On complete, non-compact manifolds and infinite graphs, Faber-Krahn inequalities have been used to estimate the rate of decay of the heat kernel. We develop this technique in the setting of finite Markov chains, proving upper and lower $L^{\infty}$ mixing time bounds via the spectral profile. This approach lets us recover and refine previous conductance-based bounds of mixing time (including the Morris-Peres result), and in general leads to sharper estimates of convergence rates. We apply this method to several models including groups with moderate growth, the fractal-like Viscek graphs, and the product group $Z_a \times Z_b$, to obtain tight bounds on the corresponding mixing times.

For real numbers $C,T_{1},T_{2},...$ we find all solutions $\mu$ to the stochastic fixed point equation $W \sim\sum_{j\ge 1}T_{j}W_{j}+C$, where $W,W_{1},W_{2},...$ are independent real-valued random variables with distribution $\mu$ and $\sim$ means equality in distribution. All solutions are infinitely divisible. The set of solutions depends on the closed multiplicative subgroup of ${ R}_{*}={ R}\backslash\{0\}$ generated by the $T_{j}$. If this group is continuous, i.e. ${R}_{*}$ itself or the positive halfline ${R}_{+}$, then all nontrivial fixed points are stable laws. In the remaining (discrete) cases further periodic solutions arise. A key observation is that the Levy measure of any fixed point is harmonic with respect to $\Lambda=\sum_{j\ge 1}\delta_{T_{j}}$, i.e. $\Gamma=\Gamma\star\Lambda$, where $\star$ means multiplicative convolution. This will enable us to apply the powerful Choquet-Deny theorem.

We study dynamic monetary risk measures that depend on bounded discrete-time processes describing the evolution of financial values. The time horizon can be finite or infinite. We call a dynamic risk measure time-consistent if it assigns to a process of financial values the same risk irrespective of whether it is calculated directly or in two steps backwards in time. We show that this condition translates into a decomposition property for the corresponding acceptance sets, and we demonstrate how time-consistent dynamic monetary risk measures can be constructed by pasting together one-period risk measures. For conditional coherent and convex monetary risk measures, we provide dual representations of Legendre--Fenchel type based on linear functionals induced by adapted increasing processes of integrable variation. Then we give dual characterizations of time-consistency for dynamic coherent and convex monetary risk measures. To this end, we introduce a concatenation operation for adapted increasing processes of integrable variation, which generalizes the pasting of probability measures. In the coherent case, time-consistency corresponds to stability under concatenation in the dual. For dynamic convex monetary risk measures, the dual characterization of time-consistency generalizes to a condition on the family of convex conjugates of the conditional risk measures at different times. The theoretical results are applied by discussing the time-consistency of various specific examples of dynamic monetary risk measures that depend on bounded discrete-time processes.

Since the pioneering works of C.C. Heyde, A.V. Nagaev, and S.V. Nagaev in 1960's and 1970's, the precise asymptotic behavior of large-deviation probabilities of sums of heavy-tailed random variables has been extensively investigated by many people, but mostly it is assumed that the random variables under discussion are independent. In this paper, we extend the study to the case of negatively dependent random variables and we find out that the asymptotic behavior of precise large deviations is insensitive to the negative dependence.

In this paper we study BSDEs with two reflecting barriers driven by a Brownian motion and an independent Poisson process. We show the existence and uniqueness of $local$ and global solutions. As an application we solve the related zero-sum Dynkin game.

In this paper we establish two-sided estimates for the density of the Feynman-Kac semigroups of stable-like processes with potentials given by signed measures belonging to the Kato class. We also provide similar estimates for the densities of two other kinds of Feynman-Kac semigroups of stable-like processes.

A random dense countable set is characterized (in distribution) by independence and stationarity. Two examples are `Brownian local minima' and `unordered infinite sample'. They are identically distributed. A framework for such concepts, proposed here, includes a wide class of random equivalence classes.

A stochastic calculus similar to Malliavin's calculus is worked out for Brownian excursions. The analogue of the Malliavin derivative in this calculus is not a differential operator, but its adjoint is (like the Skorohod integral) an extension of the Itô integral. As an application, we obtain an expression for the integrand in the stochastic integral representation of square integrable Wiener functionals; this expression is an alternative to the classical Clark-Ocone formula. Moreover, this calculus enables to construct stochastic integrals of predictable or anticipating processes (forward, backward and symmetric integrals are considered).

In this paper, we provide a scheme for simulating one-dimensional processes generated by divergence or non-divergence form operators with discontinuous coefficients. We use a space bijection to transform such a process in another one that behaves locally like a Skew Brownian motion. Indeed the behavior of the Skew Brownian motion can easily be approached by an asymmetric random walk.

