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Global Schauder estimates for a class of degenerate Kolmogorov equations

100%
Priola E.
Studia Mathematica
|
2009
|
tom 194
|
nr 2
117-153
EN
We consider a class of possibly degenerate second order elliptic operators 𝓐 on ℝⁿ. This class includes hypoelliptic Ornstein-Uhlenbeck type operators having an additional first order term with unbounded coefficients. We establish global Schauder estimates in Hölder spaces both for elliptic equations and for parabolic Cauchy problems involving 𝓐. The Hölder spaces in question are defined with respect to a possibly non-Euclidean metric related to the operator 𝓐. Schauder estimates are deduced by sharp $L^{∞}-C^{θ}$ estimates on the spatial derivatives of the associated diffusion semigroup which are of independent interest. In the proof we also use probabilistic techniques.
2
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Stochastic flow for SDEs with jumps and irregular drift term

100%
Priola E.
Banach Center Publications
|
2015
|
tom 105
|
nr 1
193-210
EN
We consider non-degenerate SDEs with a β-Hölder continuous and bounded drift term and driven by a Lévy noise L which is of α-stable type. If β > 1 - α/2 and α ∈ [1,2), we show pathwise uniqueness and existence of a stochastic flow. We follow the approach of [Priola, Osaka J. Math. 2012] improving the assumptions on the noise L. In our previous paper L was assumed to be non-degenerate, α-stable and symmetric. Here we can also recover relativistic and truncated stable processes and some classes of tempered stable processes.
3
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On a class of Markov type semigroups in spaces of uniformly continuous and bounded functions

100%
Priola E.
Studia Mathematica
|
1999
|
tom 136
|
nr 3
271-295
EN
We study a new class of Markov type semigroups (not strongly continuous in general) in the space of all real, uniformly continuous and bounded functions on a separable metric space E. Our results allow us to characterize the generators of Markov transition semigroups in infinite dimensions such as the heat and the Ornstein-Uhlenbeck semigroups.
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