Wiener process

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

A homogeneous Gaussian process 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 , , for which

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 , , can also be defined as the Gaussian process with zero expectation and covariance function

The Wiener process , , may also be defined as the homogeneous Markov process with transition function

where the transition density is the fundamental solution of the parabolic differential equation

given by the formula

The transition function is invariant with respect to translations in the phase space:

where denotes the set .

The Wiener process is the continuous analogue of the random walk of a particle which, at discrete moments of time (multiples of ), is randomly displaced by a quantity , independent of the past (, ); more precisely, if

is the random trajectory of the motion of such a particle on the interval (where is the integer part of , if and is the corresponding probability distribution in the space of continuous functions , ), then the probability distribution of the trajectory of the Wiener process , , is the limit (in the sense of weak convergence) of the distributions as .

As a function with values in the Hilbert space of all random variables with , in which the scalar product is defined by the formula

the Wiener process , , may be canonically represented as follows:

where are independent Gaussian variables:

and

are the eigenfunctions of the operator defined by the formula

in the Hilbert space of all square-integrable (with respect to Lebesgue measure) functions on .

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

which is the law of the iterated logarithm;

characterizing the modulus of continuity on ; and

When applied to the Wiener process , , the law of the iterated logarithm reads:

The distributions of the maximum , of the time at which the Brownian particle first reaches a fixed point and of the location of the maximum give insight in the nature of the movement of a Brownian particle; these distributions are given by the following formulas:

while the simultaneous density of the maximum and its location is given by:

(The laws of the Wiener process remain unchanged on transforming the phase space .) The joint distribution of the maximum point , , and of the maximum itself has the probability density

while the point by itself (with probability one there is only one maximum on the interval ) is distributed according to the arcsine law:

with the probability density:

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 this trajectory crosses the "level" (returns to its initial point) infinitely many times with probability one, however short the time ; the Brownian trajectory passes through all points (more precisely, ) with probability one (the most probable value of is of the order for large ); this trajectory, if considered on a fixed interval , tends to attain the extremal values near the end-points and .

Since a Wiener process is a homogeneous Markov process, there exists an invariant measure for it, namely:

which, since the transition function has been seen to be invariant, coincides with the Lebesgue measure on the real line: . The time which a Brownian particle spends in between the times 0 and is such that

as , with probability one for any bounded Borel sets and .

Wiener random fields, introduced by P. Lévy [3], are analogues of the Wiener process for a vector parameter .

References

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

Comments

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 [a1]–[a3] for up-to-date accounts of its properties. Of particular importance is the theory of local time. The occupation time of a Borel set on the interval is:

There exists an almost-surely jointly-continuous random field for such that

is the local time at . For fixed , sample paths of the process are increasing and continuous but singular with respect to Lebesgue measure.

See also Markov process; Stochastic differential equation.

References

[a1]	R. Durrett, "Brownian motion and martingales in analysis", Wadsworth (1984) MR0750829 Zbl 0554.60075
[a2]	I. Karatzas, S.E. Shreve, "Brownian motion and stochastic calculus", Springer (1988) MR0917065 Zbl 0638.60065
[a3]	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
[a4]	E.B. Dynkin, "Markov processes", 1, Springer (1965) (Translated from Russian) MR0193671 Zbl 0132.37901
[a5]	W. Feller, "An introduction to probability theory and its applications", 1–2, Wiley (1968–1971)
[a6]	I.I. [I.I. Gikhman] Gihman, A.V. [A.V. Skorokhod] Skorohod, "The theory of stochastic processes", III, Springer (1975) (Translated from Russian) MR0375463 Zbl 0305.60027
[a7]	T. Hida, "Brownian motion", Springer (1980) MR0566333 MR0562914 Zbl 0432.60002 Zbl 0423.60063
[a8]	F. Spitzer, "Principles of random walk", v. Nostrand (1964) MR0171290 Zbl 0119.34304
[a9]	J. Yeh, "Stochastic processes and the Wiener integral", M. Dekker (1973) MR0474528 Zbl 0277.60018
[a10]	J.L. Doob, "Classical potential theory and its probabilistic counterpart", Springer (1984) MR0731258 Zbl 0549.31001