We investigate mild solutions for stochastic evolution equations driven by a fractional Brownian motion (fBm) with Hurst parameter H ∈ (1/3, 1/2] in infinite-dimensional Banach spaces. Using elements from rough paths theory we introduce an appropriate integral with respect to the fBm. This allows us to solve pathwise our stochastic evolution equation in a suitable function space. We are grateful to M. J. Garrido-Atienza and B. Schmalfuß for helpful comments. We thank the referee for carefully reading the manuscript and for the valuable suggestions.AN acknowledges support by a DFG grant in the D-A-CH framework (KU 3333/2-1). 1 index H ∈ (1/3, 1/2]. In order to solve (1.1) we need to give a meaning of the rough integral t 0 S(t − r)G(y r )dω r . (1.2) Results in this context are available in [10] via fractional calculus and in Gubinelli et al [11], [4] [12], [13] using rough paths techniques. In this work, we combine Gubinelli's approach with the arguments employed by [10] to solve (1.1). This theory should hopefully be more simple in order to investigate the long-time behavior of such equations using a random dynamical systems approach such in [1], [8] or [2]. In this work we establish only the existence of a local mild solution. We investigate in a forthcoming paper global solutions and random dynamical systems for (1.1) as in [8]. This work should be seen as a first step in order to close the gap between rough paths and random dynamical systems in infinite-dimensional spaces. Since the fractional Brownian motion is not a semi-martingale, the construction of an appropriate integral represents a challenging problem. This has been intensively investigated and numerous results and various techniques are available, see [26], [8], [25], [16], [19] and the references specified therein. There is a huge literature where certain tools from fractional calculus (i.e. fractional/compensated fractional derivative/integral) are employed to give a pathwise meaning of the stochastic integral with respect to the fractional Brownian motion with Hurst parameter H ∈ (1/2, 1) or H ∈ (1/3, 1/2]. A different method which has been recently introduced and explored is given by the rough path approach of Gubinelli et. al. [11], [13], [12]. This goes through if H ≤ 1/2. Moreover it is suitable to define (1.2) not only with respect to the fractional Brownian motion but also to Gaussian processes for which the covariance function satisfies certain structure, see [6] or [5, Chapter 10]. An overview on the connection between rough paths and fractional calculus can be looked up in [16]. Of course, in some situations, various other techniques for H ≤ 1/3 are available. After using an appropriate integration theory with respect to the fractional Brownian motion, the next step is to analyze SDEs/SPDEs driven by this kind of noise. There is a growing interest in establishing suitable properties of the solution under several assumptions on the coefficients, consult [19], [20], [14], [7], [8], [26] and the references specified therein. To our aims we ...

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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The existence of global-in-time bounded martingale solutions to a general class of cross-diffusion systems with multiplicative Stratonovich noise is proved. The equations describe multicomponent systems from physics or biology with volume-filling effects and possess a formal gradient-flow or entropy structure. This structure allows for the derivation of almost surely positive lower and upper bounds for the stochastic processes. The existence result holds under some assumptions on the interplay between the entropy density and the multiplicative noise terms. The proof is based on a stochastic Galerkin method, a Wong-Zakai type approximation of the Wiener process, the boundedness-by-entropy method, and the tightness criterion of Brzeźniak and coworkers. Three-species Maxwell-Stefan systems and n-species biofilm models are examples that satisfy the general assumptions.