Namespaces
Variants
Actions

Difference between revisions of "White noise"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
A generalized [[Stationary stochastic process|stationary stochastic process]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097750/w0977501.png" /> with constant [[Spectral density|spectral density]]. The generalized correlation function of white noise has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097750/w0977502.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097750/w0977503.png" /> is a positive constant and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097750/w0977504.png" /> is the delta-function. The white noise process is extensively applied in describing random disturbances with a very small correlation period (e.g.  "thermal noisethermal noise"  — pulsations of the current intensity in a conductor, generated by the thermal motion of the electrons). In the spectral decomposition of white noise,
+
<!--
 +
w0977501.png
 +
$#A+1 = 54 n = 0
 +
$#C+1 = 54 : ~/encyclopedia/old_files/data/W097/W.0907750 White noise
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<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/w097750/w0977505.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
the  "elementary vibrations" <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097750/w0977506.png" /> have, on the average, the same intensity at all frequencies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097750/w0977507.png" />; more accurately, their average squared amplitude is
+
A generalized [[Stationary stochastic process|stationary stochastic process]]  $  X( t) $
 +
with constant [[Spectral density|spectral density]]. The generalized correlation function of white noise has the form $ B( t) = \sigma  ^ {2} \delta ( t) $,
 +
where  $  \sigma  ^ {2} $
 +
is a positive constant and  $  \delta ( t) $
 +
is the delta-function. The white noise process is extensively applied in describing random disturbances with a very small correlation period (e.g. "thermal noisethermal noise" — pulsations of the current intensity in a conductor, generated by the thermal motion of the electrons). In the spectral decomposition of white noise,
  
<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/w097750/w0977508.png" /></td> </tr></table>
+
$$
 +
X ( t)  = \int\limits _ {- \infty } ^  \infty  e ^ {i \lambda t }  dz ( \lambda ),
 +
$$
  
This spectral decomposition means that, for each square-integrable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097750/w0977509.png" />,
+
the  "elementary vibrations"   $  e ^ {i \lambda t }  dz ( \lambda ) $
 +
have, on the average, the same intensity at all frequencies  $  \lambda $;
 +
more accurately, their average squared amplitude 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/w097750/w09775010.png" /></td> </tr></table>
+
$$
 +
{\mathsf E} | dz ( \lambda ) |  ^ {2}  = \
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097750/w09775011.png" /> is the [[Fourier transform|Fourier transform]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097750/w09775012.png" />; a more explicit dependence of the generalized process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097750/w09775013.png" /> on the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097750/w09775014.png" /> may be described by a corresponding stochastic measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097750/w09775015.png" /> of the same type as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097750/w09775016.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097750/w09775017.png" /> is the Fourier transform of the stochastic measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097750/w09775018.png" />), viz.
+
\frac{\sigma  ^ {2} }{2 \pi }
 +
  d \lambda ,\ \
 +
- \infty < \lambda < \infty .
 +
$$
  
<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/w097750/w09775019.png" /></td> </tr></table>
+
This spectral decomposition means that, for each square-integrable function  $  \phi ( t) $,
  
Gaussian white noise <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097750/w09775020.png" />, which is the generalized derivative of [[Brownian motion|Brownian motion]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097750/w09775021.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097750/w09775022.png" />), is the basis for constructing stochastic diffusion processes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097750/w09775023.png" /> (cf. [[Diffusion process|Diffusion process]]), "controllable"  by a stochastic differential equation:
+
$$
 +
\langle  X , \phi \rangle  = \
 +
\int\limits _ {- \infty } ^  \infty  \phi ( t) X ( t)  dt  = \
 +
\int\limits _ {- \infty } ^  \infty  \widetilde \phi  ( \lambda ) dz ( \lambda ),
 +
$$
  
<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/w097750/w09775024.png" /></td> </tr></table>
+
where  $  \widetilde \phi  ( \lambda ) $
 +
is the [[Fourier transform|Fourier transform]] of  $  \phi ( t) $;  
 +
a more explicit dependence of the generalized process  $  X = \langle  X, \phi \rangle $
 +
on the function  $  \phi ( t) $
 +
may be described by a corresponding stochastic measure  $  d \eta ( t) $
 +
of the same type as  $  dz( \lambda ) $(
 +
$  d \eta ( t) $
 +
is the Fourier transform of the stochastic measure  $  dz ( \lambda ) $),
 +
viz.
 +
 
