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An equation of the form
 
An equation 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/k/k055/k055700/k0557001.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
 
 +
\frac{\partial  f }{\partial  s }
 +
  = - A _ {s} f
 +
$$
 +
 
 +
(the inverse, backward or first, equation; $  s < t $),
 +
or of the form
  
(the inverse, backward or first, equation; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055700/k0557002.png" />), or of the form
+
$$ \tag{2 }
  
<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/k/k055/k055700/k0557003.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
\frac{\partial  f }{\partial  t }
 +
  = A _ {t}  ^ {*} f
 +
$$
  
(the direct, forward or second, equation; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055700/k0557004.png" />), for the [[Transition function|transition function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055700/k0557005.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055700/k0557006.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055700/k0557007.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055700/k0557008.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055700/k0557009.png" /> being a measurable space, or its density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055700/k05570010.png" />, if it exists. For the transition function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055700/k05570011.png" /> the condition
+
(the direct, forward or second, equation; $  t > s $),  
 +
for the [[Transition function|transition function]] $  f = P ( s , x ;  t , \Gamma ) $,
 +
0 \leq  s\leq  t < \infty $,  
 +
$  x \in E $,  
 +
$  \Gamma \in \mathfrak B $,  
 +
$  ( E , \mathfrak B ) $
 +
being a measurable space, or its density $  f = p ( s , x ;  t, \Gamma ) $,  
 +
if it exists. For the transition function $  P ( s , x ;  t , \Gamma ) $
 +
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/k/k055/k055700/k05570012.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {s \uparrow t } \
 +
P ( s , x ; t , \Gamma )  = I _  \Gamma  ( x)
 +
$$
  
 
is adjoined to equation (1), and the condition
 
is adjoined to equation (1), and 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/k/k055/k055700/k05570013.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {t \downarrow s } \
 +
P ( s , x ; t , \Gamma )  = I _  \Gamma  ( x)
 +
$$
  
is adjoined to equation (2), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055700/k05570014.png" /> is the indicator function of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055700/k05570015.png" />; in this case the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055700/k05570016.png" /> is an operator acting in a function space, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055700/k05570017.png" /> acts in a space of generalized measures.
+
is adjoined to equation (2), where $  I _  \Gamma  ( x) $
 +
is the indicator function of the set $  \Gamma $;  
 +
in this case the operator $  A _ {s} $
 +
is an operator acting in a function space, while $  A _ {t}  ^ {*} $
 +
acts in a space of generalized measures.
  
For a [[Markov process|Markov process]] with a countable set of states, the transition function is completely determined by the transition probabilities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055700/k05570018.png" /> (from the state <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055700/k05570019.png" /> at instant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055700/k05570020.png" /> to the state <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055700/k05570021.png" /> at instant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055700/k05570022.png" />), for which the backward and forward Kolmogorov equations have (under certain extra assumptions) the form
+
For a [[Markov process|Markov process]] with a countable set of states, the transition function is completely determined by the transition probabilities $  p _ {ij} ( s , t ) = P ( s , i ;  t , \{ j \} ) $(
 +
from the state $  i $
 +
at instant $  s $
 +
to the state $  j $
 +
at instant $  t $),  
 +
for which the backward and forward Kolmogorov equations have (under certain extra assumptions) 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/k/k055/k055700/k05570023.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
  
<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/k/k055/k055700/k05570024.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
\frac{\partial  p _ {ij} ( s , t ) }{\partial  s }
 +
  = \
 +
\sum _ { k }
 +
\alpha _ {ik} ( s) p _ {kj} ( s , t ) ,\  s < t ,
 +
$$
 +
 
 +
$$ \tag{4 }
 +
 
 +
\frac{\partial  p  ^ {ij} ( s , t ) }{\partial  t }
 +
  = \
 +
\sum _ { k } p _ {ik} ( s , t ) \alpha _ {kj} ( t) ,\  t > s ,
 +
$$
  
 
where
 
where
  
<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/k/k055/k055700/k05570025.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
$$ \tag{5 }
 +
\alpha _ {ij} ( s)  = \
 +
\lim\limits _ {\begin{array}{c}
 +
s _ {1} \uparrow s \\
 +
s _ {2} \uparrow s
 +
\end{array}
 +
} \
 +
 
 +
\frac{p _ {ij} ( s _ {1} , s _ {2} ) - \delta _ {ij} }{s _ {2} - s _ {1} }
 +
.
 +
$$
  
 
In the case of a finite number of states, equations (3) and (4) hold, provided that the limits in (5) exist.
 
In the case of a finite number of states, equations (3) and (4) hold, provided that the limits in (5) exist.
  
