# 2002



2002
Volume 8
Issue 1

D. SvenssonA Class of Renewal Processes Driven by a Birth and Death Process
pp.1-42
This paper deals with a generalization of the class of renewal processes with
absolutely continuous life length distribution, obtained by allowing a random environment
to modulate the stochastic intensity of the renewal process. The random environment is a
birth and death process with a finite state space. The modulation is based on a set of
deterministic failure rate functions, which are associated with the different environment
states. Renewal processes in this environment (RPRE's) are constructed by using a certain
Poisson embedding technique. The coupling method is the main tool in this paper, and it
turns out to be particularly useful when the underlying deterministic failure rates are
increasing or decreasing. For such processes, domination results and stochastic monotonicity
properties are established. The existence of a stationary RPRE process is investigated by
considering an embedded regenerative process, and asymptotics, rate results and versions of
Blackwell's theorem are investigated by establishing exact couplings. Particular attention
is paid to properties not present in the standard renewal theory but which are due to the
introduction of a random environment. Asymptotic normality and some expansions of the
generalized renewal function are also considered.
Keywords: renewal and point processes, random environments, coupling, Poisson embedding,
stochastic intensity, asymptotics, stochastic domination, stochastic monotonicity

T.V. Dudnikova, A.I. Komech and H. SpohnOn a Two-Temperature Problem for Wave Equation
pp.43-80
Consider the wave equation with constant or variable coefficients in $R^3$. The
initial datum is a random function with a finite mean density of energy that also satisfies
a Rosenblatt- or Ibragimov - Linnik-type mixing condition. The random function converges to
different space-homogeneous processes as $x_3\to\pm\infty$, with the distributions  $\mu_\pm$.
We study the distribution $\mu_t$ of the random solution at a time $t\in R$. The main result
is the convergence of $\mu_t$ to a Gaussian translation-invariant measure as $t\to\infty$
that means central limit theorem for the wave equation. The proof is based on the  Bernstein
room-corridor'  argument. The application to the case of the Gibbs measures $\mu_\pm=g_\pm$
with two different temperatures $T_{\pm}$ is given. Limiting mean energy current density
formally is $-\infty\cdot (0,0,T_+ - T_-)$ for the Gibbs measures, and it is finite and
equals $-C(0,0,T_+ - T_-)$ with $C>0$ for the convolution with  a nontrivial test function.
Keywords: wave equation, random function, Gaussian measure, correlation function,

C. Boldrighini, R.A. Minlos and A. PellegrinottiDirected Polymers in Markov Random Media
pp.81-105
We consider a model of directed polymers in discrete space and time assuming a Markov
dependence of the environment in time. We extend results on the almost-sure validity  of the
Central Limit Theorem for small randomness in space dimension $\nu\geq 3$ which were
previously obtained for independent environment by relying on two main technical tools: the
analysis of the spectrum of a  kind of transfer matrix which allows to treat the averaged
model, and the explicit  construction of a multiplicative orthonormal basis in the appropriate
$L_2$ space, together with cluster estimates of cumulants of the basis functions.
Keywords: random walk, random media, Markov processes, Central Limit Theorem

C. TakacsStrong Law of Large Numbers for Branching Markov Chains
pp.107-116
We consider the Markov chains whose index-sets are infinite trees. Given a certain
state, we calculate its relative frequency up to generation $n$ of the tree and consider
the limit $n \rightarrow \infty$. In this setting we prove a strong law of large numbers
for Markov chains with finite state space and irreducible aperiodic transition matrix,
and trees with uniformly bounded degree.
Keywords: strong law of large numbers, infinite tree, branching Markov chain,
population composition

A. LejayOn the Decomposition of Excursions Measures of Processes Whose Generators
Have Diffusion Coefficients Discontinuous at One Point
pp.117-126
The coefficients in the decomposition of the excursions measure as convex combination
of excursions measures of reflected processes are computed in order to characterize the
discontinuity at one point of the diffusion coefficient. In some sense, this result extends
to general diffusions a similar one for the skew Brownian motion, and we advocate it may be
used in Monte Carlo methods for discontinuous media.
Keywords: scale function and speed measure, excursions theory, skew Brownian motion,
Monte Carlo methods

