Asymptotics for the heat kernel in multicone domains

A multi cone domain $\Omega \subseteq \mathbb{R}^n$ is an open, connected set that resembles a finite collection of cones far away from the origin. We study the rate of decay in time of the heat kernel $p(t,x,y)$ of a Brownian motion killed upon exiting $\Omega$, using both probabilistic and analytical techniques. We find that the decay is polynomial and we characterize $\lim_{t\to\infty} t^{1+\alpha}p(t,x,y)$ in terms of the Martin boundary of $\Omega$ at infinity, where $\alpha>0$ depends on the geometry of $\Omega$. We next derive an analogous result for $t^{\kappa/2}\mathbb{P}_x(T >t)$, with $\kappa = 1+\alpha - n/2$, where $T$ is the exit time form $\Omega$. Lastly, we deduce the renormalized Yaglom limit for the process conditioned on survival.Comment: 31 page

Existence and uniqueness of a quasi-stationary distribution for Markov processes with fast return from infinity

We study the long time behaviour of a Markov process evolving in $\mathbb{N}$ and conditioned not to hit 0. Assuming that the process comes back quickly from infinity, we prove that the process admits a unique quasi-stationary distribution (in particular, the distribution of the conditioned process admits a limit when time goes to infinity). Moreover, we prove that the distribution of the process converges exponentially fast in total variation norm to its quasi-stationary distribution and we provide an explicit rate of convergence. As a first application of our result, we bring a new insight on the speed of convergence to the quasi-stationary distribution for birth and death processes: we prove that these processes converge exponentially fast to a quasi-stationary distribution if and only if they have a unique quasi-stationary distribution. Also, considering the lack of results on quasi-stationary distributions for non-irreducible processes on countable spaces, we show, as a second application of our result, the existence and uniqueness of a quasi-stationary distribution for a class of possibly non-irreducible processes.Comment: 17 page

Stochastic models for a chemostat and long time behavior

We introduce two stochastic chemostat models consisting in a coupled population-nutrient process reflecting the interaction between the nutrient and the bacterias in the chemostat with finite volume. The nutrient concentration evolves continuously but depending on the population size, while the population size is a birth and death process with coefficients depending on time through the nutrient concentration. The nutrient is shared by the bacteria and creates a regulation of the bacterial population size. The latter and the fluctuations due to the random births and deaths of individuals make the population go almost surely to extinction. Therefore, we are interested in the long time behavior of the bacterial population conditioned to the non-extinction. We prove the global existence of the process and its almost sure extinction. The existence of quasi-stationary distributions is obtained based on a general fixed point argument. Moreover, we prove the absolute continuity of the nutrient distribution when conditioned to a fixed number of individuals and the smoothness of the corresponding densities

Quasi-stationary distributions for structured birth and death processes with mutations

We study the probabilistic evolution of a birth and death continuous time measure-valued process with mutations and ecological interactions. The individuals are characterized by (phenotypic) traits that take values in a compact metric space. Each individual can die or generate a new individual. The birth and death rates may depend on the environment through the action of the whole population. The offspring can have the same trait or can mutate to a randomly distributed trait. We assume that the population will be extinct almost surely. Our goal is the study, in this infinite dimensional framework, of quasi-stationary distributions when the process is conditioned on non-extinction. We firstly show in this general setting, the existence of quasi-stationary distributions. This result is based on an abstract theorem proving the existence of finite eigenmeasures for some positive operators. We then consider a population with constant birth and death rates per individual and prove that there exists a unique quasi-stationary distribution with maximal exponential decay rate. The proof of uniqueness is based on an absolute continuity property with respect to a reference measure.Comment: 39 page

Dynamic nuclear polarization enhanced solid-state NMR studies of surface modification of gamma-alumina

Dynamic nuclear polarization (DNP) gives large (>100-fold) signal enhancements in solid-state NMR spectra via the transfer of spin polarization from unpaired electrons from radicals implanted in the sample. This means that the detailed information about local molecular environment available for bulk samples from solid-state NMR spectroscopy can now be obtained for dilute species, such as sites on the surfaces of catalysts and catalyst supports. In this paper we describe a DNP-enhanced solid-state NMR study of the widely used catalyst gamma-alumina which is often modified at the surface by the incorporation of alkaline earth oxides in order to control the availability of catalytically active penta-coordinate surface Al sites. DNP-enhanced 27Al solid-state NMR allows surface sites in gamma-alumina to be observed and their 27Al NMR parameters measured. In addition changes in the availability of different surface sites can be detected after incorporation of BaO

Open multi-technology building energy management system

Energy Efficiency is one of the goals of the Smart Building initiatives. This paper presents an Open Energy Management System which consists of an ontology-based multi-technology platform and a wireless transducer network using 6LoWPAN communication technology. The system allows the integration of several building automation protocols and eases the development of different kind of services to make use of them. The system has been implemented and tested in the Energy Efficiency Research Facility at CeDInt-UPM