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

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

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

Mental Stress Provokes Ischemia in Coronary Artery Disease Subjects Without Exercise- or Adenosine-Induced Ischemia

ObjectivesThe purpose of this study was to investigate the possibility that some patients with coronary artery disease (CAD) but negative exercise or chemical stress test results might have mental stress-induced ischemia. The study population consisted solely of those with negative test results.BackgroundMental stress-induced ischemia has been reported in 20% to 70% of CAD subjects with exercise-induced ischemia. Because mechanisms of exercise and mental stress-induced ischemia may differ, we studied whether mental stress would produce ischemia in a proportion of subjects with CAD who have no inducible ischemia with exercise or pharmacologic tests.MethodsTwenty-one subjects (14 men, 7 women) with a mean age of 67 years and with a documented history of CAD were studied. All subjects had a recent negative nuclear stress test result (exercise or chemical). Subjects completed a speaking task involving role playing a difficult interpersonal situation. A total of 30 mCi 99mTc-sestamibi was injected at one minute into the speech, and imaging was started 40 min later. A resting image obtained within one week was compared with the stress image. Images were analyzed for number and severity of perfusion defects. The summed difference score based on the difference between summed stress and rest scores was calculated. Severity was assessed using a semiquantitative scoring method from zero to four.ResultsSix of 21 (29%) subjects demonstrated reversible ischemia (summed difference score ≥3) with mental stress. No subject had chest pain or electrocardiographic changes during the stressor. Mean systolic and diastolic blood pressure and heart rate all increased between resting and times of peak stress.ConclusionsMental stress may produce ischemia in some subjects with CAD and negative exercise or chemical nuclear stress test results

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

Marsupial extension in terrestrial isopods (Crustacea, Isopoda, Oniscidea)

In Oniscidea, the marsupium is a ventral pouch where the offspring develop independently of an external water source. The marsupium is formed by five pairs of overlapping oostegites that develop in the females during their reproductive period. In this study, ovigerous females of 35 species were dissected, their oostegites were extracted, and the intra-marsupial offspring were counted. Two marsupium forms were recognized: distended, in which the oostegites protrude distally in relation to the sternites; and non-distended, in which the oostegites are parallel to the sixth and seventh sternites. Armadillidium nasatum, A. vulgare, Pudeoniscus birabeni, Circoniscus gaigei and Cubaris murina, conglobating species with a non-distended marsupium, and Neotroponiscus daguerri and N. carolii, non-conglobating species with a distended marsupium, have a concavity on the ventral floor of the 6th and 7th pereionites, here called the marsupial extension. This is the first record of a marsupial extension which extends beyond the area formed by the oostegites in Oniscidea

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