Difference between revisions of "Wiener process"
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as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w097940102.png" />, with probability one for any bounded Borel sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w097940103.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w097940104.png" />. | as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w097940102.png" />, with probability one for any bounded Borel sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w097940103.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w097940104.png" />. | ||
− | Wiener random fields, introduced by P. Lévy | + | Wiener random fields, introduced by P. Lévy {{Cite|L}}, are analogues of the Wiener process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w097940105.png" /> for a vector parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w097940106.png" />. |
====References==== | ====References==== | ||
− | + | {| | |
− | + | |valign="top"|{{Ref|IM}}|| K. Itô, H.P. McKean, jr., "Diffusion processes and their sample paths" , Springer (1974) {{MR|0345224}} {{ZBL|0285.60063}} | |
− | + | |- | |
+ | |valign="top"|{{Ref|PR}}|| Yu.V. Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian) {{MR|0251754}} {{ZBL|}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|L}}|| P. Lévy, "Processus stochastiques et mouvement Brownien" , Gauthier-Villars (1965) {{MR|0190953}} {{ZBL|0137.11602}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|P}}|| V.P. Pavlov, "Brownian motion" , ''Large Soviet Encyclopaedia'' , '''4''' (In Russian) | ||
+ | |} | ||
====Comments==== | ====Comments==== | ||
− | The Wiener process is more commonly referred to as [[Brownian motion|Brownian motion]] in the Western literature. It is by far the most important construct in stochastic analysis. See | + | The Wiener process is more commonly referred to as [[Brownian motion|Brownian motion]] in the Western literature. It is by far the most important construct in stochastic analysis. See {{Cite|Du}}–{{Cite|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w097940107.png" /> on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w097940108.png" /> is: |
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w097940109.png" /></td> </tr></table> | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097940/w097940109.png" /></td> </tr></table> | ||
Line 140: | Line 146: | ||
====References==== | ====References==== | ||
− | + | {| | |
+ | |valign="top"|{{Ref|Du}}|| R. Durrett, "Brownian motion and martingales in analysis", Wadsworth (1984) {{MR|0750829}} {{ZBL|0554.60075}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|KS}}|| I. Karatzas, S.E. Shreve, "Brownian motion and stochastic calculus", Springer (1988) {{MR|0917065}} {{ZBL|0638.60065}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|RY}}|| D. Revuz, M. Yor, "Continuous martingales and Brownian motion", Springer (1990) {{MR|1725357}} {{MR|1303781}} {{MR|1083357}} {{ZBL|1087.60040}} {{ZBL|0917.60006}} {{ZBL|0804.60001}} {{ZBL|0731.60002}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|Dy}}|| E.B. Dynkin, "Markov processes", '''1''', Springer (1965) (Translated from Russian) {{MR|0193671}} {{ZBL|0132.37901}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|F}}|| W. Feller, [[Feller, "An introduction to probability theory and its applications"|"An introduction to probability theory and its applications"]], '''1–2''', Wiley (1968–1971) | ||
+ | |- | ||
+ | |valign="top"|{{Ref|GS}}|| I.I. Gihman, A.V. Skorohod, "The theory of stochastic processes", '''III''', Springer (1975) (Translated from Russian) {{MR|0375463}} {{ZBL|0305.60027}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|H}}|| T. Hida, "Brownian motion", Springer (1980) {{MR|0566333}} {{MR|0562914}} {{ZBL|0432.60002}} {{ZBL|0423.60063}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|S}}|| F. Spitzer, "Principles of random walk", v. Nostrand (1964) {{MR|0171290}} {{ZBL|0119.34304}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|Y}}|| J. Yeh, "Stochastic processes and the Wiener integral", M. Dekker (1973) {{MR|0474528}} {{ZBL|0277.60018}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|Do}}|| J.L. Doob, "Classical potential theory and its probabilistic counterpart", Springer (1984) {{MR|0731258}} {{ZBL|0549.31001}} | ||
+ | |} |
Revision as of 19:06, 2 June 2012
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 [L], are analogues of the Wiener process for a vector parameter
.
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) |
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 [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 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
[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 |
Wiener process. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wiener_process&oldid=25941