Another important class of processes for which the question of the validity of equations (1) and (2) has been studied in detail is the class of processes of diffusion type. These are defined by the condition that their transition functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055700/k05570026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055700/k05570027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055700/k05570028.png" />, satisfy the following conditions:
+
Another important class of processes for which the question of the validity of equations (1) and (2) has been studied in detail is the class of processes of diffusion type. These are defined by the condition that their transition functions $  P ( s , x ;  t , \Gamma ) $,
 +
$  x \in \mathbf R $,
 +
$  \Gamma \in \mathfrak B ( \mathbf R ) $,
 +
satisfy the following conditions:
 +
 
 +
a) for each  $  x \in \mathbf R $
 +
and  $  \epsilon > 0 $,
 +
 
 +
$$
 +
\int\limits _ {| x - y | > \epsilon }
 +
P ( s , x ;  t , d y )  = \
 +
o ( t - s ) ,
 +
$$
 +
 
 +
uniformly in  $  s $,  
 +
$  s < t $;
 +
 
 +
b) there exist functions  $  a ( s , x ) $
 +
and  $  b ( s , x ) $
 +
such that for every  $  x $
 +
and  $  \epsilon > 0 $,
  
a) for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055700/k05570029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055700/k05570030.png" />,
+
$$
 +
\int\limits _ {| x - y | \leq  \epsilon }
 +
( y - x ) P ( s , x ;  t , d y )
 +
a ( s , x ) ( t - s ) + o ( t - s ) ,
 +
$$
  
<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/k/k055/k055700/k05570031.png" /></td> </tr></table>
+
$$
 +
\int\limits _ {| x - y | \leq  \epsilon } ( y - x )  ^ {2} P ( s , x ; t
 +
, d y )  = b ( s , x ) ( t - s ) + o ( t - s ) ,
 +
$$
  
uniformly in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055700/k05570032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055700/k05570033.png" />;
+
uniformly in $  s $,
 +
$  s < t $.
 +
If the density  $  p = p ( s , x ; t , y ) $
 +
exists, then (under certain extra assumptions) the forward equation
  
b) there exist functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055700/k05570034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055700/k05570035.png" /> such that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055700/k05570036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055700/k05570037.png" />,
+
$$
  
<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/k/k055/k055700/k05570038.png" /></td> </tr></table>
+
\frac{\partial  p }{\partial  t }
 +
  = -
  
<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/k/k055/k055700/k05570039.png" /></td> </tr></table>
+
\frac \partial {\partial  y }
 +
( a p ) +
 +
\frac{1}{2}
  
uniformly in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055700/k05570040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055700/k05570041.png" />. If the density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055700/k05570042.png" /> exists, then (under certain extra assumptions) the forward equation
+
\frac{\partial  ^ {2} }{\partial  y  ^ {2} }
 +
( b p )
 +
$$
  
<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/k/k055/k055700/k05570043.png" /></td> </tr></table>
+
holds (for  $  t > s $
 +
and  $  y \in \mathbf R $)
 +
(also called the Fokker–Planck equation), while the backward equation (for  $  s < t $
 +
and  $  x \in \mathbf R $)
 +
has the form
  
holds (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055700/k05570044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055700/k05570045.png" />) (also called the Fokker–Planck equation), while the backward equation (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055700/k05570046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055700/k05570047.png" />) has the form
+
$$
 +
-
 +
\frac{\partial  p }{\partial  s }
 +
  = \
 +
a
 +
\frac{\partial  p }{\partial  x }
 +
+
  
<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/k/k055/k055700/k05570048.png" /></td> </tr></table>
+
\frac{1}{2}
 +
b
 +
\frac{\partial  ^ {2} p }{\partial  x  ^ {2} }
 +
.
 +
$$
  
 
====References====
 
====References====

Latest revision as of 22:14, 5 June 2020


2020 Mathematics Subject Classification: Primary: 60J35 [MSN][ZBL]

An equation of the form

$$ \tag{1 } \frac{\partial f }{\partial s } = - A _ {s} f $$

(the inverse, backward or first, equation; $ s < t $), or of the form

$$ \tag{2 } \frac{\partial f }{\partial t } = A _ {t} ^ {*} f $$

(the direct, forward or second, equation; $ t > s $), for the transition function $ f = P ( s , x ; t , \Gamma ) $, $ 0 \leq s\leq t < \infty $, $ x \in E $, $ \Gamma \in \mathfrak B $, $ ( E , \mathfrak B ) $ being a measurable space, or its density $ f = p ( s , x ; t, \Gamma ) $, if it exists. For the transition function $ P ( s , x ; t , \Gamma ) $ the condition

$$ \lim\limits _ {s \uparrow t } \ P ( s , x ; t , \Gamma ) = I _ \Gamma ( x) $$

is adjoined to equation (1), and the condition

$$ \lim\limits _ {t \downarrow s } \ P ( s , x ; t , \Gamma ) = I _ \Gamma ( x) $$

is adjoined to equation (2), where $ I _ \Gamma ( x) $ is the indicator function of the set $ \Gamma $; in this case the operator $ A _ {s} $ is an operator acting in a function space, while $ A _ {t} ^ {*} $ acts in a space of generalized measures.

