Itô formula for mild solutions of SPDEs with Gaussian and non-Gaussian noise and applications to stability properties

Sergio Albeverio; Leszek Gawarecki; Vidyadhar Mandrekar; Barbara Rüdiger; Barun Sarkar

Itô formula for mild solutions of SPDEs with Gaussian and non-Gaussian noise and applications to stability properties

Sergio Albeverio, Leszek Gawarecki, Vidyadhar Mandrekar, Barbara Rüdiger, Barun Sarkar

Kettering University

Research output: Contribution to journal › Article › peer-review

Abstract

<p> We use Yosida approximation to &filig;nd an Itô formula for mild solutions { Xx(t), t ≥ 0 }of SPDEs with Gaussianand non-Gaussian coloured noise, the non Gaussian noise being de&filig;ned through compensated Poisson random measure associated to a Lévy process. The functions to which we apply such Itô formula are in C1,2([0,T]×H), as in the case considered for SDEs in [19]. Using this Itô formula we prove exponential stability and exponential ultimate boundedness properties in mean square sense for mild solutions. We also compare such Itô formula to an Itô formula for mild solutions introduced by Ichikawain [15], and an Itô formula written in terms of the semigroup of the drift operator [6] which we extend before to the non Gaussian case.</p>

Original language	American English
Journal	arXiv:1612.09440 [math.PR]
Volume	1
State	Published - Dec 30 2016

Keywords

Stochastic Partial Diﬀerential Equations
Mild solutions
Itô Formula
Generator of a Semigroup
Yosida approximation
exponential stability

Disciplines

Mathematics

Access to Document

https://arxiv.org/pdf/1612.09440

Cite this

@article{19274f85cec94728a3c6c34eaa957c81,

title = "It{\^o} formula for mild solutions of SPDEs with Gaussian and non-Gaussian noise and applications to stability properties",

abstract = " We use Yosida approximation to \&filig;nd an It\ô formula for mild solutions \{ Xx(t), t \≥ 0 \}of SPDEs with Gaussianand non-Gaussian coloured noise, the non Gaussian noise being de\&filig;ned through compensated Poisson random measure associated to a L\évy process. The functions to which we apply such It\ô formula are in C1,2([0,T]\×H), as in the case considered for SDEs in [19]. Using this It\ô formula we prove exponential stability and exponential ultimate boundedness properties in mean square sense for mild solutions. We also compare such It\ô formula to an It\ô formula for mild solutions introduced by Ichikawain [15], and an It\ô formula written in terms of the semigroup of the drift operator [6] which we extend before to the non Gaussian case.",

keywords = "Stochastic Partial Diﬀerential Equations, Mild solutions, It{\^o} Formula, Generator of a Semigroup, Yosida approximation, exponential stability",

author = "Sergio Albeverio and Leszek Gawarecki and Vidyadhar Mandrekar and Barbara R{\"u}diger and Barun Sarkar",

year = "2016",

month = dec,

day = "30",

language = "American English",

volume = "1",

journal = "arXiv:1612.09440 [math.PR]",

}

TY - JOUR

T1 - Itô formula for mild solutions of SPDEs with Gaussian and non-Gaussian noise and applications to stability properties

AU - Albeverio, Sergio

AU - Gawarecki, Leszek

AU - Mandrekar, Vidyadhar

AU - Rüdiger, Barbara

AU - Sarkar, Barun

PY - 2016/12/30

Y1 - 2016/12/30

N2 - We use Yosida approximation to &filig;nd an Itô formula for mild solutions { Xx(t), t ≥ 0 }of SPDEs with Gaussianand non-Gaussian coloured noise, the non Gaussian noise being de&filig;ned through compensated Poisson random measure associated to a Lévy process. The functions to which we apply such Itô formula are in C1,2([0,T]×H), as in the case considered for SDEs in [19]. Using this Itô formula we prove exponential stability and exponential ultimate boundedness properties in mean square sense for mild solutions. We also compare such Itô formula to an Itô formula for mild solutions introduced by Ichikawain [15], and an Itô formula written in terms of the semigroup of the drift operator [6] which we extend before to the non Gaussian case.

AB - We use Yosida approximation to &filig;nd an Itô formula for mild solutions { Xx(t), t ≥ 0 }of SPDEs with Gaussianand non-Gaussian coloured noise, the non Gaussian noise being de&filig;ned through compensated Poisson random measure associated to a Lévy process. The functions to which we apply such Itô formula are in C1,2([0,T]×H), as in the case considered for SDEs in [19]. Using this Itô formula we prove exponential stability and exponential ultimate boundedness properties in mean square sense for mild solutions. We also compare such Itô formula to an Itô formula for mild solutions introduced by Ichikawain [15], and an Itô formula written in terms of the semigroup of the drift operator [6] which we extend before to the non Gaussian case.

KW - Stochastic Partial Diﬀerential Equations

KW - Mild solutions

KW - Itô Formula

KW - Generator of a Semigroup

KW - Yosida approximation

KW - exponential stability

UR - https://digitalcommons.kettering.edu/mathematics_facultypubs/6

UR - https://arxiv.org/pdf/1612.09440

M3 - Article

VL - 1

JO - arXiv:1612.09440 [math.PR]

JF - arXiv:1612.09440 [math.PR]

ER -

Itô formula for mild solutions of SPDEs with Gaussian and non-Gaussian noise and applications to stability properties

Abstract

Keywords

Disciplines

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