Abstract. Conditions are given, sufficient for the distribution of an Ornstein-Uhlenbeck process with Lévy noise to be absolutely continuous or to possess a smooth density. For the processes with non-degenerate drift coefficient, these conditions are a necessary ones. A multidimensional analogue for the non-degeneracy condition on the drift coefficient is introduced.
We obtain a sufficient condition of smoothness for the distribution density of a multidimensional Ornstein-Uhlenbeck process with Lévy noise, i.e., for the solution of a linear stochastic differential equation with Lévy noise.
We propose a new method of stochastic control for stochastic processes with Lévy noise based on time-change transformations. Applying this method, we prove that the integral minorization condition holds for Markov processes defined by stochastic equations with Lévy noise and obtain the explicit estimates for the rate of convergence in the ergodic theorem.
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