We use the Malliavin calculus for Poisson processes in order to compute sensitivities for European and Asian options with underlying following a jump type diffusion. The main point is to settle an integration by parts formula (similar to the one in the Malliavin calculus) for a general multidimensional random variable which has an absolutely continuous law with differentiable density. We give an explicit expression of the differential operators involved in this formula and this permits to simulate them and consequently to run a Monte Carlo algorithm

Consider reinforced random walk on a graph that looks like a doubly infinite ladder. All edges have initial weight 1, and the reinforcement convention is to add $\delta > 0$ to the weight of an edge upon first crossing, with no reinforcement thereafter. This paper proves recurrence for all $\delta > 0$. In so doing, we introduce a more general class of processes, termed multiple-level reinforced random walks.

Editor's Note. A draft of this paper was written in 1994. The paper is one of the first to make any progress on this type of reinforcement problem. It has motivated a substantial number of new and sometimes quite difficult studies of reinforcement models in pure and applied probability. The persistence of interest in models related to this has caused the original unpublished manuscript to be frequently cited, despite its lack of availability and the presence of errors. The opportunity to rectify this situation has led us to the somewhat unusual step of publishing a result that may have already entered the mathematical folklore.

We consider a branching system of $N$ Brownian particles evolving independently in a domain $D$ during any time interval between boundary hits. As soon as one particle reaches the boundary it is killed and one of the other particles splits into two independent particles, the complement of the set $D$ acting as a catalyst or hard obstacle. Identifying the newly born particle with the one killed upon contact with the catalyst, we determine the exact law of the tagged particle as $N$ approaches infinity. In addition, we show that any finite number of labelled particles become independent in the limit. Both results can be seen as scaling limits of a genome population undergoing redistribution present in the Fleming-Viot dynamics.

In the two-type Richardson model on a graph $G=(V,E)$, each vertex is at a given time in state $0$, $1$ or $2$. A $0$ flips to a $1$ (resp.\ $2$) at rate $\lambda_1$ ($\lambda_2$) times the number of neighboring $1$'s ($2$'s), while $1$'s and $2$'s never flip. When $G$ is infinite, the main question is whether, starting from a single $1$ and a single $2$, with positive probability we will see both types of infection reach infinitely many sites. This has previously been studied on the $d$-dimensional cubic lattice $Z^d$, $d\geq 2$, where the conjecture (on which a good deal of progress has been made) is that such coexistence has positive probability if and only if $\lambda_1=\lambda_2$. In the present paper examples are given of other graphs where the set of points in the parameter space which admit such coexistence has a more surprising form. In particular, there exist graphs exhibiting coexistence at some value of $\frac{\lambda_1}{\lambda_2} \neq 1$ and non-coexistence when this ratio is brought closer to $1$.

We consider continuous and discrete (1+1)-dimensional wetting models which undergo a localization/delocalization phase transition. Using a simple approach based on Renewal Theory we determine the precise asymptotic behavior of the partition function, from which we obtain the scaling limits of the models and an explicit construction of the infinite volume measure in all regimes, including the critical one.

This paper extends the notion of the $\Lambda$-coalescent of Pitman (1999) to the spatial setting. The partition elements of the spatial $\Lambda$-coalescent migrate in a (finite) geographical space and may only coalesce if located at the same site of the space. We characterize the $\Lambda$-coalescents that come down from infinity, in an analogous way to Schweinsberg (2000). Surprisingly, all spatial coalescents that come down from infinity, also come down from infinity in a uniform way. This enables us to study space-time asymptotics of spatial $\Lambda$-coalescents on large tori in $d\geq 3$ dimensions. Some of our results generalize and strengthen the corresponding results in Greven et al. (2005) concerning the spatial Kingman coalescent.

Fragmentation processes of exchangeable partitions have already been studied by several authors. This paper deals with fragmentations of exchangeable compositions, i.e. partitions of $\mathbb{N}$ in which the order of the blocks matters. We will prove that such a fragmentation is bijectively associated to an interval fragmentation. Using this correspondence, we then study two examples: Ruelle's interval fragmentation and the interval fragmentation derived from the standard additive coalescent.