A.B. Varakin and A.Yu. VeretennikovOn Parameter Estimation for "Polynomial Ergodic"
Markov Chains with Polynomial Growth Loss Functions
pp.127-144
We establish the Hajek - Le Cam asymptotic efficiency of maximum likelihood estimators
for "polynomial ergodic" Markov regular experiments in the class of loss functions with a
polynomial growth.
Keywords: asymptotic normality, polynomial loss function, maximum likelihood estimation,
Hajek - Le Cam efficiency

2002
Volume 8
Issue 2

V. BaladiFinite-Dimensional Functional Analysis Applied to Transfer Operators for
Infinite-Dimensional Maps
pp.149-154
We describe a simple approach to perturbative analysis of Perron - Frobenius operators
and study the Floquet spectrum of the transfer operators of weakly coupled analytic maps
on an infinite lattice: we are able to go beyond the first spectral gap and to exhibit
smooth curves of eigenvalues and eigenvectors as functions of the crystal momenta. This
talk given on January 23, 2001, at the session on Rapidity of convergence to equilibrium
or stationary states, Journees Systemes Aleatoires Inhomogenes (Universite de Cergy-Pontoise,
France) describes joint work with H.H. Rugh, Cergy-Pontoise. The detailed proofs of the
results announced here, as well as further statements and references, may be found in
[V. Baladi and H.-H. Rugh, Floquet spectrum of weakly coupled map lattices, Commun. Math.
Phys., 2001, v. 220, 561-582].
Keywords: coupled map lattices, transfer operator, Floquet spectrum

C. LiveraniComputing the Rate of Decay of Correlations in Expanding and Hyperbolic Systems
pp.155-162
I discuss a general approach allowing to accurately investigate the statistical
properties of expanding and hyperbolic dynamical systems.
Keywords: Dynamical systems, transfer operators, statistical properties

D. TalayStochastic Hamiltonian Systems: Exponential Convergence to the Invariant Measure,
and Discretization by the Implicit Euler Scheme
pp.163-198
In this paper we carefully study the large time behaviour of
$$u(t,x,y) := E_{x,y}f(X_t,Y_t)-\int f d\mu,$$
where $(X_t,Y_t)$ is the solution of a stochastic Hamiltonian dissipative system with
non globally Lipschitz coefficients, $\mu$ its unique invariant law, and $f$ a smooth
function with polynomial growth at infinity. Our aim is to prove the exponential decay
to 0 of $u(t,x,y)$ and all its derivatives when $t$ goes to infinity, for all $(x,y)$ in
$R^{2d}$.
We apply our precise estimates on $u(t,x,y)$ to analyze the convergence rate of a
probabilistic numerical method based upon the implicit Euler discretization scheme which
approximates $\int f d\mu$.
Keywords: stochastic differential equations, stochastic Hamiltonian systems, parabolic
partial differential equation, invariant measure, Euler method, simulation

J.C. Mattingly and A.M. StuartGeometric Ergodicity of Some Hypo-Elliptic Diffusions
for Particle Motions
pp. 199-214
Two degenerate SDEs arising in statistical physics are studied. The first is a Langevin
equation with state-dependent noise and damping. The second is the equation of motion for
a particle obeying Stokes' law in a Gaussian random field; this field is chosen to mimic
certain features of turbulence. Both equations are hypo-elliptic and smoothness of
probability densities may be established. By developing appropriate Lyapunov functions and
by studying the necessary control problems, geometric ergodicity is proved.
Keywords: geometric ergodicity, stochastic differential equations, Langevin equation,
synthetic turbulence, hypoelliptic and degenerate diffusions

W. KrauthDisks on a Sphere and Two-Dimensional Glasses
pp.215-219
Talk given at the conference on Inhomogeneous Random Systems' at the University of
Cergy-Pontoise, France, (23 January 2001).
I describe the classic circle-packing problem on a sphere, and the analytic and
numerical approaches that have been used to study it. I then present a very simple
Markov-chain Monte Carlo algorithm, which succeeds in finding the best solutions known
today. The behavior of the algorithm is put into the context of the statistical physics
of glasses.
Keywords: Monte Carlo methods, packing of circles, glass transition

