Stable cylindrical Lévy processes and the stochastic Cauchy problem

Riedle, Markus

doi:10.1214/18-ecp134

Cited by 13 publications

(11 citation statements)

References 16 publications

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“…In this example, we assume that U = V and B = Id in equation (3.1). Let L be the canonical α-stable cylindrical Lévy process for α ∈ (0, 2), which is defined in [19] by requiring that its characteristic function is of the form…”

Section: Consequently Fatou's Lemma Guarantees For Eachmentioning

confidence: 99%

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The stochastic Cauchy problem driven by a cylindrical Lévy process

Kumar

Riedle²

2020

Electron. J. Probab.

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In this work, we derive sufficient and necessary conditions for the existence of a weak and mild solution of an abstract stochastic Cauchy problem driven by an arbitrary cylindrical Lévy process. Our approach requires to establish a stochastic Fubini result for stochastic integrals with respect to cylindrical Lévy processes. This approach enables us to conclude that the solution process has almost surely scalarly square integrable paths. Further properties of the solution such as the Markov property and stochastic continuity are derived.

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Section: Consequently Fatou's Lemma Guarantees For Eachmentioning

confidence: 99%

“…Assume that there exists an orthonormal basis (e k ) k∈AE of U and an increasing se- [19]. For example, a sufficient assumption for the validity of (3.33) is .…”

Section: Consequently Fatou's Lemma Guarantees For Eachmentioning

confidence: 99%

The stochastic Cauchy problem driven by a cylindrical Lévy process

Kumar

Riedle²

2020

Electron. J. Probab.

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“…This was mainly accomplished in the work [6] by Brzeźniak and Zabczyk on existence and regularity of solutions by modelling the stable noise as a subordinated cylindrical Brownian motion. In the setting of generalised processes, the linear equation driven by a standard α-stable process was considered by Riedle [34].…”

Section: Introductionmentioning

confidence: 99%

Stochastic evolution equations driven by cylindrical stable noise

Kosmala¹,

Riedle²

2021

Preprint

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We prove existence and uniqueness of a mild solution of a stochastic evolution equation driven by a standard α-stable cylindrical Lévy process defined on a Hilbert space for α ∈ (1, 2). The coefficients are assumed to map between certain domains of fractional powers of the generator present in the equation. The solution is constructed as a weak limit of the Picard iteration using tightness arguments. Existence of strong solution is obtained by a general version of the Yamada-Watanabe theorem.

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“…In recent years, there has been an increasing interest into the study of stochastic partial differential equations driven by cylindrical noise (e.g. Brzeźniak and Zabczyk, 2010;Kosmala and Riedle, 2021;Kumar and Riedle, 2020;Liu and Zhai, 2016;Priola and Zabczyk, 2011;Riedle, 2015Riedle, , 2018. Motivated by these developments, in this work we introduce a construction for the stochastic integral with respect to some classes of cylindrical martingale-valued measures.…”

Section: Introductionmentioning

confidence: 99%

Stochastic integration in Hilbert spaces with respect to cylindrical martingale-valued measures

Alvarado-Solano¹,

Fonseca-Mora²

2021

ALEA

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In this work we introduce a theory of stochastic integration for operator-valued integrands with respect to some classes of cylindrical martingale-valued measures in Hilbert spaces. The integral is constructed via the radonification of cylindrical martingales by a Hilbert-Schmidt operator theorem and unifies several other theories of stochastic integration in Hilbert spaces. In particular, our theory covers the theory of stochastic integration with respect to a Hilbert space valued Lévy process with second moments, with respect to a cylindrical Lévy processes with (weak) second moments and with respect to a Lévy-valued random martingale measures with finite second moment. As an application of our theory of integration we prove existence and uniqueness of solutions for stochastic stochastic partial differential equations driven by multiplicative cylindrical martingale-valued measure noise with rather general coefficients. Existence and uniqueness of solutions in the presence of multiplicative Lévy noise (with no moments assumptions) is also proved.

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Stable cylindrical Lévy processes and the stochastic Cauchy problem

Cited by 13 publications

References 16 publications

The stochastic Cauchy problem driven by a cylindrical Lévy process

The stochastic Cauchy problem driven by a cylindrical Lévy process

Stochastic evolution equations driven by cylindrical stable noise

Stochastic integration in Hilbert spaces with respect to cylindrical martingale-valued measures

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