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Transition semigroups for stochastic semilinear equations on Hilbert spaces

100%

Chojnowska-Michalik A.

Instytut Matematyczny Polskiej Akademii Nauk

EN

A large class of stochastic semilinear equations with measurable nonlinear term on a Hilbert space H is considered. Assuming the corresponding nonsymmetric Ornstein-Uhlenbeck process has an invariant measure μ, we prove in the $L^{p}(H,μ)$ spaces the existence of a transition semigroup $(P_{t})$ for the equations. Sufficient conditions are provided for hyperboundedness of $P_{t}$ and for the Log Sobolev Inequality to hold; and in the case of a bounded nonlinear term, sufficient and necessary conditions are obtained. We prove the existence, uniqueness and some regularity of an invariant density for $(P_{t})$. A characterization of the domain of the generator is also given. The main tools are the Girsanov transform and Miyadera perturbations.

2

Exponential ergodicity of semilinear equations driven by Lévy processes in Hilbert spaces

61%

Chojnowska-Michalik A., Goldys B.

Banach Center Publications

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2015

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tom 105

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nr 1

59-72

EN

We study convergence to the invariant measure for a class of semilinear stochastic evolution equations driven by Lévy noise, including the case of cylindrical noise. For a certain class of equations we prove the exponential rate of convergence in the norm of total variation. Our general result is applied to a number of specific equations driven by cylindrical symmetric α-stable noise and/or cylindrical Wiener noise. We also consider the case of a "singular" Wiener process with unbounded covariance operator. In particular, in the equation with diagonal pure α-stable cylindrical noise introduced by Priola and Zabczyk we generalize results from Priola, Shirikyan, Xu and Zabczyk (2012). In the proof we use an idea of Maslowski and Seidler (1999).

3

Stochastic differential equations in Hilbert spaces

48%

Chojnowska-Michalik A.

Banach Center Publications

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1979

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tom 5

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nr 1

53-74

Ograniczanie wyników

2 Banach Center Publications

3 Chojnowska-Michalik A.

1 Goldys B.

1 2015

1 2001

1 1979

Transition semigroups for stochastic semilinear equations on Hilbert spaces

Exponential ergodicity of semilinear equations driven by Lévy processes in Hilbert spaces

Stochastic differential equations in Hilbert spaces