 +
$$
 +
\langle  X , \phi \rangle  = \int\limits _ {- \infty } ^  \infty  \phi ( t)  d \eta ( t).
 +
$$
 +
 
 +
Gaussian white noise  $  X( t) $,
 +
which is the generalized derivative of [[Brownian motion|Brownian motion]]  $  \eta ( t) $(
 +
$  X( t) = \eta  ^  \prime  ( t) $),
 +
is the basis for constructing stochastic diffusion processes  $  Y( t) $(
 +
cf. [[Diffusion process|Diffusion process]]),  "controllable" by a stochastic differential equation:
 +
 
 +
$$
 +
Y  ^  \prime  ( t)  = a ( t, Y ( t)) + \sigma ( t, Y ( t)) \cdot \eta  ^  \prime  ( t).
 +
$$
  
 
These equations are often written in the form of differentials:
 
These equations are often written in the form of differentials:
  
<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/w097750/w09775025.png" /></td> </tr></table>
+
$$
 +
dY ( t)  = a ( t, Y ( t))  dt + \sigma ( t, Y ( t))  d \eta ( t).
 +
$$
  
Another important model involving the use of white noise is the stochastic process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097750/w09775026.png" /> which describes the behaviour of a stable vibrating system acted upon by stationary random perturbations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097750/w09775027.png" />, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097750/w09775028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097750/w09775029.png" />, do not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097750/w09775030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097750/w09775031.png" />. A very simple example of this is the system
+
Another important model involving the use of white noise is the stochastic process $  Y( t) $
 +
which describes the behaviour of a stable vibrating system acted upon by stationary random perturbations $  X( t) $,  
 +
when $  Y( s) $,  
 +
$  s < t $,  
 +
do not depend on $  X( u) $,  
 +
$  u > t $.  
 +
A very simple example of this is the system
  
<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/w097750/w09775032.png" /></td> </tr></table>
+
$$
 +
P \left (
 +
\frac{d}{dt}
 +
\right ) Y ( t)  = X ( t),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097750/w09775033.png" /> is a polynomial with roots in the left half-plane; after the damping of the  "transient processes" , the process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097750/w09775034.png" /> is given by
+
where $  P( z) $
 +
is a polynomial with roots in the left half-plane; after the damping of the  "transient processes" , the process $  Y( t) $
 +
is given by
  
<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/w097750/w09775035.png" /></td> </tr></table>
+
$$
 +
Y ( t)  = \int\limits
 +
\frac{1}{P ( i \lambda ) }
 +
  dz ( \lambda ).
 +
$$
  
 
In practical applications, in the description of the so-called [[Shot effect|shot effect]] process, white noise of the form
 
In practical applications, in the description of the so-called [[Shot effect|shot effect]] process, white noise of the form
  
<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/w097750/w09775036.png" /></td> </tr></table>
+
$$
 +
X ( t)  = \sum _ { k } \delta ( t - \tau _ {k} )
 +
$$
  
plays an important role (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097750/w09775037.png" /> varies between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097750/w09775038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097750/w09775039.png" /> and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097750/w09775040.png" /> form a Poisson process); more accurately, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097750/w09775041.png" /> is the generalized derivative of a Poisson process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097750/w09775042.png" />. The shot effect process itself has the form
+
plays an important role ( $  k $
 +
varies between $  - \infty $
 +
and $  \infty $
 +
and the $  \dots \tau _ {-} 1 , \tau _ {0} , \tau _ {1} \dots $
 +
form a Poisson process); more accurately, $  X( t) $
 +
is the generalized derivative of a Poisson process $  \eta ( t) $.  
 +
The shot effect process itself has the form
  
<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/w097750/w09775043.png" /></td> </tr></table>
+
$$
 +
Y ( t)  = \
 +
\int\limits _ {- \infty } ^  \infty  c ( t, s) X ( s)  ds  = \
 +
\int\limits _ {- \infty } ^  \infty  c ( t, s)  d \eta ( s ) =
 +
$$
  
