# 2010

### 2010
Volume 16
Issue 1

#### V.A. MalyshevForeword - After 15 Years
pp. 1-2

15 years of MPRF is a good occasion to express our biggest gratitude to the
editorial board, to the authors, to the referees and to everyone else without
whom the journal could not exist. We thank also the authors who submitted
invited papers for the anniversary issues (there will be two) of the journal.
It is a moment to say how the journal appeared. Once I wondered who
are the owners and the real bosses (those who periodically change
main editors) of the existing mathematical journals. Instead of pursuing
this direction, the idea to launch a new journal appeared to be more
useful. However, I still think that any journal should have one who
is personally responsible for the scientific policy. The authors of
the rejected papers should know personally one who rejected their
papers. Seemingly, it is the main editor, often he has to play mediator
between the author and some unknown referees. But as far as I know,
often the main editor is not completely free in his decisions. That
is why I prefer smaller journals, owned and managed by a working mathematician.
It could be useful also that the author knew the names of his referees.
But now the world is still far from this.
MPRF was the first and still remains the unique international mathematical
journal, which is owned, edited and published in Russia. International
means that it is published only in English, there is no Russian translation
of MPRF. Moreover, it is really a private journal, all other journals
are owned by grand institutions and societies. While launching the
journal, I underwent big pressure from some institutions who wanted
to own it and from others who did not want such kind of journals to exist.
Next important remark. Now there appeared the so called ratings, impact
factors etc. of papers, journals, and even authors. MPRF did not make
any effort to get these scores for the only reason - when the science
wants to be similar to a business enterprise and to use PR agencies,
the science starts degenerate, and scientific politics prevails.
I want to thank personally my colleagues and former colleagues whose
efforts allowed the journal to appear. Vadim Scherbakov was the first
general director of our small publishing company POLYMAT, followed now
by Elena Petrova. Elena Petrova is also the executive editor who governs all
current activity of the journal. Without Flora Spieksma we could hardly
establish connections with the distributors, that allowed to get first
subscriptions (for some years I was the only financial sponsor of
the journal). She also, for some period, took responsibility for the
English language. Sergej Popov, Alexander Gajrat, Anatoli Manita
designed the journal style, its cover etc.
At the beginning we suffered from the lack of submissions and in the
first issues we had to publish mostly the papers of my Moscow laboratory.
Soon we received some papers which used words "stochastic, Markov"
etc. but were so ridiculous that I thought it was the test for our editorial
board. I rejected them immediately without any refereeing process.
Now a difficult era for the academic science seems to come. Too many
fields of mathematics, too many interesting problems, but no global
goals. Insufficient number of scientists to cover even most interesting
fields. There are several big communities which aggressively propagate
their field of interest, making no attention to others. That is why
wider papers and reviews, explaining connections between different
areas of mathematics and other disciplines, are welcome to MPRF, and
we ask the authors to spare no pains to write them.
#### A. van Enter and E. Verbitskiy
Erasure Entropies and Gibbs Measures
pp. 3-14

Recently Verdu and Weissman introduced erasure entropies, which are meant
to measure the information carried by one or more symbols given all of the
remaining symbols in the realization of a random process or field.
A natural relation to Gibbs measures has also been observed. In this short
note we study this relation further, review a few earlier contributions from
statistical mechanics, and provide the formula for the erasure entropy
of a Gibbs measure in terms of the corresponding potential. For some
2-dimensional Ising models, for which Verdu and Weissman suggested a
numerical procedure, we show how to obtain an exact formula for the erasure entropy.
Keywords: erasure entropy, Gibbs measures, solvable Ising models,
conditional entropy
#### G. Basile and A. BovierConvergence of a Kinetic Equation to a Fractional Diffusion Equation
pp. 15-44