In the modified bootstrap percolation model, sites in the cube $\{1,\ldots,L\}^d$ are initially declared active independently with probability $p$. At subsequent steps, an inactive site becomes active if it has at least one active nearest neighbour in each of the $d$ dimensions, while an active site remains active forever. We study the probability that the entire cube is eventually active. For all $d\geq 2$ we prove that as $L\to\infty$ and $p\to 0$ simultaneously, this probability converges to $1$ if $L\geq\exp \cdots \exp \frac{\lambda+\epsilon}{p}$, and converges to $0$ if $L\leq\exp \cdots \exp \frac{\lambda-\epsilon}{p}$, for any $\epsilon>0$. Here the exponential function is iterated $d-1$ times, and the threshold $\lambda$ equals $\pi^2/6$ for all $d$.

We introduce a class of iterated processes called $\alpha$-time Brownian motion for $0 < \alpha \leq 2$. These are obtained by taking Brownian motion and replacing the time parameter with a symmetric $\alpha$-stable process. We prove a Chung-type law of the iterated logarithm (LIL) for these processes which is a generalization of LIL proved in Hu (1995) for iterated Brownian motion. When $\alpha =1$ it takes the following form $$ \liminf_{T\to\infty}\ T^{-1/2}(\log\log T) \sup_{0\leq t\leq T}|Z_{t}|=\pi^{2}\sqrt{\lambda_{1}} \quad a.s. $$ where $\lambda_{1}$ is the first eigenvalue for the Cauchy process in the interval $[-1,1].$ We also define the local time $L^{*}(x,t)$ and range $R^{*}(t)=|{x: Z(s)=x \text{ for some } s\leq t}|$ for these processes for $1 <\alpha <2$. We prove that there are universal constants $c_{R},c_{L}\in (0,\infty) $ such that $$ \limsup_{t\to\infty}\frac{R^{*}(t)}{(t/\log \log t)^{1/2\alpha}\log \log t}= c_{R} \quad a.s. $$ $$ \liminf_{t\to\infty} \frac{\sup_{x\in {R}}L^{*}(x,t)}{(t/\log \log t)^{1-1/2\alpha}}= c_{L} \quad a.s. $$

We study a model of $ N $ mutually repellent Brownian motions under confinement to stay in some bounded region of space. Our model is defined in terms of a transformed path measure under a trap Hamiltonian, which prevents the motions from escaping to infinity, and a pair-interaction Hamiltonian, which imposes a repellency of the $N$ paths. In fact, this interaction is an $N$-dependent regularisation of the Brownian intersection local times, an object which is of independent interest in the theory of stochastic processes. The time horizon (interpreted as the inverse temperature) is kept fixed. We analyse the model for diverging number of Brownian motions in terms of a large deviation principle. The resulting variational formula is the positive-temperature analogue of the well-known Gross-Pitaevskii formula, which approximates the ground state of a certain dilute large quantum system; the kinetic energy term of that formula is replaced by a probabilistic energy functional. This study is a continuation of the analysis in [ABK06] where we considered the limit of diverging time (i.e., the zero-temperature limit) with fixed number of Brownian motions, followed by the limit for diverging number of motions.

Given $(M_t)_{t\in \mathbb{R}_+}$ and $(M^*_t)_{t\in \mathbb{R}_+}$ respectively a forward and a backward martingale with jumps and continuous parts, we prove that $E[\phi (M_t+M^*_t)]$ is non-increasing in $t$ when $\phi$ is a convex function, provided the local characteristics of $(M_t)_{t\in \mathbb{R}_+}$ and $(M^*_t)_{t\in \mathbb{R}_+}$ satisfy some comparison inequalities. We deduce convex concentration inequalities and deviation bounds for random variables admitting a predictable representation in terms of a Brownian motion and a non-necessarily independent jump component

Consider the following forest-fire model where the possible locations of trees are the sites of a cubic lattice. Each site has two possible states: 'vacant' or 'occupied'. Vacant sites become occupied according to independent rate 1 Poisson processes. Independently, at each site ignition (by lightning) occurs according to independent rate lambda Poisson processes. When a site is ignited, its occupied cluster becomes vacant instantaneously. If the lattice is one-dimensional or finite, then with probability one, at each time the state of a given site only depends on finitely many Poisson events; a process with the above description can be constructed in a standard way. If the lattice is infinite and multi-dimensional, in principle, the state of a given site can be influenced by infinitely many Poisson events in finite time.

With the help of the methods developed in our previous article [Schmitz, to appear in Annales de l'I.H.P., in press], we highlight condition $(T)$ as a source of new examples of 'ballistic' diffusions in a random environment when $d>1$ ('ballistic' means that a strong law of large numbers with non-vanishing limiting velocity holds). In particular we are able to treat the case of non-constant diffusion coefficients, a feature that causes problems. Further we recover the ballistic character of two important classes of diffusions in a random environment by simply checking condition $(T)$. This not only points out to the broad range of examples where condition $(T)$ can be checked, but also fortifies our belief that condition $(T)$ is a natural contender for the characterisation of ballistic diffusions in a random environment when $d>1$.