<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/w097750/w09775044.png" /></td> </tr></table>
+
$$
 +
= \
 +
\sum _ { k } c ( t, \tau _ {k} ),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097750/w09775045.png" /> is some weight function satisfying the condition
+
where $  c( t, s) $
 +
is some weight function satisfying the condition
  
<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/w097750/w09775046.png" /></td> </tr></table>
+
$$
 +
\int\limits _ {- \infty } ^  \infty  | c( t, s ) |  ^ {2}  ds  < \infty ;
 +
$$
  
in addition, the average value of the generalized process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097750/w09775047.png" /> is
+
in addition, the average value of the generalized process $  X = \langle  X, \phi \rangle $
 +
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/w097750/w09775048.png" /></td> </tr></table>
+
$$
 +
a ( \phi )  = a \int\limits _ {- \infty } ^  \infty  \phi ( t)  dt.
 +
$$
  
Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097750/w09775049.png" /> is the parameter of the Poisson law (see above), and the stochastic measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097750/w09775050.png" /> in the spectral representation
+
Here, $  a $
 +
is the parameter of the Poisson law (see above), and the stochastic measure $  dz ( \lambda ) $
 +
in the spectral representation
  
<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/w097750/w09775051.png" /></td> </tr></table>
+
$$
 +
X ( t)  = a + \int\limits _ {- \infty } ^  \infty  e ^ {i \lambda t }  dz ( \lambda )
 +
$$
  
 
of this process is such that
 
of this process is such that
  
<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/w097750/w09775052.png" /></td> </tr></table>
+
$$
 +
{\mathsf E} | dz ( \lambda ) |  ^ {2}  =
 +
\frac{a}{2 \pi }
 +
  d \lambda .
 +
$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.V. [Yu.V. Prokhorov] Prohorov,  Yu.A. Rozanov,  "Probability theory, basic concepts. Limit theorems, random processes" , Springer  (1969)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.V. [Yu.V. Prokhorov] Prohorov,  Yu.A. Rozanov,  "Probability theory, basic concepts. Limit theorems, random processes" , Springer  (1969)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
See [[#References|[a1]]] for applications of white noise as the limit of  "wide bandwidth"  noise in physical systems and [[#References|[a2]]] for the relationship between differential equations with white noise inputs and the stochastic differential equations of Itô calculus (cf. also [[Itô formula|Itô formula]]; [[Stochastic differential equation|Stochastic differential equation]]). See also [[Stratonovich integral|Stratonovich integral]] for further information on this topic. Further important topics are the analysis of white noise regarded as a generalized random function [[#References|[a3]]], i.e. a probability on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097750/w09775053.png" /> of tempered distributions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097750/w09775054.png" /> (cf. [[White noise analysis|White noise analysis]]), and application of white noise theory in non-linear filtering [[#References|[a4]]], where  "white noise"  is interpreted in terms of finitely-additive Gaussian measures on cylinder sets of a separable Hilbert space.
+
See [[#References|[a1]]] for applications of white noise as the limit of  "wide bandwidth"  noise in physical systems and [[#References|[a2]]] for the relationship between differential equations with white noise inputs and the stochastic differential equations of Itô calculus (cf. also [[Itô formula|Itô formula]]; [[Stochastic differential equation|Stochastic differential equation]]). See also [[Stratonovich integral|Stratonovich integral]] for further information on this topic. Further important topics are the analysis of white noise regarded as a generalized random function [[#References|[a3]]], i.e. a probability on the space $  {\mathcal S}  ^  \prime  $
 +
of tempered distributions on $  [ 0, \infty ) $(
 +
cf. [[White noise analysis|White noise analysis]]), and application of white noise theory in non-linear filtering [[#References|[a4]]], where  "white noise"  is interpreted in terms of finitely-additive Gaussian measures on cylinder sets of a separable Hilbert space.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.J. Kushner,  "Approximation and weak convergence methods for random processes, with applications to stochastic systems theory" , M.I.T.  (1984)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Ikeda,  S. Watanabe,  "Stochastic differential equations and diffusion processes" , North-Holland &amp; Kodansha  (1988)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  T. Hida,  "Brownian motion" , Springer  (1980)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Kallianpur,  R.L. Karandikar,  "White noise theory of prediction, filtering and smoothing" , Gordon &amp; Breach  (1988)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  I.M. Gel'fand,  N.Ya. Vilenkin,  "Generalized functions. Applications of harmonic analysis" , '''4''' , Acad. Press  (1968)  pp. Chapt. III  (Translated from Russian)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  T. Hida (ed.)  H.-H. Kuo (ed.)  J. Potthoff (ed.)  L. Streid (ed.) , ''White noise analysis - mathematics and applications'' , World Sci.  (1990)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.J. Kushner,  "Approximation and weak convergence methods for random processes, with applications to stochastic systems theory" , M.I.T.  (1984)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Ikeda,  S. Watanabe,  "Stochastic differential equations and diffusion processes" , North-Holland &amp; Kodansha  (1988)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  T. Hida,  "Brownian motion" , Springer  (1980)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Kallianpur,  R.L. Karandikar,  "White noise theory of prediction, filtering and smoothing" , Gordon &amp; Breach  (1988)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  I.M. Gel'fand,  N.Ya. Vilenkin,  "Generalized functions. Applications of harmonic analysis" , '''4''' , Acad. Press  (1968)  pp. Chapt. III  (Translated from Russian)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  T. Hida (ed.)  H.-H. Kuo (ed.)  J. Potthoff (ed.)  L. Streid (ed.) , ''White noise analysis - mathematics and applications'' , World Sci.  (1990)</TD></TR></table>

