Nonlinear stochastic partial differential equations of hyperbolic type driven by Lévy-type noises

“…From all these deliberations we finally conclude Proof. The proof in [64,Section 4] and [14,Section 3] adapts to our setting. We denote by S the set of all stopping times such that τ ∈ S if and only if there exists a process (u(t)) t∈[0,τ ) such that (u, τ ) is the unique local pathwise mild solution of (3.24).…”

Section: Remark 327mentioning

confidence: 99%

“…For example, if the determinisitc PDE part is a cross-diffusion system with an entropy structure [37] and if the noise is multiplicative, we expect that the maximal local pathwise mild solution obtained in Theorem 3.28 is indeed a global one. We plan to investigate this in a future work using for instance using Khashminski's test for non-explosion; see for example [64,Lemma 4.1], [16,Theorem 3.2] or [46,Section 5].…”

Section: Remark 329mentioning

confidence: 99%

Pathwise mild solutions for quasilinear stochastic partial differential equations

Kuehn

Neamţu

2020

Journal of Differential Equations

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Stochastic partial differential equations (SPDEs) have become a key modelling tool in applications. Yet, there are many classes of SPDEs, where the existence and regularity theory for solutions is not completely developed. Here we contribute to this aspect and prove the existence of mild solutions for a broad class of quasilinear Cauchy problems, includingamong others -cross-diffusion systems as a key application. Our solutions are local-in-time and are derived via a fixed point argument in suitable function spaces. The key idea is to combine the theory of deterministic quasilinear parabolic partial differential equations (PDEs) with recent theory of evolution semigroups. We also show, how to apply our theory to the Shigesada-Kawasaki-Teramoto (SKT) model. Furthermore, we provide examples of blow-up and ill-posed operators, which can occur after finite-time.Here we aim to develop a theory for quasilinear stochastic evolution equations (1.1) using a semigroup approach. The main theme is to extend the very general deterministic theory of quasilinear Cauchy problems [3,61,63]. The key idea is to employ a modified definition of mild solutions [57] for the quasilinear case in comparison to the more classical parabolic SPDE setting [54]. Before we describe our approach in more detail, we briefly review some other techniques and solution concepts used for (certain subclasses of) the SPDE (1.1). Instead of mild solutions, one may instead use weak, or martingale, solutions [13,20,24,35] of (1.1); here weak solution is interpreted in the classical PDE sense while these solutions are also sometimes referred to as strong solutions from a probabilistic perspective [55]. There are also several works exploiting the additional assumption of monotone coefficients [43] particularly in the case of the stochastic porous medium equation [9,10], where A(u) = ∆(a(u)) for a maximal monotone map a and F ≡ 0. Other approaches to quasilinear SPDEs are based upon a gradient structure [29], approximation methods [38,39], kinetic solutions [20,30], or directly looking at strong (in the PDE sense) solutions [36].One may ask, why one might want to prove the existence of pathwise mild solutions obtained by a suitable variations-of-constants/Duhamel formula [34] instead of working with weak solutions obtained in a formulation via test functions [26]? One reason is that a mild formulation is often more natural to work with in the context of (random) dynamical systems for SPDEs [18,34]. In fact, many classical results regarding dynamics and long-time behavior of semilinear SPDEs are often crucially based upon the mild formulation and semigroups [34]. We expect this theory to generalize a lot easier also in the quasilinear case if one does not have to work with weak(er) solutions. If we take the SKT system again as a motivation, then there are deterministic results regarding the existence of attractors using weak [53] as well as mild [62] solutions concepts. We intend to investigate the existence of random attractors for the stochastic SKT equation ...

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“…Many interesting and important results have been studied for linear and non-linear SPDEs driven by 2990 MIN NIU AND BIN XIE jump noises. Firstly, the fundamental problem of the existence and uniqueness of solutions is wildly studied [1,3,4,17,21,26,27,28]. On the other hand, the other important properties for SPDEs driven by jump noises are also actively studied, such as the large deviation principle [6], the moderate deviation principle [10], invariant measure [17], exponential ergodicity [9,24] and so on.…”

mentioning

confidence: 99%

Comparison theorem and correlation for stochastic heat equations driven by Lévy space-time white noises

Niu¹,

Xie²

2019

Discrete &Amp; Continuous Dynamical Systems - B

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Two properties of stochastic heat equations driven by impulsive noises, which are also called Lévy space-time white noises, are mainly investigated in this paper. We first study the comparison theorem for two stochastic heat equations driven by same noises under some sufficient condition, which is proved via the application of Itô's formula. In particular, we obtain the nonnegativity of solutions with non-negative initial data. Then, we investigate the positive correlation of the solutions as the application of the comparison theorem. We prove that the total masses of two solutions relative to two different stochastic heat equations with same noise become nearly uncorrelated after a long time.

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Nonlinear stochastic partial differential equations of hyperbolic type driven by Lévy-type noises

Cited by 8 publications

References 25 publications

The stochastic nonlinear Schrödinger equations driven by pure jump noise

The stochastic nonlinear Schrödinger equations driven by pure jump noise

Pathwise mild solutions for quasilinear stochastic partial differential equations

Comparison theorem and correlation for stochastic heat equations driven by Lévy space-time white noises

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