Degenerate stochastic partial differential equations of divergence and non-divergence forms are considered in non-smooth domains. Existence and uniqueness results are given in weighted Sobolev spaces, and Holder estimates of the solutions are presented.

Recently, several authors have studied maps where a function, describing the local diffusion matrix of a diffusion process with a linear drift towards an attraction point, is mapped into the average of that function with respect to the unique invariant measure of the diffusion process, as a function of the attraction point. Such mappings arise in the analysis of infinite systems of diffusions indexed by the hierarchical group, with a linear attractive interaction between the components. In this context, the mappings are called renormalization transformations. We consider such maps for catalytic Wright-Fisher diffusions. These are diffusions on the unit square where the first component (the catalyst) performs an autonomous Wright-Fisher diffusion, while the second component (the reactant) performs a Wright-Fisher diffusion with a rate depending on the first component through a catalyzing function. We determine the limit of rescaled iterates of renormalization transformations acting on the diffusion matrices of such catalytic Wright-Fisher diffusions.

We study random walks on supercritical percolation clusters on wedges in $Z^3$, and show that the infinite percolation cluster is (a.s.) transient whenever the wedge is transient. This solves a question raised by O. Häggström and E. Mossel. We also show that for convex gauge functions satisfying a mild regularity condition, the existence of a finite energy flow on $Z^2$ is equivalent to the (a.s.) existence of a finite energy flow on the supercritical percolation cluster. This answers a question of C. Hoffman.

In the case of a rarefaction fan in a non-stationary Hammersley process, we explicitly calculate the asymptotic behavior of the process as we move out along a ray, and the asymptotic distribution of the angle within the rarefaction fan of a second class particle and a dual second class particle. Furthermore, we consider a stationary Hammersley process and use the previous results to show that trajectories of a second class particle and a dual second class particles touch with probability one, and we give some information on the area enclosed by the two trajectories, up until the first intersection point. This is linked to the area of influence of an added Poisson point in the plane.

Abstract. We consider stationary solutions of the three dimensional Navier--Stokes equations (NS3D) with periodic boundary conditions and driven by an external force which might have a deterministic and a random part. The random part of the force is white in time and very smooth in space. We investigate smoothness properties in space of the stationary solutions. Classical technics for studying smoothness of stochastic PDEs do not seem to apply since global existence of strong solutions is not known. We use the Kolmogorov operator and Galerkin approximations. We first assume that the noise has spatial regularity of order $p$ in the $L^2$ based Sobolev spaces, in other words that its paths are in $H^p$. Then we prove that at each fixed time the law of the stationary solutions is supported by $H^{p+1}$. Then, using a totally different technic, we prove that if the noise has Gevrey regularity then at each fixed time, the law of a stationary solution is supported by a Gevrey space. Some informations on the Kolmogorov dissipation scale are deduced.

We establish the precise asymptotics of the quantization and entropy coding errors for fractional Brownian motion with respect to the supremum norm and $L^p [0,1]$-norm distortions. We show that all moments in the quantization problem lead to the same asymptotics. Using a general principle, we conclude that entropy coding and quantization coincide asymptotically. Under supremum-norm distortion, our proof uses an explicit construction of efficient codebooks based on a particular entropy constrained coding scheme.

We consider the behaviour of a continuous super-Brownian motion catalysed by a random medium with infinite overall density under the hydrodynamic scaling of mass, time, and space. We show that, in supercritical dimensions, the scaled process converges to a macroscopic heat flow, and the appropriately rescaled random fluctuations around this macroscopic flow are asymptotically bounded, in the sense of log-Laplace transforms, by generalised stable Ornstein-Uhlenbeck processes. The most interesting new effect we observe is the occurrence of an index-jump from a Gaussian situation to stable fluctuations of index $1+\gamma$, where $\gamma \in (0,1)$ is an index associated to the medium.

We study the evolution of the interface for the one-dimensional voter model. We show that if the random walk kernel associated with the voter model has finite $\gamma$-th moment for some $\gamma > 3$, then the evolution of the interface boundaries converge weakly to a Brownian motion under diffusive scaling. This extends recent work of Newman, Ravishankar and Sun. Our result is optimal in the sense that finite $\gamma$-th moment is necessary for this convergence for all $\gamma \in (0,3)$. We also obtain relatively sharp estimates for the tail distribution of the size of the equilibrium interface, extending earlier results of Cox and Durrett, and Belhaouari, Mountford and Valle.