Latest revision as of 08:29, 6 June 2020


A generalized stationary stochastic process $ X( t) $ with constant spectral density. The generalized correlation function of white noise has the form $ B( t) = \sigma ^ {2} \delta ( t) $, where $ \sigma ^ {2} $ is a positive constant and $ \delta ( t) $ is the delta-function. The white noise process is extensively applied in describing random disturbances with a very small correlation period (e.g. "thermal noisethermal noise" — pulsations of the current intensity in a conductor, generated by the thermal motion of the electrons). In the spectral decomposition of white noise,

$$ X ( t) = \int\limits _ {- \infty } ^ \infty e ^ {i \lambda t } dz ( \lambda ), $$

the "elementary vibrations" $ e ^ {i \lambda t } dz ( \lambda ) $ have, on the average, the same intensity at all frequencies $ \lambda $; more accurately, their average squared amplitude is

$$ {\mathsf E} | dz ( \lambda ) | ^ {2} = \ \frac{\sigma ^ {2} }{2 \pi } d \lambda ,\ \ - \infty < \lambda < \infty . $$

This spectral decomposition means that, for each square-integrable function $ \phi ( t) $,

$$ \langle X , \phi \rangle = \ \int\limits _ {- \infty } ^ \infty \phi ( t) X ( t) dt = \ \int\limits _ {- \infty } ^ \infty \widetilde \phi ( \lambda ) dz ( \lambda ), $$

where $ \widetilde \phi ( \lambda ) $ is the Fourier transform of $ \phi ( t) $; a more explicit dependence of the generalized process $ X = \langle X, \phi \rangle $ on the function $ \phi ( t) $ may be described by a corresponding stochastic measure $ d \eta ( t) $ of the same type as $ dz( \lambda ) $( $ d \eta ( t) $ is the Fourier transform of the stochastic measure $ dz ( \lambda ) $), viz.

$$ \langle X , \phi \rangle = \int\limits _ {- \infty } ^ \infty \phi ( t) d \eta ( t). $$

Gaussian white noise $ X( t) $, which is the generalized derivative of Brownian motion $ \eta ( t) $( $ X( t) = \eta ^ \prime ( t) $), is the basis for constructing stochastic diffusion processes $ Y( t) $( cf. Diffusion process), "controllable" by a stochastic differential equation:

$$ Y ^ \prime ( t) = a ( t, Y ( t)) + \sigma ( t, Y ( t)) \cdot \eta ^ \prime ( t). $$

