A Note on Evolution Systems of Measures for Time-Dependent Stochastic Differential Equations

We consider a stochastic equation in ℝ

n

with time-dependent coefficients assuming that it has a unique solution and denote by

P

s,t

,

s

<

t

the corresponding transition semigroup. Then we consider a family of measures (

ν

t

)

t

∈ℝ

such that

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$$ \smallint _{\mathbb{R}^d } P_{s,t} \varphi \left( x \right)\nu _s \left( {dx} \right) = \smallint _{\mathbb{R}^d } \varphi \left( x \right)\nu _t \left( {dx} \right),s \leqslant t $$

, for all continuous and bounded functions

ϕ

. The family (ν

t

)

t

∈ℝ

is called an

evolution system of measures indexed by

ℝ. It plays the role of a probability invariant measure for autonomous systems. In this paper we generalize the Krylov-Bogoliubov criterion to prove the existence of an evolution system of measures. Moreover, we study some properties of the corresponding Kolmogorov operator proving in particular that it is dissipative with respect to the measure

ν

(

dt, dx

) =

ν

t

(

dx

)

dt

.