Random attractors for stochastic evolution equations driven by fractional Brownian motion

The main goal of this article is to prove the existence of a random attractor for a stochastic evolution equation driven by a fractional Brownian motion with $H\in (1/2,1)$. We would like to emphasize that we do not use the usual cohomology method, consisting of transforming the stochastic equation into a random one, but we deal directly with the stochastic equation. In particular, in order to get adequate a priori estimates of the solution needed for the existence of an absorbing ball, we will introduce stopping times to control the size of the noise. In a first part of this article we shall obtain the existence of a pullback attractor for the non-autonomous dynamical system generated by the pathwise mild solution of an nonlinear infinite-dimensional evolution equation with non--trivial H\"older continuous driving function. In a second part, we shall consider the random setup: stochastic equations having as driving process a fractional Brownian motion with $H\in (1/2,1)$. Under a smallness condition for that noise we will show the existence and uniqueness of a random attractor for the stochastic evolution equation

Mikrochemische Zinkbestimmung im Serum und Urin

Peer Reviewe

The random case of Conley's theorem

The well-known Conley's theorem states that the complement of chain recurrent set equals the union of all connecting orbits of the flow $\phi$ on the compact metric space $X$, i.e. $X-\mathcal{CR}(\phi)=\bigcup [B(A)-A]$, where $\mathcal{CR}(\phi)$ denotes the chain recurrent set of $\phi$, $A$ stands for an attractor and $B(A)$ is the basin determined by $A$. In this paper we show that by appropriately selecting the definition of random attractor, in fact we define a random local attractor to be the $\omega$-limit set of some random pre-attractor surrounding it, and by considering appropriate measurability, in fact we also consider the universal $\sigma$-algebra $\mathcal F^u$-measurability besides $\mathcal F$-measurability, we are able to obtain the random case of Conley's theorem.Comment: 15 page

Calcite distribution and orientation in the tergite exocuticle of the isopods porcellio scaber and armadillidium vulgare (Oniscidea, Crustacea) - A combined FE-SEM, polarized SCm-RSI and EBSD study

The crustacean cuticle is a bio-composite consisting of hierarchically organized chitin-protein fibres, reinforced with calcite, amorphous calcium carbonate and phosphates. Comparative studies revealed that the structure and composition of tergite cuticle of terrestrial isopods is adapted to the habitat of the animals, and to their behavioural patterns to avoid predation. In this contribution we use FE-SEM, polarized SCm-RSI and EBSD to investigate micro- and nano-patterns of mineral phase distribution and crystal orientation within the tergite cuticle of the two terrestrial isopod species Armadillidium vulgare and Porcellio scaber. The results show that the proximal regions of the exocuticle contain both calcite and ACC, with ACC located within the pore canals. Calcite forms hierarchically organised mesocrystalline aggregates of similar crystallographic orientation. Surprisingly, c-axis orientation preference is horizontal in regard to the local cuticle surface for both species, in contrast to mollusc and brachiopod shell structures in which the c-axis is always perpendicular to the shell surface. The overall sharpness of calcite crystal orientation is weak compared to that of mollusc shells. However, there are considerable differences in texture sharpness between the two isopod species. In the thick cuticle of the slow-walking A. vulgare calcite is more randomly oriented resulting in more isotropic mechanical properties of the cuticle. In contrast, the rather thin and more flexible cuticle of the fast- running P. scaber texture sharpness is stronger with a preference of c-axis orientation being parallel to the bilateral symmetry-plane of the animal, leading to more anisotropic mechanical properties of the cuticle. These differences may represent adaptations to different external and/or internal mechanical loads the cuticle has to resist during predatory attempts

Smooth stable and unstable manifolds for stochastic partial differential equations

Invariant manifolds are fundamental tools for describing and understanding nonlinear dynamics. In this paper, we present a theory of stable and unstable manifolds for infinite dimensional random dynamical systems generated by a class of stochastic partial differential equations. We first show the existence of Lipschitz continuous stable and unstable manifolds by the Lyapunov-Perron's method. Then, we prove the smoothness of these invariant manifolds

Random attractors for degenerate stochastic partial differential equations

We prove the existence of random attractors for a large class of degenerate stochastic partial differential equations (SPDE) perturbed by joint additive Wiener noise and real, linear multiplicative Brownian noise, assuming only the standard assumptions of the variational approach to SPDE with compact embeddings in the associated Gelfand triple. This allows spatially much rougher noise than in known results. The approach is based on a construction of strictly stationary solutions to related strongly monotone SPDE. Applications include stochastic generalized porous media equations, stochastic generalized degenerate p-Laplace equations and stochastic reaction diffusion equations. For perturbed, degenerate p-Laplace equations we prove that the deterministic, infinite dimensional attractor collapses to a single random point if enough noise is added.Comment: 34 pages; The final publication is available at http://link.springer.com/article/10.1007%2Fs10884-013-9294-

The random case of Conley's theorem: III. Random semiflow case and Morse decomposition

In the first part of this paper, we generalize the results of the author \cite{Liu,Liu2} from the random flow case to the random semiflow case, i.e. we obtain Conley decomposition theorem for infinite dimensional random dynamical systems. In the second part, by introducing the backward orbit for random semiflow, we are able to decompose invariant random compact set (e.g. global random attractor) into random Morse sets and connecting orbits between them, which generalizes the Morse decomposition of invariant sets originated from Conley \cite{Con} to the random semiflow setting and gives the positive answer to an open problem put forward by Caraballo and Langa \cite{CL}.Comment: 21 pages, no figur

Sea-land transitions in isopods: pattern of symbiont distribution in two species of intertidal isopods Ligia pallasii and Ligia occidentalis in the Eastern Pacific

Studies of microbial associations of intertidal isopods in the primitive genus Ligia (Oniscidea, Isopoda) can help our understanding of the formation of symbioses during sea-land transitions, as terrestrial Oniscidean isopods have previously been found to house symbionts in their hepatopancreas. Ligia pallasii and Ligia occidentalis co-occur in the high intertidal zone along the Eastern Pacific with a large zone of range overlap and both species showing patchy distributions. In 16S rRNA clone libraries mycoplasma-like bacteria (Firmicutes), related to symbionts described from terrestrial isopods, were the most common bacteria present in both host species. There was greater overall microbial diversity in Ligia pallasii compared with L. occidentalis. Populations of both Ligia species along an extensive area of the eastern Pacific coastline were screened for the presence of mycoplasma-like symbionts with symbiont-specific primers. Symbionts were present in all host populations from both species but not in all individuals. Phylogenetically, symbionts of intertidal isopods cluster together. Host habitat, in addition to host phylogeny appears to influence the phylogenetic relation of symbionts