A linear Boltzmann equation is interpreted as the forward equation
for the probability density of a Markov process $(K(t)$, $Y(t))$ on
$(\T\times\R)$, where $\T$ is the one-dimensional torus. $K(t)$ is
a autonomous reversible jump process, with waiting times between two jumps
with finite expectation value but infinite variance. $Y(t)$ is an additive
functional of $K$, defined as $\int_0^t v(K(s)) ds$, where $|v|\sim 1$ for
small $k$. We prove that the rescaled process $N^{-2/3}Y(Nt)$ converges
in distribution to a symmetric Levy process, stable with index $\alpha=3/2$.
Keywords: anomalous diffusion, Levy process, Boltzmann equation, coupled
oscillators, kinetic limit, heat conductance
#### C. Maes and B. WynantsOn a Response Formula and Its Interpretation
pp. 45-58

We present a physically inspired generalization of equilibrium response
formulae, the fluctuation-dissipation theorem, to Markov jump processes possibly
describing interacting particle systems out-of-equilibrium, following the recent work of Baiesi, Maes and Wynants
[M. Baiesi, C. Maes and B. Wynants, Fluctuations and response of nonequilibrium
states. Phys. Rev. Lett., 2009, v. 103, 010602.
Nonequilibrium linear response for Markov dynamics, I. Jump processes and overdamped diffusions.
J. Stat. Phys., v. 137, 1094-1116]. Here, the time-dependent perturbation adding a potential $V$ with
small amplitude $h_t$ changes the rates $W(x,y)$ for the transition $x\rightarrow y$ into
\[
W_t(x,y) = W(x,y) \exp \big\{ h_t \big( bV(y)-aV(x) \big) \big\}
\]
as first considered by Diezemann
[G. Diezemann, Fluctuation-dissipation relations for Markov processes.
Phys. Rev. E, 2005, v. 72, 011104]
$a$, $b$ are constants. We observe that the linear response relation shows a
reciprocity symmetry in the nonequilibrium stationary regime and we interpret
the connection with dynamical fluctuation theory.
Keywords: fluctuation response relation, nonequilibrium dynamics
#### V. Jaksic, Y. Pautrat and C.-A. PilletA Non-commutative Levy - Cramer Continuity Theorem
pp. 59-78

The classical Levy - Cramer continuity theorem asserts that the convergence
of the characteristic functions implies the weak convergence of the corresponding
probability measures. We extend this result to the setting of non-commutative
probability theory and discuss some applications.
Keywords: non-commutative probability, Levy - Cramer theorem
#### L.T. Rolla, V. Sidoravicius, D. Surgailis and M.E. VaresThe Discrete and Continuum Broken Line Process
pp. 79-116

In this work we introduce the discrete-space broken line process (with
discrete and continuous parameter values) and derive some of its properties.
We explore polygonal Markov field techniques developed by Arak - Surgailis.
The discrete version is presented first and a natural generalization to a
continuous object living on the discrete lattice is then proposed and studied
The broken lines also resemble the Young diagram and the Hammersley process
and are useful for computing last passage percolation values and finding maximal
oriented paths. For a class of passage time distributions there is a family
of boundary conditions that make the process stationary and self-dual.
For such distributions there is a law of large numbers and the process
extends to the infinite lattice. A proof of Burke's theorem emerges from
the construction. We present a simple proof of the explicit law of large
numbers for last passage percolation as an application. Finally we show that
the exponential and geometric distributions are the only non-trivial ones
that yield self-duality.
Keywords: spatial random processes, Hammersley process, last passage
percolation, time constant, broken line process
#### J. FritzHyperbolic Scaling Limits: The Method of Compensated Compactness
pp. 117-138

Hydrodynamic limit of various models with hyperbolic (Euler) scaling law is
discussed, we are mainly interested in the limiting behavior of the microscopic
systems in a regime of shocks. In the absence of an effective coupling
anadvanced method of PDE theory: compensated compactness is required. We
consider some deterministic and Ginzburg - Landau models of classical
statistical mechanics; the proof of several recent results is outlined.
Microscopic systems living on the infinite line are preferred.
Keywords: interacting exclusions, hyperbolic scaling, Lax entropy pairs,
compensated compactness, logarithmic Sobolev inequalities, relaxation schemes
#### L. Avena, F. den Hollander and F. RedigLarge Deviation Principle for One-Dimensional Random Walk in Dynamic
Random Environment: Attractive Spin-Flips and Simple Symmetric Exclusion
pp. 139-168