E. JanvresseApproach to Equilibrium for Kac Master Equation
pp. 221-232
We consider the random walk on $S^{n-1}(1)$, the $(n-1)$-dimensional sphere of radius
1, generated by random rotations on randomly selected coordinate planes i,j with
$1 \le i < j \le n$. This dynamics was used by M. Kac as a model for the spatially
homogeneous Boltzmann equation. If we assume that the initial distribution is of product
form, Kac proved that this property remains valid for all time in the limit
$n \rightarrow \infty$. In modern terminology, Kac proved the "propagation of chaos". Once
propagation of chaos is proved, it is straightforward to show that the marginal density
of a particle satisfies the analog of a Boltzmann equation. Clearly, the spectral properties
of the collision operator of the Boltzmann equation is of critical importance to understand
it. Since this collision operator is generated by a Kac process, a very basic property is
the size of the spectral gap, which Kac conjectured to be of order $1/n$. After recalling
the idea of the proof of the Kac conjecture and the generalization to the same walk on
$SO(n)$, we discuss other ways of measuring the rate of convergence to equilibrium for the
Kac master equation (L.S.I., entropy dissipation bound).
Keywords: convergence to equilibrium, spectral gap, Kac model, Boltzmann equation

S.G. Bobkov, I. Gentil and M. LedouxHypercontractivity of Hamilton - Jacobi Equations
pp.233-235
Following the equivalence between logarithmic Sobolev inequalities and hypercontractivity
showed by L. Gross, we prove that logarithmic Sobolev inequalities are related similarly
to hypercontractivity of solutions of Hamilton - Jacobi equations. Given a bounded Lipschitz
function $f$, solutions of the Hamilton - Jacobi initial value problem
$$\left \{ \begin{array}{rcll} \frac{\partial v}{\partial t} + \frac{1}{2} \, |\nabla v|^2 & = & 0& {\rm in}\,\, R^n \times(0,\infty ), \\ v & = & f& {\rm on} \,\, R^n \times \{t=0\}, \end{array} \right.$$
are described by the Hopf - Lax representation formula as the infimum-convolution of $f$
$$Q_t f(x) = \inf _{y \in R^n} \Big [ f(y) + \frac{1}{2t} \, |x-y|^2 \Big ], \quad t>0, \, x \in R^n.$$
Our main result is the following theorem.
Let $\mu$ be a probability measure on the Borel sets of $R^n$ absolutely continuous
with respect to Lebesgue measure such that for some $\rho >0$ and all smooth enough
functions $\varphi$ on $R^n$ with $\int \varphi ^2 d\mu =1$,
$$\label{(1)} \rho \int \varphi ^2 \log \varphi ^2 d\mu \leq 2 \int |\nabla \varphi |^2 d\mu .$$
Then for every bounded measurable function $f$ on $R^n$, every $t > 0$ and every $a \in R$,
$$\label{(2)} {\big\| \,{\rm e}^{Q_t f} \big\|}_{a + \rho t} \leq {\big\|\, {\rm e}^f\big\|}_a$$
(where the norms are understood with respect to $\mu$). Conversely, if (\ref{(2)}) holds
for all $t> 0$ and some $a \not= 0$, then the logarithmic Sobolev inequality (\ref{(1)})
holds.
When $a=0$, (\ref{(2)}) actually amounts to the infimum-convolution inequality
$$\int {\rm e}^{\,\rho \,Q_1 f} d\mu \leq {\rm e}^{\,\rho \int f d\mu }$$
holding for every bounded (or integrable) function $f$. This inequality is known to be the
Monge - Kantorovitch - Rubinstein dual version of the transportation cost inequality
$$\rho \, W_2(\mu ,\nu)^2 \leq H(\nu \, | \, \mu )$$
holding for all probability measures $\nu$ absolutely continuous with respect to $\mu$
with Radon - Nikodym derivative $d\nu / d\mu$. Here $W_2$ is the Wasserstein distance
$$W_2 (\mu , \nu)^2 = \inf \int \int {\frac{1}{2}} \, |x-y|^2 d\pi (x,y)$$
where the infimum is running over all probability measures $\pi$ on $R^n \times R^n$ with
respective marginals $\mu$ and $\nu$ and $H(\nu \, | \, \mu )$ is the relative entropy,
or informational divergence, of $\nu$ with respect to $\mu$. This approach thus provides
a clear view of the connection between logarithmic Sobolev inequalities and transportation
cost inequalities investigated recently by F. Otto and C. Villani. Extensions to the
Riemannian setting and applications to transportation cost and concentration inequalities,
HWI inequalities and isoperimetry complete this work.
Keywords: hypercontractivity, logarithmic Sobolev inequality, transportation cost
inequality, Hamilton - Jacobi equations

