P. Pavlov, "Brownian motion" , Large Soviet Encyclopaedia , 4 (In Russian)

The Wiener process is more commonly referred to as Brownian motion in the Western literature. It is by far the most important construct in stochastic analysis. See [Du][RY] for up-to-date accounts of its properties. Of particular importance is the theory of local time. The occupation time of a Borel set $B \subset \mathbf R$ on the interval $[ 0, t]$ is:

$$\Gamma _ {t} ( B) = \int\limits _ { 0 } ^ { t } I _ {B} ( X( s)) ds .$$

There exists an almost-surely jointly-continuous random field $L ( t, x)$ for $( t, x) \in \mathbf R _ {+} \times \mathbf R$ such that

$$\Gamma _ {t} ( B) = 2 \int\limits _ { B } L( t, x) dx;$$

$L( t, x)$ is the local time at $x$. For fixed $x \in \mathbf R$, sample paths of the process $t \mapsto L( t, x)$ are increasing and continuous but singular with respect to Lebesgue measure.

#### References

 [Du] R. Durrett, "Brownian motion and martingales in analysis", Wadsworth (1984) MR0750829 Zbl 0554.60075 [KS] I. Karatzas, S.E. Shreve, "Brownian motion and stochastic calculus", Springer (1988) MR0917065 Zbl 0638.60065 [RY] D. Revuz, M. Yor, "Continuous martingales and Brownian motion", Springer (1990) MR1725357 MR1303781 MR1083357 Zbl 1087.60040 Zbl 0917.60006 Zbl 0804.60001 Zbl 0731.60002 [Dy] E.B. Dynkin, "Markov processes", 1, Springer (1965) (Translated from Russian) MR0193671 Zbl 0132.37901 [F] W. Feller, "An introduction to probability theory and its applications", 1–2, Wiley (1968–1971) [GS] I.I. Gihman, A.V. Skorohod, "The theory of stochastic processes", III, Springer (1975) (Translated from Russian) MR0375463 Zbl 0305.60027 [H] T. Hida, "Brownian motion", Springer (1980) MR0566333 MR0562914 Zbl 0432.60002 Zbl 0423.60063 [S] F. Spitzer, "Principles of random walk", v. Nostrand (1964) MR0171290 Zbl 0119.34304 [Y] J. Yeh, "Stochastic processes and the Wiener integral", M. Dekker (1973) MR0474528 Zbl 0277.60018 [Do] J.L. Doob, "Classical potential theory and its probabilistic counterpart", Springer (1984) MR0731258 Zbl 0549.31001
How to Cite This Entry:
Wiener process. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wiener_process&oldid=55109
This article was adapted from an original article by Yu.A. Rozanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article