Consider a one-dimensional shift-invariant attractive spin-flip system
in equilibrium, constituting a dynamic random environment, together with
a nearest-neighbor random walk that on occupied sites has a local drift
to the right but on vacant sites has a local drift to the left. In
[L. Avena, F. den Hollander and F. Redig, Law of large numbers for a class
of random walks in dynamic random environments. EURANDOM Report 2009-032]
we proved a law of large numbers for dynamic random environments satisfying
a space-time mixing property called cone-mixing. If an attractive spin-flip
system has a finite average coupling time at the origin for two copies starting
from the all-occupied and the all-vacant configuration, respectively, then it
is cone-mixing.
In the present paper we prove a large deviation principle for the empirical
speed of the random walk, both quenched and annealed, and exhibit some properties
of the associated rate functions. Under an exponential space-time mixing condition
for the spin-flip system, which is stronger than cone-mixing, the two rate
functions have a unique zero, i.e., the slow-down phenomenon known to be possible
in a static random environment does not survive in a fast mixing dynamic random
environment. In contrast, we show that for the simple symmetric exclusion dynamics,
which is not cone-mixing (and which is not a spin-flip system either), slow-down
does occur.
Keywords: dynamic random environment, random walk, quenched vs. annealed
large deviation principle, slow-down
#### A.M. VershikDynamics of Metrics in Measure Spaces and Their Asymptotic Invariants
pp. 169-184

We discuss the Kolmogorov's entropy and Sinai's definition of it; and then
define a deformation of the entropy, called scaling entropy; this is also
a metric invariant of the measure preserving actions of the group, which is
more powerful than the ordinary entropy. To define it, we involve the notion
of the $\epsilon$-entropy of a metric in a measure space, also suggested by
A.N. Kolmogorov slightly earlier. We suggest to replace the techniques
of measurable partitions, conventional in entropy theory, by that of iterations
of metrics or semi-metrics. This leads us to the key idea of this paper
which as we hope is the answer on the old question: what is the natural
context in which one should consider the entropy of measure-preserving
actions of groups? the same question about its generalizations - scaling
entropy, and more general problems of ergodic theory. Namely, we propose
a certain research program, called asymptotic dynamics of metrics
in a measure space, in which, for instance, the generalized entropy
is understood as the asymptotic Hausdorff dimension of a sequence of metric
spaces associated with dynamical system. As may be supposed, the metric
isomorphism problem for dynamical systems as a whole also gets a new geometric
interpretation.
Keywords: scaling entropy, metric compact with measure, asymptotic geometry, filtrations
#### W. Kager, H. Liu and R. MeesterExistence and Uniqueness of the Stationary Measure in the Continuous
Abelian Sandpile
pp. 185-204

Let $\Lambda \subset \mathbb{Z}^d$ be finite. We study the following sandpile
model on $\Lambda$. The height at any given vertex $x \in \Lambda$ is a positive
real number, and additions are uniformly distributed on some interval
$[a,b] \subset [0,1]$. The threshold value is $1$; when the height at a given
vertex exceeds $1$, it topples, that is, its height is reduced by $1$, and
the heights of all its neighbours in $\Lambda$ increase by $1/2d$. We first
establish that the uniform measure $\mu$ on the so called `allowed configurations'
is invariant under the dynamics. When $a < b$, we show with coupling ideas
that starting from any initial configuration of heights, the process converges
in distribution to $\mu$, which therefore is the unique invariant measure
for the process. When $a=b$, that is, when the addition amount is non-random,
and $a \notin \mathbb{Q}$, it is still the case that $\mu$ is the unique
invariant probability measure, but in this case we use random ergodic theory
to prove this; this proof proceeds in a very different way. Indeed, the coupling
approach cannot work in this case since we also show the somewhat
surprising fact that when $a=b\notin \mathbb{Q}$, the process does
not converge in distribution at all starting from any initial configuration.
Keywords: continuous Abelian sandpile model, invariant measures,
measure-preserving transformations, ergodic theory
#### F. DunlopSpace-time Correlations of a Gaussian Interface
pp. 205-222

