Criterion on stability for Markov processes applied to a model with jumps

We formulate and prove a new criterion for stability of e-processes. In particular we show that any e-process which is averagely bounded and concentrating is asymptotically stable. This general result is applied to a stochastic process with jumps that is a continuous counterpart of the chain considered in Szarek (Ann. Probab. 34:1849-1863, 2006)

Existence and uniqueness for stochastic 2D Euler flows with bounded vorticity

The strong existence and the pathwise uniqueness of solutions with (Formula presented.) -vorticity of the 2D stochastic Euler equations are proved. The noise is multiplicative and it involves the first derivatives. A Lagrangian approach is implemented, where a stochastic flow solving a nonlinear flow equation is constructed. The stability under regularizations is also proved

Stochastic attractors for shell phenomenological models of turbulence

Recently, it has been proposed that the Navier-Stokes equations and a relevant linear advection model have the same long-time statistical properties, in particular, they have the same scaling exponents of their structure functions. This assertion has been investigate rigorously in the context of certain nonlinear deterministic phenomenological shell model, the Sabra shell model, of turbulence and its corresponding linear advection counterpart model. This relationship has been established through a "homotopy-like" coefficient $\lambda$ which bridges continuously between the two systems. That is, for $\lambda=1$ one obtains the full nonlinear model, and the corresponding linear advection model is achieved for $\lambda=0$. In this paper, we investigate the validity of this assertion for certain stochastic phenomenological shell models of turbulence driven by an additive noise. We prove the continuous dependence of the solutions with respect to the parameter $\lambda$. Moreover, we show the existence of a finite-dimensional random attractor for each value of $\lambda$ and establish the upper semicontinuity property of this random attractors, with respect to the parameter $\lambda$. This property is proved by a pathwise argument. Our study aims toward the development of basic results and techniques that may contribute to the understanding of the relation between the long-time statistical properties of the nonlinear and linear models

Nucleus-specific abnormalities of GABAergic synaptic transmission in a genetic model of absence seizures

Human and experimental studies indicate that molecular genetic changes in GABAA receptors may underlie the expression of spike-and-waves discharges (SWDs) occurring during absence seizures. However, the full spectrum of the genetic defects underlying these seizures has only been partially elucidated, the expression and functional profiles of putative abnormal protein(s) within the thalamocortical network are undefined, and the pathophysiological mechanism(s) by which these proteins would lead to absence paroxysms are poorly understood. Here we investigated GABAA inhibitory postsynaptic currents (IPSCs) in key thalamocortical areas, i.e., the somatosensory cortex, ventrobasal thalamus (VB) and nucleus reticularis thalami (NRT), in preseizure genetic absence epilepsy rats from Strasbourg (GAERS), a well-established genetic model of typical absence seizures that shows no additional neurological abnormalities, and compared their properties to age-matched non-epileptic controls (NECs). Miniature GABAA IPSCs of VB and cortical layers II/III neurons were similar in GAERS and NEC, whereas in GAERS NRT neurons they had 25% larger amplitude, 40% faster decay. In addition, baclofen was significantly less effective in decreasing the frequency of NRT mIPSCs in GAERS than in NEC, whereas no difference was observed for cortical and VB mIPSCS between the two strains. Paired-pulse depression was 45% smaller in GAERS NRT, but not in VB, and was insensitive to GABAB antagonists. These results point to subtle, nucleus-specific, GABAA receptor abnormalities underlying SWDs of typical absence seizures rather than a full block of these receptors across the whole thalamocortical network, and their occurrence prior to seizure onset suggests that they might be of epileptogenic significance