Wiener process
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
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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
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The Wiener process ,
, may also be defined as the homogeneous Markov process with transition function
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where the transition density is the fundamental solution of the parabolic differential equation
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given by the formula
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The transition function is invariant with respect to translations in the phase space:
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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
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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
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the Wiener process ,
, may be canonically represented as follows:
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where are independent Gaussian variables:
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and
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are the eigenfunctions of the operator defined by the formula
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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:
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which is the law of the iterated logarithm;
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characterizing the modulus of continuity on ; and
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When applied to the Wiener process ,
, the law of the iterated logarithm reads:
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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:
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while the simultaneous density of the maximum and its location
is given by:
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(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
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while the point by itself (with probability one there is only one maximum on the interval
) is distributed according to the arcsine law:
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with the probability density:
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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:
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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
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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) |
[2] | Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian) |
[3] | P. Lévy, "Processus stochastiques et mouvement Brownien" , Gauthier-Villars (1965) |
[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:
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There exists an almost-surely jointly-continuous random field for
such that
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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) |
[a2] | I. Karatzas, S.E. Shreve, "Brownian motion and stochastic calculus" , Springer (1988) |
[a3] | D. Revuz, M. Yor, "Continuous martingales and Brownian motion" , Springer (1990) |
[a4] | E.B. Dynkin, "Markov processes" , 1 , Springer (1965) (Translated from Russian) |
[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) |
[a7] | T. Hida, "Brownian motion" , Springer (1980) |
[a8] | F. Spitzer, "Principles of random walk" , v. Nostrand (1964) |
[a9] | J. Yeh, "Stochastic processes and the Wiener integral" , M. Dekker (1973) |
[a10] | J.L. Doob, "Classical potential theory and its probabilistic counterpart" , Springer (1984) |
Wiener process. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wiener_process&oldid=23675