The serial harness introduced by Hammersley
[J.M. Hammersley, Harnesses. In: Proc. Fifth Berkeley Sympos. Mathematical
Statistics and Probability, 1966, Vol. III, 89-117]
is equivalent, in the Gaussian case, to the Gaussian Solid-On-Solid interface
model with parallel heat bath dynamics. Here we consider sub-lattice parallel
dynamics, and give exact results about relaxation dynamics, based on the
equivalence to the infinite time limit of a time periodic random field.
We also give a numerical comparison to the harness process in continuous time
studied by Hsiao
[C.-T. Hsiao, Stochastic processes with Gaussian interaction of components.
Z. Wahrsch. Verw. Geb., 1982, v. 59, 39-53]
and by Ferrari, Niederhauser and Pechersky
[P.A. Ferrari and B.M. Niederhauser, Harness processes and harmonic crystals.
Stoch. Process. Appl., 2006, v. 116, 939-956, math.PR/0312402;
P.A. Ferrari, B.M. Niederhauser and E.A. Pechersky, Harness processes and
non-homogeneous crystals. J. Stat. Phys., 2007, v. 128, 1159-1176,
math.PR/0409301].
Keywords: random surface, interface dynamics, harness.

### 2010
Volume 16
Issue 2

#### The second part of invited papers of the 15th anniversary issue of Markov Processes and Related Fields
M. Campanino, D. Ioffe and O. LouidorFinite Connections for Supercritical Bernoulli Bond Percolation in 2D
pp. 225-266

Two vertices x and y are said to be finitely connected if they belong to the
same cluster and this cluster is finite. We derive sharp asymptotics of finite
connections for super-critical Bernoulli bond percolation on $Z^2$. These asymptotics
are based on a detailed fluctuation analysis of long finite super-critical clusters or,
more precisely, of dual open (sub-critical) loops which surround such clusters.
Keywords: Bernoulli bond percolation, random walk representation, interacting random walks,
Ornstein - Zernike decay of correlations, local limit theorems
#### A. Rybko, S. Shlosman and A. VladimirovAbsence of Breakdown of the Poisson Hypothesis. I. Closed Networks at Low Load
pp. 267-285

We prove that the general mean-field type networks at low load behave in accordance
with the Poisson Hypothesis. That means that the network equilibrates in time
independent of its size. This is a 'high-temperature' counterpart of our earlier result,
where we have shown that at high load the relaxation time can diverge with the size of the network
('low-temperature'). In other words, the phase transitions in the networks can happen
at high load, but cannot take place at low load.
Keywords: coupled dynamical systems, non-linear Markov processes, stable
attractor, phase transition, long-range order
#### S. Miracle-SoleOn the Theory of Cluster Expansions
pp. 287-294

A short exposition with complete proofs of the theory of cluster expansions
for an abstract polymer system is presented.
Keywords: lattice systems, cluster expansions, cluster properties
#### K.R. DuffyMean Field Markov Models of Wireless Local Area Networks
pp. 295-328

In 1998, Giuseppe Bianchi introduced a mean field Markov model of
the fundamental medium access control protocol used in Wireless
Local Area Networks (WLANs). Due to the model's intuitive appeal
and the accuracy of its predictions, since then there has been a
vast body of material published that extends and analyzes models
of a similar character. As the majority of this development has
taken place within the culture and nomenclature of the telecommunications
community, the aim of the present article is to review this work
in a way that makes it accessible to probabilists. In doing so, we
hope to illustrate why this modeling approach has proved so popular,
to explain what is known rigorously, and to draw attention to
outstanding questions of a mathematical nature whose solution would
be of interest to the telecommunications community. For non-saturated
WLANs, these questions include rigorous support for its fundamental
decoupling approximation, determination of the properties of the
self-consistent equations and the identification of the queueing
stability region.
Keywords: wireless local area networks, mean field models, Markov chains
#### C. Boeinghoff, E.E. Dyakonova, G. Kersting and V.A. VatutinBranching Processes in Random Environment which Extinct
at a Given Moment
pp. 329-350

