A new approach to stochastic evolution equations with adapted drift

Pronk, Matthijs; Veraar, Mark

doi:10.1016/j.jde.2014.02.014

Cited by 21 publications

(119 citation statements)

References 40 publications

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“…This formula has been introduced as so-called "pathwise mild solution" in a much more general setting (cf. [17]) and manifests a way to pass around the difficulty of defining a mild-solution-like stochastic convolution in the case where the integrand cannot be assumed to be adapted. This indeed occurs if the operator A(t) depends on the underlying probability space.…”

Section: Ornstein-uhlenbeck Decomposition Of the L 2 -Componentmentioning

confidence: 99%

“…As a consequence of the randomness of the linear drift term it is not possible to directly define a stochastic convolution of the (possibly non-adapted) semigroup with the driving Wiener process as required for a mild-solution concept. We therefore represent the Ornstein-Uhlenbeck process via a stochastic convolution of new type (recently introduced in [17]) that is applicable to general adapted random drifts and allows us to derive a locally uniform (in time) pathwise control on the Ornstein-Uhlenbeck component. A similar approach based on a pathwise control of the stochastic convolution has also been taken in [19].…”

Section: Introductionmentioning

confidence: 99%

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Front Propagation in Stochastic Neural Fields: A Rigorous Mathematical Framework

Krüger¹,

Stannat²

2014

SIAM J. Appl. Dyn. Syst.

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Abstract. We develop a complete and rigorous mathematical framework for the analysis of stochastic neural field equations under the influence of spatially extended additive noise. By comparing a solution to a fixed deterministic front profile it is possible to realise the difference as strong solution to an L 2 (R)-valued SDE. A multiscale analysis of this process then allows us to obtain rigorous stability results. Here a new representation formula for stochastic convolutions in the semigroup approach to linear function-valued SDE with adapted random drift is applied. Additionally, we introduce a dynamic phaseadaption process of gradient type.

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Section: Ornstein-uhlenbeck Decomposition Of the L 2 -Componentmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Front Propagation in Stochastic Neural Fields: A Rigorous Mathematical Framework

Krüger¹,

Stannat²

2014

SIAM J. Appl. Dyn. Syst.

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“…To deal with rough noises an alternative to regularity structures is to stay closer to a paracontrolled approach [31], which has recently been proposed for certain quasilinear SPDEs [8,27,50,51].Since the linear operator A depends on the solution itself, which will be in our case a stochastic process, we cannot apply the standard fixed-point argument as in [3,63]. Namely, if we denote with U u the random evolution operator generated by A(u), one naturally expects that the mild solution of (1.1) should be given by the variation-of-constants formula u(t) = U u (t, 0)u 0 + t 0 U u (t, s)F (s, u(s)) ds +t 0 U u (t, s)σ(s, u(s)) dW (s).(1.2)As already observed in [57], and justified in Sections 2 and 3, the random evolution operator…”

mentioning

confidence: 73%

“…A way out of this situation is to introduce a new concept of mild solution for (1.1), which is based on the integration-by-parts formula for stochastic convolutions. This is motivated in [57,Sec. 4] as well as in Appendix A here for convenience.…”

Section: Introductionmentioning

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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“…In the nonautonomous case, one can also use the forward integral of Russo-Vallois [153] to define the convolution (2.18) and to construct a solution to the corresponding SPDE which coincides with the pathwise mild solution as argued in [151,Section 4.5]. The statement can be proved using fixed-point arguments as in [123].…”

Section: The Quasilinear Casementioning

confidence: 99%