Kolmogorov equation

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