C. VillaniOn the Trend to Equilibrium for Kinetic Equations
pp.237-250
We summarize recent works about the convergence to thermodynamical equilibrium for the
Boltzmann equation modelling dilute gases.
Keywords: kinetic equations, Boltzmann equation, long-time behavior, logarithmic
Sobolev inequalities

N. CancriniRelaxation to Equilibrium of Spin Exchange Dynamics for Lattice Gases
pp.251-270
We review recent results on the relaxation to equilibrium of spin exchange dynamics
reversible with respect to the canonical Gibbs measure of a lattice gas model.
Keywords: Kawasaki dynamics, spectral gap, logarithmic Sobolev constant

F. Baffioni, I. Merola and E. PresuttiThe Liquid and Vapor Phases in Particle Models with Kac Potentials
pp. 271-318
We consider here the model introduced in [J.L. Lebowitz, A. Mazel and E. Presutti,
Liquid-vapor phase transitions for systems with finite-range interactions, J. Stat. Phys.,
1999, v. 94, N5-6, 955-1025] to study phase transitions for point particles in the
continuum. The mean field phase diagram of the model in the $(\beta,\lambda)$-plane,
$\lambda$ the chemical potential, $\beta$ the inverse temperature, consists of a smooth
curve $\lambda=\lambda(\beta)$, $\beta>\beta_c>0$, where two phases (liquid and vapor)
coexist, elsewhere the phase is unique. In the mentioned paper it has been proved that
phase transitions (in the sense of non uniqueness of DLR measures) persist when the mean
field interaction is replaced by a Kac potential with small but fixed scaling parameter
$\gamma$ (mean field being derived in the limit $\gamma\to 0$). In particular it is shown
that for any $\beta>\beta_c$, there is a phase transition at $\lambda=\lambda(\beta,\ga)$
for any $\gamma$ small enough with $\lambda(\beta,\gamma)\to\lambda(\beta)$ as $\gamma\to 0$.
Here we complete the analysis of the phase diagram by showing that if
$\lambda\ne\lambda(\beta)$, then, for all $\gamma$ small enough, there is a unique DLR
measure.
Keywords: Pirogov - Sinai theory, Peierls estimates, Vaserstein distance, disagreement
percolation

T. BodineauOn the van der Waals Theory of Surface Tension
pp.319-338
In this paper, the works on the justification of the van der Waals theory of surface
tension in the context of the Kac Ising models are reviewed. The second part of the paper
is devoted to a coarse grained definition of the surface tension for Kac Ising models which
is appropriate for the $L^1$ approach of phase coexistence.
Keywords: Ising model

L. TrioloFree Energy Functionals in the Mesoscopic Description of Phase Transitions
pp.339-350
A ferromagnetic spin system and a system of point particles in the continuum, both
interacting with a long-range Kac-type potentials, are considered in a suitable scaled
space coordinates (mesoscopic limit). The main issue is the description of the connection
between the two phases which are present at low temperature, and the role of associated
free-energy  functionals is pointed out.
Keywords: nonlocal equations, fronts between phases

