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Harnack inequalities with power \(\pmb {p\in (1,+\infty )}\) for transition semigroups in Hilbert spaces

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Abstract

We consider the stochastic differential equation

$$\begin{aligned} \left\{ \begin{array}{ll} dX(t)=[AX(t)+F(X(t))]dt+C^{1/2}dW(t), &{} t>0,\\ X(0)=x \in {\mathcal {X}}, \end{array}\right. \end{aligned}$$

where \({\mathcal {X}}\) is a separable Hilbert space, \(\{W(t)\}_{t\ge 0}\) is a \({\mathcal {X}}\)-cylindrical Wiener process, A and C are suitable operators on \({\mathcal {X}}\) and \(F:{{\text {Dom}}}(F)\subseteq {\mathcal {X}}\rightarrow {\mathcal {X}}\) is a smooth enough function. We establish a Harnack inequality with power \(p \in (1,+\infty )\) for the transition semigroup \(\{P(t)\}_{t\ge 0}\) associated with the stochastic problem above, under less restrictive conditions than those considered in the literature. Some applications to these inequalities are shown.

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Notes

  1. We say that a \({\mathcal {K}}\)-valued process \(\psi \) is progressively measurable if, as a transformation from \(\Omega \times [0,t]\) equipped with the \(\sigma \)-field \(\mathcal {F}_t \times \mathcal {B}([0,t])\) into \(({\mathcal {K}}, \mathcal {B}({\mathcal {K}}))\) is measurable for any \([0,t]\subseteq I\).

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Acknowledgements

S.F. has been partially supported by the OK-INSAID project ARS01-00917. The authors are members of G.N.A.M.P.A. of the Italian Istituto Nazionale di Alta Matematica (INdAM) and have been partially supported by the PRIN 2015 MIUR project 2015233N54. The authors are grateful to Alessandra Lunardi for many helpful conversations. We would like to thank the anonymous referee for their valuable comments and suggestions which allow us to improve the reading of our paper.

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Authors and Affiliations

  1. Dipartimento di Matematica e Fisica “Ennio De Giorgi”, Università del Salento, Via per Arnesano, 73100, Lecce, Italy

    Luciana Angiuli & Simone Ferrari

  2. Dipartimento di Matematica, Universita degli Studi di Pavia, Via Adolfo Ferrata 5, 27100, Pavia, Italy

    Davide A. Bignamini

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  1. Luciana Angiuli
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Correspondence to Davide A. Bignamini.

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Angiuli, L., Bignamini, D.A. & Ferrari, S. Harnack inequalities with power \(\pmb {p\in (1,+\infty )}\) for transition semigroups in Hilbert spaces. Nonlinear Differ. Equ. Appl. 30, 6 (2023). https://doi.org/10.1007/s00030-022-00812-0

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