Abstract:
A stochastic dynamics $({\bf X}(t))_{t\ge0}$ of a classical continuous system is a stochastic process which takes values in the space $\Gamma$ of all locally finite subsets (configurations) in $\Bbb R$ and which has a Gibbs measure $\mu$ as an invariant measure. We assume that $\mu$ corresponds to a symmetric pair potential $\phi(x-y)$. An important class of stochastic dynamics of a classical continuous system is formed by diffusions. Till now, only one type of such dynamics--the so-called gradient stochastic dynamics, or interacting Brownian particles--has been investigated. By using the theory of Dirichlet forms, we construct and investigate a new type of stochastic dynamics, which we call infinite interacting diffusion particles. We introduce a Dirichlet form ${\cal E}_\mu^\Gamma$ on $L^2(\Gamma;\mu)$, and under general conditions on the potential $\phi$, prove its closability. For a potential $\phi$ having a ``weak'' singularity at zero, we also write down an explicit form of the generator of ${\cal E}_\mu^\Gamma$ on the set of smooth cylinder functions. We then show that, for any Dirichlet form ${\cal E}_\mu^\Gamma$, there exists a diffusion process that is properly associated with it. Finally, we study a scaling limit of interacting diffusions in terms of convergence of the corresponding Dirichlet forms, and we also show that these scaled processes are tight in $C([0,\infty),{\cal D}')$, where ${\cal D}'$ is the dual space of ${\cal D}{:=}C_0^\infty({\Bbb R})$.

Abstract:
We study properties of the semigroup $(e^{-tH})_{t\ge 0}$ on the space $L^ 2(\Gamma_X,\pi)$, where $\Gamma_X$ is the configuration space over a locally compact second countable Hausdorff topological space $X$, $\pi$ is a Poisson measure on $\Gamma_X$, and $H$ is the generator of the Glauber dynamics. We explicitly construct the corresponding Markov semigroup of kernels $(P_t)_{t\ge 0}$ and, using it, we prove the main results of the paper: the Feller property of the semigroup $(P_t)_{t\ge 0}$ with respect to the vague topology on the configuration space $\Gamma_X$, and the ergodic property of $(P_t)_{t\ge 0}$. Following an idea of D. Surgailis, we also give a direct construction of the Glauber dynamics of a continuous infinite system of free particles. The main point here is that this process can start in every $\gamma\in\Gamma_X$, will never leave $\Gamma_X$ and has cadlag sample paths in $\Gamma_X$.

Abstract:
We study the problem of identification of a proper state-space for the stochastic dynamics of free particles in continuum, with their possible birth and death. In this dynamics, the motion of each separate particle is described by a fixed Markov process $M$ on a Riemannian manifold $X$. The main problem arising here is a possible collapse of the system, in the sense that, though the initial configuration of particles is locally finite, there could exist a compact set in $X$ such that, with probability one, infinitely many particles will arrive at this set at some time $t>0$. We assume that $X$ has infinite volume and, for each $\alpha\ge1$, we consider the set $\Theta_\alpha$ of all infinite configurations in $X$ for which the number of particles in a compact set is bounded by a constant times the $\alpha$-th power of the volume of the set. We find quite general conditions on the process $M$ which guarantee that the corresponding infinite particle process can start at each configuration from $\Theta_\alpha$, will never leave $\Theta_\alpha$, and has cadlag (or, even, continuous) sample paths in the vague topology. We consider the following examples of applications of our results: Brownian motion on the configuration space, free Glauber dynamics on the configuration space (or a birth-and-death process in $X$), and free Kawasaki dynamics on the configuration space. We also show that if $X=\mathbb R^d$, then for a wide class of starting distributions, the (non-equilibrium) free Glauber dynamics is a scaling limit of (non-equilibrium) free Kawasaki dynamics.

Abstract:
We study the problem of identification of a proper state-space for the stochastic dynamics of free particles in continuum, with their possible birth and death. In this dynamics, the motion of each separate particle is described by a fixed Markov process M on a Riemannian manifold X. The main problem arising here is a possible collapse of the system, in the sense that, though the initial configuration of particles is locally finite, there could exist a compact set in X such that, with probability one, infinitely many particles will arrive at this set at some time t>0. We assume that X has infinite volume and, for each α≥1, we consider the set Θα of all infinite configurations in X for which the number of particles in a compact set is bounded by a constant times the α-th power of the volume of the set. We find quite general conditions on the process M which guarantee that the corresponding infinite particle process can start at each configuration from Θα, will never leave Θα, and has cadlag (or, even, continuous) sample paths in the vague topology. We consider the following examples of applications of our results: Brownian motion on the configuration space, free Glauber dynamics on the configuration space (or a birth-and-death process in X), and free Kawasaki dynamics on the configuration space. We also show that if X=Rd, then for a wide class of starting distributions, the (non-equilibrium) free Glauber dynamics is a scaling limit of (non-equilibrium) free Kawasaki dynamics.

Abstract:
We construct Gibbs perturbations of the Gamma process on $\mathbbm{R}^d$, which may be used in applications to model systems of densely distributed particles. First we propose a definition of Gibbs measures over the cone of discrete Radon measures on $\mathbbm{R}^d$ and then analyze conditions for their existence. Our approach works also for general L\'evy processes instead of Gamma measures. To this end, we need only the assumption that the first two moments of the involved L\'evy intensity measures are finite. Also uniform moment estimates for the Gibbs distributions are obtained, which are essential for the construction of related diffusions. Moreover, we prove a Mecke type characterization for the Gamma measures on the cone and an FKG inequality for them.