O. PenroseStatistical Mechanics of Nonlinear Elasticity
pp.351-364
A method is suggested for defining a deformation-dependent free energy in microscopic
terms for a deformed elastic solid and applied to a simple microscopic model of such a solid.
Some of the convexity and continuity properties of this free energy function are derived.
Keywords: nonlinear elasticity, restricted ensembles in statistical mechanics

L. BertiniDynamic Fluctuations for Kac and Related Models
pp.365-381
This note is an extended version of a talk given at the conference Homogeneisation en
mecanique statistique pour des potentiels a longue portee, Universite de Cergy-Pontoise
(2001). We recall the definition of dynamical models with Kac potential and review their
scaling limits; particular emphasis is dedicated to the fluctuations phenomena which are
also analyzed for the (simpler) Ginzburg - Landau models. Some open problems are also discussed.
Keywords: Kac potential, dynamical fluctuation phenomena, Ginzburg - Landau models,
interface dynamics

M. ZahradnikCluster Expansions of Small Contours in Abstract Pirogov - Sinai Models
p.381
We develop the Pirogov - Sinai theory for "abstract Pirogov - Sinai models" where
configuration is already  represented as a system of matching compatible contours. The
Hamiltonian is then expressed in terms of contour energies and an additional external
field, the latter acting not on single spins but more generally on their "clusters", in
the space ("colored" by various "local ground states") outside of contours. This general
model is formulated in such a way that
i) its general structure does not change when cluster expansion of its "smallest
possible" contours resp. contour systems is applied;
ii) it includes, after suitable preparation, many interesting lattice spin models,
notably the reformulation of the Ising Kac ferromagnet studied in the paper
[A. Bovier and M. Zahradnik, Cluster Expansions and Pirogov - Sinai Theory for Long Range
Spin Systems, Markov Processes Relat. Fields, 2002, v. 8, N 3, 443-478]. In fact the result of the
latter paper - a model with expanded restricted ensembles living outside of suitably
defined contours - is the main testing  example for the usefulness of this general method,
further elaborating and simplifying the method of the paper [M. Zahradnik, A short course
in the Pirogov - Sinai theory, Rendiconti Mat. Appl., 1997, v. 18, N 7, 411-486].
Keywords: low temperature Gibbs states, contours, cluster expansion, contour
functional, abstract Pirogov - Sinai model with cluster field, Pirogov - Sinai theory

2002
Volume 8
Issue 3

M. ZahradnikCluster Expansions of Small Contours in Abstract Pirogov - Sinai Models
pp.383-441
We develop the Pirogov - Sinai theory for "abstract Pirogov - Sinai models" where
configuration is already  represented as a system of matching compatible contours. The
Hamiltonian is then expressed in terms of contour energies and an additional external
field, the latter acting not on single spins but more generally on their "clusters", in
the space ("colored" by various "local ground states") outside of contours. This general
model is formulated in such a way that
i) its general structure does not change when cluster expansion of its "smallest
possible" contours resp. contour systems is applied;
ii) it includes, after suitable preparation, many interesting lattice spin models,
notably the reformulation of the Ising Kac ferromagnet studied in the paper
[A. Bovier and M. Zahradnik, Cluster Expansions and Pirogov - Sinai Theory for Long Range
Spin Systems, Markov Processes Relat. Fields, 2002, v. 8, N 3, 443-478]. In fact the result of the
latter paper - a model with expanded restricted ensembles living outside of suitably
defined contours - is the main testing  example for the usefulness of this general method,
further elaborating and simplifying the method of the paper [M. Zahradnik, A short course
in the Pirogov - Sinai theory, Rendiconti Mat. Appl., 1997, v. 18, N 7, 411-486].
Keywords: low temperature Gibbs states, contours, cluster expansion, contour
functional, abstract Pirogov - Sinai model with cluster field, Pirogov - Sinai theory