For a Markov process with a countable set of states, the transition function is completely determined by the transition probabilities $ p _ {ij} ( s , t ) = P ( s , i ; t , \{ j \} ) $( from the state $ i $ at instant $ s $ to the state $ j $ at instant $ t $), for which the backward and forward Kolmogorov equations have (under certain extra assumptions) the form

$$ \tag{3 } \frac{\partial p _ {ij} ( s , t ) }{\partial s } = \ \sum _ { k } \alpha _ {ik} ( s) p _ {kj} ( s , t ) ,\ s < t , $$

$$ \tag{4 } \frac{\partial p ^ {ij} ( s , t ) }{\partial t } = \ \sum _ { k } p _ {ik} ( s , t ) \alpha _ {kj} ( t) ,\ t > s , $$

where

$$ \tag{5 } \alpha _ {ij} ( s) = \ \lim\limits _ {\begin{array}{c} s _ {1} \uparrow s \\ s _ {2} \uparrow s \end{array} } \ \frac{p _ {ij} ( s _ {1} , s _ {2} ) - \delta _ {ij} }{s _ {2} - s _ {1} } . $$

In the case of a finite number of states, equations (3) and (4) hold, provided that the limits in (5) exist.

Another important class of processes for which the question of the validity of equations (1) and (2) has been studied in detail is the class of processes of diffusion type. These are defined by the condition that their transition functions $ P ( s , x ; t , \Gamma ) $, $ x \in \mathbf R $, $ \Gamma \in \mathfrak B ( \mathbf R ) $, satisfy the following conditions:

a) for each $ x \in \mathbf R $ and $ \epsilon > 0 $,

$$ \int\limits _ {| x - y | > \epsilon } P ( s , x ; t , d y ) = \ o ( t - s ) , $$

uniformly in $ s $, $ s < t $;

b) there exist functions $ a ( s , x ) $ and $ b ( s , x ) $ such that for every $ x $ and $ \epsilon > 0 $,

$$ \int\limits _ {| x - y | \leq \epsilon } ( y - x ) P ( s , x ; t , d y ) = a ( s , x ) ( t - s ) + o ( t - s ) , $$

$$ \int\limits _ {| x - y | \leq \epsilon } ( y - x ) ^ {2} P ( s , x ; t , d y ) = b ( s , x ) ( t - s ) + o ( t - s ) , $$

uniformly in $ s $, $ s < t $. If the density $ p = p ( s , x ; t , y ) $ exists, then (under certain extra assumptions) the forward equation

$$ \frac{\partial p }{\partial t } = - \frac \partial {\partial y } ( a p ) + \frac{1}{2} \frac{\partial ^ {2} }{\partial y ^ {2} } ( b p ) $$

holds (for $ t > s $ and $ y \in \mathbf R $) (also called the Fokker–Planck equation), while the backward equation (for $ s < t $ and $ x \in \mathbf R $) has the form

$$ - \frac{\partial p }{\partial s } = \ a \frac{\partial p }{\partial x } + \frac{1}{2} b \frac{\partial ^ {2} p }{\partial x ^ {2} } . $$

References

[K] A.N. Kolmogorov, "Ueber die analytischen Methoden in der Wahrscheinlichkeitsrechnung" Math. Ann. , 104 (1931) pp. 415–458
[GS] I.I. Gihman, A.V. Skorohod, "The theory of stochastic processes" , 2 , Springer (1979) (Translated from Russian) MR0651014 MR0651015 Zbl 0404.60061

Comments

Besides in a probabilistic context, Kolmogorov equations of course also occur as equations modelling real diffusions, such as the diffusion of molecules of a substance through porous material or the spread of some property through a biological structured population.

See also (the editorial comments to) Einstein–Smoluchowski equation.

References

[L] P. Lévy, "Processus stochastiques et mouvement Brownien" , Gauthier-Villars (1965) MR0190953 Zbl 0137.11602
[D] E.B. Dynkin, "Markov processes" , 1 , Springer (1965) pp. Sect. 5.26 (Translated from Russian) MR0193671 Zbl 0132.37901
[F] W. Feller, "An introduction to probability theory and its applications" , 1 , Wiley (1966) pp. Chapt. XV.13
How to Cite This Entry:
Kolmogorov equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kolmogorov_equation&oldid=47515
This article was adapted from an original article by A.N. Shiryaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article