Let $\{Z_{n}, n\geq 0\}$ be a critical branching process in random environment
and let $T$ be its moment of extinction. Under the annealed approach we prove,
as $n\rightarrow \infty$, a limit theorem for the number of particles in the
process at moment $n$ given $T=n+1$ and a functional limit theorem for the properly scaled
process $\{ Z_{nt}, \delta \leq t \leq 1-\delta \} $ given $T = n+1$
and $\delta \in (0,1/2)$.
Keywords: branching process, random environment, random walk, change of
measure, survival probability, functional limit theorem
#### I.M. MacPhee, M.V. Menshikov and A.R. WadeAngular Asymptotics for Multi-Dimensional Non-Homogeneous
Random Walks with Asymptotically Zero Drift
pp. 351-388

We study the first exit time $\tau$ from an arbitrary cone with apex
at the origin by a non-homogeneous random walk (Markov chain) on $\Z^d$
($d \geq 2$) with mean drift that is asymptotically zero. Specifically,
if the mean drift at $\bx \in \Z^d$ is of magnitude $O(\| \bx\|^{-1})$,
we show that $\tau<\infty$ a.s. for any cone. On the other hand, for an
appropriate drift field with mean drifts of magnitude $\| \bx\|^{-\beta}$,
$\beta \in (0,1)$, we prove that our random walk has a limiting (random)
direction and so eventually remains in an arbitrarily narrow cone.
The conditions imposed on the random walk are minimal: we assume only a uniform
bound on $2$nd moments for the increments and a form of weak isotropy.
We give several illustrative examples, including a random walk in random environment model.
Keywords: asymptotic direction, exit from cones, inhomogeneous random
walk, perturbed random walk, random walk in random environment
#### P. Bremaud and S. FossErgodicity of a Stress Release Point Process Seismic Model with Aftershocks
pp. 389-408

We prove ergodicity of a point process earthquake model combining the classical stress release
model for primary shocks with the Hawkes model for aftershocks.
Keywords: earthquakes, secondary earthquakes, point process, stochastic intensity, ergodicity,
Harris chains, Foster's theorem
#### O.J. Boxma, H. Jonsson, J.A.C. Resing and S. ShneerAn Alternating Risk Reserve Process - Part I
pp. 409-424

We consider an alternating risk reserve process with a threshold
dividend strategy. The process can be in two different states and the
state of the process can only change just after claim arrival instants.
If at such an instant the capital is below the threshold, the system is
set to state $1$ (paying no dividend), and if the capital is above the
threshold, the system is set to state~$2$ (paying dividend). Our interest
is in the survival probabilities. In the case of exponentially
distributed claim sizes, survival probabilities are found by solving a
system of integro-differential equations. In the case of generally
distributed claim sizes, they are expressed in the survival
probabilities of the corresponding standard risk reserve processes.
Keywords: insurance risk models, alternating risk reserve process,
survival probabilities
#### O.J. Boxma, H. Jonsson, J.A.C. Resing and S. ShneerAn Alternating Risk Reserve Process - Part II
pp. 425-446