A. Bovier and M. ZahradnikCluster Expansions  and Pirogov - Sinai Theory for Long Range Spin Systems
pp.443-478
We investigate the low temperature phases  of lattice spin systems  with  interactions
of Kac type, that is  interactions that are
weak but long range in such a way that the total interaction of one
spin with all the others  is
of order unity. In particular we develop a systematic approach to
convergent  low temperature expansions in situations where interactions are
weak but long range.  This leads to a reformulation of the model
in terms of a
generalized abstract Pirogov - Sinai model, that is a representation in terms of
contours interacting through  cluster fields. The main point
of our approach is that all quantities in the contour representation
satisfy estimates that are uniform in the range of the interaction and
depend only on the overall interaction strength. The extension of the
Pirogov - Sinai theory to such models developed in [M. Zahradnik, Cluster expansions of small contours in abstract
Pirogov - Sinai models, Markov Processes Relat. Fields, 2002, v.8, N3, 383-441] allows then the investigation of the low-temperature phase diagram of these models.
Keywords: low temperature Gibbs states, discrete spin lattice models of Kac - Ising type
restricted ensembles with low density constraints, cluster expansion,  contours,  Pirogov - Sinai theory

A.C.D. van Enter, I. Medved' and K. NetocnyChaotic Size Dependence in the Ising Model with Random Boundary Conditions
pp.479-508
We study the nearest-neighbour Ising model with a class of random
boundary conditions, chosen from a symmetric i.i.d. distribution.
We show for dimensions 4 and higher that almost surely the only
limit points for a sequence of increasing cubes are the plus and
the minus state. For d=2 and d=3 we prove a similar result for
sparse sequences of increasing cubes. This question was raised by
Newman and Stein. Our results imply that the Newman - Stein
metastate is concentrated on the plus and the minus state.
Keywords: random boundary conditions, metastates, contour models,
local limit theorem, central limit theorem

A.I. PetrovCritical S0L systems
pp.509-526
L systems  were first introduced by  Aristid Lindenmayer as models in developmental biology in the late
1960s. One can find  surveys on L systems  in journals in biology,
molecular genetics, semiotics, artificial intelligence, theory of formal languages.
A stochastic version of the 0L systems was introduced in
[P. Eichhorst and W.J. Savitch, Growth functions of stochastic Lindenmayer systems.
Information and Control, 1980, v.45, 217-228] and
[T. Yokomori, Stochastic characterizations of EOL languages, Information and Control, 1980, v.45, 26-33] but statistical physics point
of view (thermodynamic limit, cluster expansions technique,
etc.) started only recently, see
[V.A. Malyshev, Random grammars, Russian Math. Surveys, 1998, v.53, N2, 107-134]. In this paper we
undertake more detailed study of the long time behaviour of the
critical S0L systems. The supercritical case was considered in
[F.I. Karpelevich, V.A. Malyshev, A.I. Petrov, S.A. Pirogov, A.N. Rybko,
Context free evolution of words, Rapport de Recherche INRIA, 2002, No. 4413
and
[A.I. Petrov, Context free random grammars: supercritical case with nonzero extinction probability. To appear in Theory Probab. and Appl., 2002].
This paper can be read independently of these two papers.
Keywords: random grammars, 0L systems, context free,
branching process, critical  area, thermodynamic limit

B.L. GranovskyThe Simplest Nearest-Neighbour Spin Systems on Regular Graphs: Time Dynamics of the Mean Coverage Function
pp.527-546
We establish  a characterization of the class of the simplest
nearest neighbour spin systems possessing the mean coverage
function (mcf) that obeys a  second order differential equation,
and derive explicit expressions for the mcf's of  the above
models. Based on these  expressions, the problem of ergodicity of
the  models is studied and bounds for their spectral gaps are
obtained.
Keywords: interacting spin systems, mean coverage
function, voter models, ergodicity, spectral gap

2002
Volume 8
Issue 4

T.M. LiggettNegative Correlations and Particle Systems
pp.547-564
We consider the symmetric exclusion process and systems of independent Markov chains. For each of these, we prove
that certain classes of distributions with negative dependence
are preserved by the evolution. We also show by example that
the class of negatively associated distributions is not
preserved by the symmetric exclusion process.
Keywords: correlation inequalities, symmetric exclusion
process, negative association, independent particle systems