These equations are often written in the form of differentials:

$$ dY ( t) = a ( t, Y ( t)) dt + \sigma ( t, Y ( t)) d \eta ( t). $$

Another important model involving the use of white noise is the stochastic process $ Y( t) $ which describes the behaviour of a stable vibrating system acted upon by stationary random perturbations $ X( t) $, when $ Y( s) $, $ s < t $, do not depend on $ X( u) $, $ u > t $. A very simple example of this is the system

$$ P \left ( \frac{d}{dt} \right ) Y ( t) = X ( t), $$

where $ P( z) $ is a polynomial with roots in the left half-plane; after the damping of the "transient processes" , the process $ Y( t) $ is given by

$$ Y ( t) = \int\limits \frac{1}{P ( i \lambda ) } dz ( \lambda ). $$

In practical applications, in the description of the so-called shot effect process, white noise of the form

$$ X ( t) = \sum _ { k } \delta ( t - \tau _ {k} ) $$

plays an important role ( $ k $ varies between $ - \infty $ and $ \infty $ and the $ \dots \tau _ {-} 1 , \tau _ {0} , \tau _ {1} \dots $ form a Poisson process); more accurately, $ X( t) $ is the generalized derivative of a Poisson process $ \eta ( t) $. The shot effect process itself has the form

$$ Y ( t) = \ \int\limits _ {- \infty } ^ \infty c ( t, s) X ( s) ds = \ \int\limits _ {- \infty } ^ \infty c ( t, s) d \eta ( s ) = $$

$$ = \ \sum _ { k } c ( t, \tau _ {k} ), $$

where $ c( t, s) $ is some weight function satisfying the condition

$$ \int\limits _ {- \infty } ^ \infty | c( t, s ) | ^ {2} ds < \infty ; $$

in addition, the average value of the generalized process $ X = \langle X, \phi \rangle $ is

$$ a ( \phi ) = a \int\limits _ {- \infty } ^ \infty \phi ( t) dt. $$

Here, $ a $ is the parameter of the Poisson law (see above), and the stochastic measure $ dz ( \lambda ) $ in the spectral representation

$$ X ( t) = a + \int\limits _ {- \infty } ^ \infty e ^ {i \lambda t } dz ( \lambda ) $$

of this process is such that

$$ {\mathsf E} | dz ( \lambda ) | ^ {2} = \frac{a}{2 \pi } d \lambda . $$

References

[1] Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian)

Comments

See [a1] for applications of white noise as the limit of "wide bandwidth" noise in physical systems and [a2] for the relationship between differential equations with white noise inputs and the stochastic differential equations of Itô calculus (cf. also Itô formula; Stochastic differential equation). See also Stratonovich integral for further information on this topic. Further important topics are the analysis of white noise regarded as a generalized random function [a3], i.e. a probability on the space $ {\mathcal S} ^ \prime $ of tempered distributions on $ [ 0, \infty ) $( cf. White noise analysis), and application of white noise theory in non-linear filtering [a4], where "white noise" is interpreted in terms of finitely-additive Gaussian measures on cylinder sets of a separable Hilbert space.

References

[a1] H.J. Kushner, "Approximation and weak convergence methods for random processes, with applications to stochastic systems theory" , M.I.T. (1984)
[a2] N. Ikeda, S. Watanabe, "Stochastic differential equations and diffusion processes" , North-Holland & Kodansha (1988)
[a3] T. Hida, "Brownian motion" , Springer (1980)
[a4] G. Kallianpur, R.L. Karandikar, "White noise theory of prediction, filtering and smoothing" , Gordon & Breach (1988)
[a5] I.M. Gel'fand, N.Ya. Vilenkin, "Generalized functions. Applications of harmonic analysis" , 4 , Acad. Press (1968) pp. Chapt. III (Translated from Russian)
[a6] T. Hida (ed.) H.-H. Kuo (ed.) J. Potthoff (ed.) L. Streid (ed.) , White noise analysis - mathematics and applications , World Sci. (1990)
How to Cite This Entry:
White noise. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=White_noise&oldid=19248
This article was adapted from an original article by Yu.A. Rozanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article