We consider an alternating risk reserve process with a threshold
dividend strategy. The process can be in two different states and the
state of the process can only change at the arrival instants of an
independent Poisson observer. Whether or not a change then occurs
depends on the value of the risk reserve w.r.t. the barrier.
If at such an instant the capital is below the threshold, the system is
set to state 1 (paying no dividend), and if the capital is above the
threshold, the system is set to state 2 (paying dividend).
In each of the two states, the process is described by different
premium rates, Poisson claim arrival intensities, and claim
size distributions. For this model we determine the survival
probabilities, distinguishing between the initial state being $1$
or 2, and the process starting below or above the barrier.
In the case of exponentially distributed claim sizes, survival
probabilities are found by solving a system of integro-differential
equations. In the case of generally distributed claim sizes,
they are expressed in the survival probabilities of the
corresponding standard risk reserve processes. We perform several
numerical experiments, including a comparison with the case
in which state changes can only occur just after claim arrival instants;
that case is treated in Part I.
Keywords: insurance risk models, alternating risk reserve process,
survival probabilities

### 2010
Volume 16
Issue 3

#### I. PapageorgiouThe Logarithmic Sobolev Inequality in Infinite
Dimensions for Unbounded Spin Systems on the Lattice
with Non-Quadratic Interactions
pp. 447-484

We are interested in the Logarithmic Sobolev inequality for the
infinite volume Gibbs measure with no quadratic interactions.
We consider unbounded spin systems on the one-dimensional lattice
with interactions that go beyond the usual strict convexity and
without uniform bound on the second derivative. We assume that
the one-dimensional single-site measure with boundaries satisfies
the Log-Sobolev inequality uniformly in the boundary conditions
and we determine conditions under which the Log-Sobolev inequality
can be extended to the infinite volume Gibbs measure.
Keywords: Logarithmic Sobolev inequality, Gibbs measure, infinite
dimensions, spin systems
#### G. Fayolle and K. RaschelOn the Holonomy or Algebraicity
of Generating Functions Counting Lattice Walks in the Quarter-Plane
pp. 485-496

In two recent works
[M. Bousquet-Melou and M. Mishna, Walks with small steps in the quarter plane.
In: "Algorithmic Probability and Combinatorics", special volume of the
Contemporary Mathematics series of the Amer. Math. Soc., 2010, v. 520,
pp. 1-40; A. Bostan and M. Kauers, The complete generating function for
Gessel walks is algebraic. Proc. Amer. Math. Soc., 2009, v. 138, N9, 3063-3078],
it has been shown that the counting generating functions (CGF) for the 23
walks with small steps confined in a quarter-plane and associated with a finite
group of birational transformations are holonomic, and even algebraic in 4 cases -
in particular for the so-called Gessel's walk. It turns out that the type of functional
equations satisfied by these CGF appeared in a probabilistic context almost 40 years
ago. Then a method of resolution was proposed in
[G. Fayolle, R. Iasnogorodski and V. Malyshev, Random Walks in the Quarter-Plane,
Applications of Mathematics (New York), vol. 40, 1999, Springer-Verlag, Berlin],
involving at once algebraic techniques and reduction to boundary value problems.
Recently this method has been developed in a combinatorics framework in
[K. Raschel, Counting walks in a quadrant: a unified approach via boundary
value problems. Preprint http://arxiv.org/abs/1003.1362, 2010],
where a thorough study of the explicit expressions for the CGF is proposed.
The aim of this paper is to derive the nature of the bivariate CGF by a direct
use of some general theorems given in [G. Fayolle, R. Iasnogorodski and V. Malyshev].
Keywords: generating function, piecewise homogeneous lattice walk,
quarter-plane, universal covering, Weierstrass elliptic functions, automorphism
#### A. Faggionato, D. Gabrielli and M.R. CrivellariAveraging and Large Deviation Principles for Fully-Coupled Piecewise
Deterministic Markov Processes and Applications to Molecular Motors
pp. 497-548

We consider Piecewise Deterministic Markov Processes (PDMPs) with a finite
set of discrete states. In the regime of fast jumps between discrete states,
we prove a law of large numbers and a large deviation principle. In the
regime of fast and slow jumps, we analyze a coarse-grained process associated
to the original one and prove its convergence to a new PDMP with effective
force fields and jump rates. In all the above cases, the continuous variables
evolve slowly according to ODEs. Finally, we discuss some applications related
to the mechanochemical cycle of macromolecules, including strained-dependent
power-stroke molecular motors. Our analysis covers the case of fully-coupled
slow and fast motions.
Keywords: piecewise deterministic Markov process,
averaging principle, large deviations, molecular motors
#### N. Berglund and B. GentzThe Eyring - Kramers Law for Potentials with Nonquadratic Saddles
pp. 549-598