G. Ben Arous, O. Hryniv and S. MolchanovPhase Transition for the Spherical Hierarchical Model
pp.565-594
We present the whole spectrum of the limit theorems for the total
magnetization in the hierarchical version of the spherical model in
dimensions dim > 2.
Keywords: spherical model, hierarchical interactions, phase transition

P. MathieuLog-Sobolev and Spectral Gap Inequalities for the Knapsack Markov Chain
pp.595-610
We study a random walk moving on a discrete cube of
large dimension and subject to a linear constraint. Under the assumption
that more than the half of the
points in the cube satisfy the constraint,
we prove log-Sobolev and  Poincare inequalities with constants
of the correct order.
Keywords: knapsack problem, log-Sobolev inequalities, spectral gap,
generalized Poincare inequalities

R. Liptser, V. Spokoiny and A.Yu. VeretennikovFreidlin - Wentzell Type Large Deviations for Smooth Processes
pp.611-636
We establish large deviation principle for the family of vector-valued random processes
$X^\varepsilon,\varepsilon\to 0$ defined by ordinary differential equations
(under $0<\kappa<1/2$)
$$\dot{X}_t^\varepsilon=F(X_t^\varepsilon)+\varepsilon^{1/2-\kappa} G(X_t^\varepsilon)\dot{W}^\varepsilon_t,$$
where $\dot{W}^\varepsilon_t={\varepsilon}^{-1/2}g(\xi_{t/\varepsilon})$,
$\xi_t$ is a vector-valued ergodic diffusion satisfying, so
called, recurrence condition'' and $g$ is a vector-function with
zero barycenter with respect to the invariant measure of $(\xi_t)$.
A choice of $\kappa<1/2$ provides the rate function of Freidlin - Wentzell type.
Keywords: moderate deviations, Poisson decomposition, Puhalskii
theorem

D. SvenssonA Random Environment Generalization of Lorden's Renewal Inequality
pp.637-649
The aim of this paper is to obtain a generalization of Lorden's renewal
inequality for a class of renewal processes in random environments.
These processes (called RPRE's) are generalizations of the classical
renewal processes with absolutely continuous
life length distribution, and are obtained by allowing a
random environment to modulate the stochastic intensity.
The first part of this paper presents a
probabilistic proof of Lorden's classical renewal inequality.
The main ideas from this proof are
generalized for regenerative point processes and
a certain Lorden type inequality is obtained.  Finally,
this inequality is applied to RPRE's and the aimed generalization of
Lorden's inequality is obtained. It takes a particularly
transparent form when the RPRE's considered are of DFR or IFR type.
Keywords: renewal processes, regenerative point processes,
random environments, coupling, Poisson embedding, failure rates,
stochastic intensity, DFR, IFR

M. Gonzalez, M. Molina and M. MotaBisexual Galton - Watson Branching Processes with Immigration of Females and Males. Asymptotic Behaviour
pp.651-663
A bisexual Galton - Watson branching process allowing the
immigration of females and males is introduced and its asymptotic
behaviour is investigated. Under certain assumptions on the
immigration probability distribution, conditions for the almost
sure convergence of the suitably normed underlying Markov chains
are provided.
Keywords: bisexual Galton - Watson process,
branching process with immigration, almost sure convergence

A. Chen and K. LiuPiecewise Birth-Death Processes
pp.665-698
A natural generalization  of the ordinary birth-death processes,
the Piecewise Birth-Death Process is considered. The close relationship
between this kind of processes and the well-known ordinary birth-death
processes is revealed.
Regularity criteria for such processes are established. Properties of such structure are further presented. In particular,
conditions for recurrence and positive recurrence are obtained. Equilibrium
distributions are given  for the ergodic case. Some useful properties of
the transient Piecewise Birth-Death process are also provided.
Probabilistic meanings of the analytic results are explained.  These
results are then illustrated through several examples.
Keywords: birth-death processes, piecewise birth-death
processes, uniqueness, recurrence, positive recurrence, equilibrium
distributions, regularity, Feller minimal processes