Forgotten initial data in Markov and conditional Markov evolutions

Abstract: 

In various stochastic models it is useful to know some quantitative 
results on weak dependence of random variables. A good measure of this 
dependence may be provided by beta-mixing coefficient. For recurrent 
Markov processes, bounds for this coefficient have been studied. 
In the first part of the talk basic results of this type will be 
presented. Applications include not only limit theorems, but also 
some Green's function bounds and Poisson equations in unbounded domains.

The second part will be devoted to conditional Markov dynamics in filtering 
problems. As in any Bayesian setting, it is important to have theoretical 
results which would justify the use of prior distribution with errors. 
Difficulties arise due to non-linearity and due to possible non-compactness 
of state spaces. A solution of a long-standing problem will be presented.

The key techniques is stochastic analysis. Harnack inequality for parabolic 
PDEs plays a significant role. Positive operator methods are also important.
The second part of the setting is statistical, however, the presentation will 
be still within the stochastic analysis style.