The Eyring - Kramers law describes the mean transition time of an overdamped
Brownian particle between local minima in a potential landscape. In the
weak-noise limit, the transition time is to leading order exponential in the
potential difference to overcome. This exponential is corrected by a prefactor
which depends on the principal curvatures of the potential at the starting
minimum and at the highest saddle crossed by an optimal transition path. The
Eyring - Kramers law, however, does not hold whenever one or more of these
principal curvatures vanishes, since it would predict a vanishing or infinite transition
time. We derive the correct prefactor up to multiplicative errors that tend to
one in the zero-noise limit. As an illustration, we discuss the case of a
symmetric pitchfork bifurcation, in which the prefactor can be expressed in
terms of modified Bessel functions, as well as bifurcations with two vanishing
eigenvalues. The corresponding transition times are studied in a full
neighbourhood of the bifurcation point. These results extend work by Bovier,
Eckhoff, Gayrard and Klein
[A. Bovier, M. Eckhoff, V. Gayrard and M. Klein, Metastability in reversible
diffusion processes. I. Sharp asymptotics for capacities and exit times.
J. Euro. Math. Soc., 2004, v. 6, N 4, pp. 399-424],
who rigorously analysed the case of quadratic saddles, using methods from potential theory.
Keywords: stochastic differential equations, exit problem, transition times,
most probable transition path, large deviations, Wentzell - Freidlin theory,
metastability, potential theory, capacities, subexponential asymptotics,
pitchfork bifurcation
#### P.J. Fitzsimmons and D.M. WroblewskiMartingale Functions of Brownian Motion and its Local Time
pp. 599-608

We characterize the class of local martingales of the form $H(B_t,L_t)$ for a
standard one-dimensional Brownian motion $B=(B_t)_{t\ge 0}$ and its local
time at $0$, $L=(L_t)_{t\ge 0}$. The main result is closely related to work
of J. Obloj, who studied the local martingales of the form $H(B_t,\ov B_t)$,
where $\ov B_t = \sup_{0\le s\le t} B_s$.
Keywords: Brownian motion, local martingale, local time

### 2010
Volume 16
Issue 4

#### Inhomogeneous Random Systems.
Concentration Inequalities. Random Matrices
pp. 609-611

The present issue of "Markov Processes and Related Fields" contains
papers presented at the meeting "Inhomogeneous Random Systems" held at the
Institut Henri Poincare, Paris, on January 27-28, 2009. These meetings, which
concentrate each year on different topics, bring together an interdisciplinary
audience of mathematicians and physicists.
#### M. LedouxIntroduction to Concentration Inequalities
pp. 613-614

This conference was the opportunity to present some of the most recent
developments in the area, emphasizing in particular the multiple scopes
of measure concentration. The speakers indeed presented aspects of measure
concentration and its applications in various settings, from isoperimetric and
functional aspects to random matrices and dynamical systems. The contributions
gathered in this issue reflect this variety and the lively recent developments
in the area.
#### C. RobertoIsoperimetry for Product of Probability Measures: Recent Results
pp. 617-634

We present recent results on the isoperimetric problem for product of probability measures.
For distributions with tails between exponential and Gaussian we state a dimension free result.
We sketch its proof that relies on a functional inequality of Poincar\'e type and on a
semi-group argument. Also, we give isoperimetric and concentration results, depending on the dimension, for measures
with heavy tails.
Keywords: isoperimetry, Poincare type inequality, concentration of measure
phenomenon, heavy tails distributions, Gaussian measure
#### N. Gozlan and C. LeonardTransport Inequalities. A Survey
pp. 635-736

