On the continuity of local times of Borel right Markov processes

The problem of finding a necessary and sufficient condition for the continuity of the local times for a general Markov process is still open. Barlow and Hawkes have completely treated the case of the L\'{e}vy processes, and Marcus and Rosen have solved the case of the strongly symmetric Markov processes. We treat here the continuity of the local times of Borel right processes. Our approach unifies that of Barlow and Hawkes and of Marcus and Rosen, by using an associated Gaussian process, that appears as a limit in a CLT involving the local time process.Comment: Published at http://dx.doi.org/10.1214/009117906000000980 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

A characterization of the infinitely divisible squared Gaussian processes

We show that, up to multiplication by constants, a Gaussian process has an infinitely divisible square if and only if its covariance is the Green function of a transient Markov process.Comment: Published at http://dx.doi.org/10.1214/009117905000000684 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Inverting Ray-Knight identity

We provide a short proof of the Ray-Knight second generalized Theorem, using a martingale which can be seen (on the positive quadrant) as the Radon-Nikodym derivative of the reversed vertex-reinforced jump process measure with respect to the Markov jump process with the same conductances. Next we show that a variant of this process provides an inversion of that Ray-Knight identity. We give a similar result for the Ray-Knight first generalized Theorem.Comment: 18 page

Decompositions of infinitely divisible nonnegative processes

International audienceWe establish decomposition formulas for nonnegative infinitely divisible processes. They allow to give an explicit expression of their Lévy measure. In the special case of infinitely divisible permanental processes, one of these decompositions represents a new isomorphism theorem involving the local time process of a transient Markov process. We obtain in this case the expression of the Lévy measure of the total local time process which is in itself a new result on the local time process. Finally, we identify a determining property of the local times for their connection with permanental processes

Dynkin's isomorphism theorem and the stochastic heat equation

Consider the stochastic heat equation \partial_t u = \sL u + \dot{W}, where \sL is the generator of a [Borel right] Markov process in duality. We show that the solution is locally mutually absolutely continuous with respect to a smooth perturbation of the Gaussian process that is associated, via Dynkin's isomorphism theorem, to the local times of the replica-symmetric process that corresponds to \sL.In the case that \sL is the generator of a L\'evy process on $\R^d$, our result gives a probabilistic explanation of the recent findings of Foondun et al

A family of integral representations for the brownian variables

ABSTRACT. – The natural filtration of a real Brownian motion and its excursion filtration are sharing a fundamental property: the property of integral representation. As a consequence, every Brownian variable admits two distinct integral representations. We show here that there are other integral representations of the Brownian variables. They make use of a stochastic flow studied by Bass and Burdzy. Our arguments are inspired by Rogers and Walsh’s results on stochastic integration with respect to the Brownian local times. 2003 Éditions scientifiques et médicales Elsevier SAS MSC: 60H05; 60G4

Role of Cerebrovascular Cells in Tau Processing Following Traumatic Brain Injury

Repetitive exposure to mild traumatic brain injuries (r-mTBI) sustained through the participation in contact sports can lead to chronic post-concussive symptoms and the development of neurodegenerative diseases such as Alzheimer’s disease and Chronic Traumatic Encephalopathy (CTE). A primary hallmark of CTE is the accumulation of pathogenic tau in neurons and astrocytes that surround small blood vessels in the brain. Chronic exposure to r-mTBI leads to elevated levels of extracellular tau and pathogenic tau accumulation in neurons, ultimately resulting in neuronal death. While the mechanisms responsible for pathogenic tau elimination from the brain are unclear, our prior work demonstrated that cells associated with the cerebrovasculature can interact with extracellular tau and may contribute to the removal of extracellular tau from the brain. In this thesis, I examined the mechanisms through which the cerebrovascular cells eliminate extracellular tau from the brain and how those processes are impacted by r-mTBI. I demonstrated that brain vascular mural cells (pericytes and smooth muscle cells) progressively degenerate following exposure to r-mTBI consistent with what is observed in individuals with AD. This mural cell dysfunction impairs the ability of the cerebrovessels to interact with tau. Furthermore, I found that the cerebrovasculature can eliminate extracellular tau from the brain through caveolae-mediated endothelial transcytosis, which is impaired following chronic exposure to r-mTBI. The diminished tau transit across the blood-brain barrier following brain injury may be a contributing factor in the pathogenic tau accumulation observed in CTE. A significant genetic risk factor for neurodegenerative diseases including AD and CTE is possession of the E4 isoform of Apolipoprotein E (ApoE). Astrocytes are the predominant source of ApoE in the brain, though there is very little understanding regarding their interactions with extracellular tau, particularly after exposure to head trauma. While the ApoE4 isoform has been associated with increased tau accumulation and cerebrovascular dysfunction after TBI, investigations into these associations are limited. The current studies found that while astrocytes internalize and release tau back into the extracellular space under normal conditions, these processes become dysfunctional following r-mTBI leading to astrocytic tau accumulation, which is further exacerbated by the ApoE4 isoform. In summary, I identified the factors responsible for the elimination of extracellular tau across the BBB, which are impaired after head trauma. Therapeutic interventions that restore these processes may ameliorate the chronic accumulation of tau associated with neurodegenerative disease. These findings may be particularly important for individuals with the ApoE4 isoform, who are more susceptible to the pathophysiological sequelae of tau accumulation, particularly after exposure to r-mTBI