Abstract:
We construct a new equilibrium dynamics of infinite particle systems in a Riemannian manifold $X$. This dynamics is an analog of the Kawasaki dynamics of lattice spin systems. The Kawasaki dynamics now is a process where interacting particles randomly hop over $X$. We establish conditions on the {\it a priori} explicitly given symmetrizing measure and the generator of this dynamics, under which a corresponding conservative Markov processes exists. We also outline two types of scaling limit of the equilibrium Kawasaki dynamics: one leading to an equilibrium Glauber dynamics in continuum (a birth-and-death process), and the other leading to a diffusion dynamics of interacting particles (in particular, the gradient stochastic dynamics).

Abstract:
We provide an $N/V$-limit for the infinite particle, infinite volume stochastic dynamics associated with Gibbs states in continuous particle systems on $\mathbb R^d$, $d \ge 1$. Starting point is an $N$-particle stochastic dynamic with singular interaction and reflecting boundary condition in a subset $\Lambda \subset {\mathbb R}^d$ with finite volume (Lebesgue measure) $V = |\Lambda| < \infty$. The aim is to approximate the infinite particle, infinite volume stochastic dynamic by the above $N$-particle dynamic in $\Lambda$ as $N \to \infty$ and $V \to \infty$ such that $N/V \to \rho$, where $\rho$ is the particle density.

Abstract:
We prove a priori estimates and, as sequel, existence of Euclidean Gibbs states for quantum lattice systems. For this purpose we develop a new analytical approach, the main tools of which are: first, a characterization of the Gibbs states in terms of their Radon-Nikodym derivatives under shift transformations as well as in terms of their logarithmic derivatives through integration by parts formulae, and second, the choice of appropriate Lyapunov functionals describing stabilization effects in the system. The latter technique becomes applicable since on the basis of the integration by parts formulae the Gibbs states are characterized as solutions of an infinite system of partial differential equations. Our existence result generalize essentially all previous ones. In particular, superquadratic growth of the interaction potentials is allowed and $N$-particle interactions for $N\in \mathbb{N}\cup \{\infty \}$ are included. We also develop abstract frames both for the necessary single spin space analysis and for the lattice analysis apart from their applications to our concrete models. Both types of general results obtained in these two frames should be also of their own interest in infinite dimensional analysis.

Abstract:
X-ray pulsations with a 6.85 s period were recently detected in the SMC and were subsequently identified as originating from the Be/X-ray binary system XTE J0103-728. The recent localization of the source of the X-ray emission has made a targeted search for radio pulsations from this source possible. The detection of pulsed radio emission from XTE J0103-728 would make it only the second system after PSR B1259-63 that is both a Be/X-ray binary and a radio pulsar. We observed XTE J0103-728 in Feb 2008 with the Parkes 64-m radio telescope soon after the identification of the source of X-ray pulsations was reported in order to search for corresponding radio pulsations. We used a continuous 6.4 hour observation with a 256 MHz bandwidth centered at 1390 MHz using the center beam of the Parkes multibeam receiver. In the subsequent data analysis, which included a folding search, a Fourier search, a fast-folding algorithm search, and a single-pulse search, no pulsed signals were found for trial dispersion measures (DMs) between 0 and 800 pc cm^-3. This DM range easily encompasses the expected values for sources in the SMC. We place an upper limit of ~45 mJy kpc^2 on the luminosity of periodic radio emission from XTE J0103-728 at the epoch of our observation, and we compare this limit to a range of luminosities measured for PSR B1259-63, the only Be/X-ray binary currently known to emit radio pulses. We also compare our limit to the radio luminosities of neutron stars having similarly long spin periods to XTE J0103-728. Since the radio pulses from PSR B1259-63 are eclipsed and undetectable during the portion of the orbit near periastron, repeated additional radio search observations of XTE J0103-728 may be valuable if it is undergoing similar eclipsing and if such observations are able to sample the orbital phase of this system well.

Abstract:
High-precision timing of millisecond pulsars (MSPs) over years to decades is a promising technique for direct detection of gravitational waves at nanohertz frequencies. Time-variable, multi-path scattering in the interstellar medium is a significant source of noise for this detector, particularly as timing precision approaches 10 ns or better for MSPs in the pulsar timing array. For many MSPs the scattering delay above 1 GHz is at the limit of detectability; therefore, we study it at lower frequencies. Using the LOFAR (LOw-Frequency ARray) radio telescope we have analyzed short (5-20 min) observations of three MSPs in order to estimate the scattering delay at 110-190 MHz, where the number of scintles is large and, hence, the statistical uncertainty in the scattering delay is small. We used cyclic spectroscopy, still relatively novel in radio astronomy, on baseband-sampled data to achieve unprecedented frequency resolution while retaining adequate pulse phase resolution. We detected scintillation structure in the spectra of the MSPs PSR B1257+12, PSR J1810+1744, and PSR J2317+1439 with diffractive bandwidths of $6\pm 3$, $2.0\pm 0.3$, and $\sim 7$ kHz, respectively, where the estimate for PSR J2317+1439 is reliable to about a factor of 2. For the brightest of the three pulsars, PSR J1810+1744, we found that the diffractive bandwidth has a power-law behavior $\Delta\nu_d \propto \nu^{\alpha}$, where $\nu$ is the observing frequency and $\alpha = 4.5\pm 0.5$, consistent with a Kolmogorov inhomogeneity spectrum. We conclude that this technique holds promise for monitoring the scattering delay of MSPs with LOFAR and other high-sensitivity, low-frequency arrays like SKA-Low.