This is a survey of recent developments in the area of transport
inequalities. We investigate their consequences in terms of
concentration and deviation inequalities and sketch their links
with other functional inequalities and also large deviation theory.
Keywords: transport inequalities, optimal transport, relative entropy,
Fisher information, concentration of measure, deviation inequalities,
logarithmic Sobolev inequalities, inf-convolution inequalities, large deviations
#### F. Redig and F. WangTransformations of One-Dimensional Gibbs Measures with Infinite Range Interaction
pp. 737-752

We study single-site stochastic and deterministic transformations
of one-dimensional Gibbs measures in the uniqueness regime with
infinite-range interactions. We prove conservation of Gibbsianness
and give quantitative estimates on the decay of the transformed
potential. As examples, we consider exponentially decaying potentials,
and potentials decaying as a power-law.
Keywords: Gibbs measures, potential, Kozlov theorem, house-of-cards coupling,
renormalization group transformation
#### B. ZegarlinskiLinear and Nonlinear Concentration Phenomena
pp. 753-782

In this work we consider variety of themes including Linear versus Nonlinear,
Extensive versus Intensive (i.e. dimension independent and where dependence
on dimension is relevant), conjugation of measures, interplay of Entropy and
Nonlinearity, and others.
Keywords: linear and nonlinear Markov semigroups, coercive inequalities,
nonlinear probability, entropic switch
#### A. EdelmanThe Random Matrix Technique of Ghosts and Shadows
pp. 783-790

We propose to abandon the notion that a random matrix exists only
if it can be sampled. Much of today's applied finite random matrix
theory concerns real or complex random matrices ($\beta=1,2$). The
"threefold way" so named by Dyson in 1962
[F.J. Dyson, The threefold way. Algebraic structures of symmetry groups and
ensembles in Quantum Mechanics. J. Math. Phys., 1963, v. 3, pp. 1199-1215]
adds quaternions ($\beta=4$). While it is true there are only three real
division algebras ($\beta$="dimension over the reals"), this mathematical
fact while critical in some ways, in other ways is irrelevant and
perhaps has been over interpreted over the decades.
We introduce the notion of a "ghost" random matrix quantity that exists for
every beta, and a shadow quantity which may be real or complex which allows
for computation. Any number of computations have successfully given reasonable
answers to date though difficulties remain in some cases.
Though it may seem absurd to have a "three and a quarter" dimensional or "pi"
dimensional algebra, that is exactly what we propose and what we compute with.
In the end $\beta$ becomes a noisiness parameter rather than a dimension.
Keywords: random matrix theory, ghost random variables
#### P. SebaParking in the City: An Example of Limited Resource Sharing
pp. 793-802

During the attempt to park a car in the city the drivers have to
share limited resources (the available roadside). We show that this
fact leads to a predictable distribution of the distances between
the cars that depends on the length of the street segment used for
the collective parking. We demonstrate in addition that the
individual parking maneuver is guided by generic psychophysical
perceptual correlates. Both predictions are compared with the actual
parking data collected in the city of Hradec Kralove (Czech Republic).
Keywords: parking process, distances between cars distribution
#### F. BornemannOn the Numerical Evaluation of Distributions
in Random Matrix Theory: A Review
pp. 803-866

In this paper we review and compare the numerical evaluation of those
probability distributions in random matrix theory that are analytically
represented in terms of Painleve transcendents or Fredholm determinants.
Concrete examples for the Gaussian and Laguerre (Wishart) $\beta$-ensembles
and their various scaling limits are discussed. We argue that the numerical
approximation of Fredholm determinants is the conceptually more simple and
efficient of the two approaches, easily generalized to the computation of
joint probabilities and correlations. Having the means for extensive numerical
explorations at hand, we discovered new and surprising determinantal formulae
for the k-th largest (or smallest) level in the edge scaling limits of the Orthogonal
and Symplectic Ensembles; formulae that in turn led to improved numerical
evaluations. The paper comes with a toolbox of Matlab functions that facilitates
further mathematical experiments by the reader.
Keywords: random matrix theory, numerical approximation, Painleve transcendents, Fredholm determinants