This paper states a law of large numbers for a random walk in a random iid environment on $Z^d$, where the environment follows some Dirichlet distribution. Moreover, we give explicit bounds for the asymptotic velocity of the process and also an asymptotic expansion of this velocity at low disorder.

We prove the following results about the images and multiple points of an $N$-parameter, $d$-dimensional Brownian sheet $B =\{B(t)\}_{t \in R_+^N}$: (1) If $\text{dim}_H F \leq d/2$, then $B(F)$ is almost surely a Salem set. (2) If $N \leq d/2$, then with probability one $\text{dim}_H B(F) = 2 \text{dim} F$ for all Borel sets of $R_+^N$, where "$\text{dim}_H$" could be everywhere replaced by the "Hausdorff," "packing," "upper Minkowski," or "lower Minkowski dimension." (3) Let $M_k$ be the set of $k$-multiple points of $B$. If $N \leq d/2$ and $ Nk > (k-1)d/2$, then $\text{dim}_H M_k = \text{dim}_p M_k = 2 Nk - (k-1)d$, a.s. The Hausdorff dimension aspect of (2) was proved earlier; see Mountford (1989) and Lin (1999). The latter references use two different methods; ours of (2) are more elementary, and reminiscent of the earlier arguments of Monrad and Pitt (1987) that were designed for studying fractional Brownian motion. If $N>d/2$ then (2) fails to hold. In that case, we establish uniform-dimensional properties for the $(N,1)$-Brownian sheet that extend the results of Kaufman (1989) for 1-dimensional Brownian motion. Our innovation is in our use of the sectorial local nondeterminism of the Brownian sheet (Khoshnevisan and Xiao, 2004).

Let $f_{n,h}$ be a kernel density estimator of a continuous and bounded $d$-dimensional density $f$. Let $\psi(t)$ be a positive continuous function such that $\|\psi f^\beta\| _\infty < \infty$ for some $0< \beta < 1/2$. We are interested in the rate of consistency of such estimators with respect to the weighted sup-norm determined by $\psi$. This problem has been considered by Gin, Koltchinskii and Zinn (2004) for a deterministic bandwidth $h_n$. We provide ``uniform in $h$'' versions of some of their results, allowing us to determine the corresponding rates of consistency for kernel density estimators where the bandwidth sequences may depend on the data and/or the location.

Inspired by the fundamental work of T.J. Lyons, we develop a theory of curvilinear integrals along a new kind of enriched paths in $R^d$. We apply these methods to the fractional Brownian Motion, and prove a support theorem for SDE driven by the Skorohod fBM of Hurst parameter $H > 1/4$.

A coagulation model on a finite spatial grid is considered. Particles of discrete masses jump randomly between sites and, while located at the same site, stick together according to some coagulation kernel. The asymptotic behavior (for increasing particle numbers) of this model is studied in the situation when the coagulation kernel grows sufficiently fast so that the phenomenon of gelation is observed. Weak accumulation points of an appropriate sequence of measure-valued processes are characterized in terms of solutions of a nonlinear equation. A natural description of the behavior of the gel is obtained by using the one-point compactification of the size space. Two aspects of the limiting equation are of special interest. First, for a certain class of coagulation kernels, this equation differs from a naive extension of Smoluchowski's coagulation equation. Second, due to spatial inhomogeneity, an equation for the time evolution of the gel mass density has to be added. The jump rates are assumed to vanish with increasing particle masses so that the gel is immobile. Two different gel growth mechanisms (active and passive gel) are found depending on the type of the coagulation kernel.

This work introduces the extended Skorokhod problem (ESP) and associated extended Skorokhod map (ESM) that enable a pathwise construction of reflected diffusions that are not necessarily semimartingales. Roughly speaking, given the closure $G$ of an open connected set in ${\mathbb R}^J$, a non-empty convex cone $d(x) \subset {\mathbb R}^J$ specified at each point $x$ on the boundary $\partial G$, and a cadlag trajectory $\psi$ taking values in ${\mathbb R}^J$, the ESM $\bar \Gamma$ defines a constrained version $\phi$ of $\psi$ that takes values in $G$ and is such that the increments of $\phi - \psi$ on any interval $[s,t]$ lie in the closed convex hull of the directions $d(\phi(u)), u \in (s,t]$. When the graph of $d(\cdot)$ is closed, the following three properties are established: (i) given $\psi$, if $(\phi,\eta)$ solve the ESP then $(\phi,\eta)$ solve the corresponding Skorokhod problem (SP) if and only if $\eta$ is of bounded variation; (ii) given $\psi$, any solution $(\phi,\eta)$ to the ESP is a solution to the SP on the interval $[0,\tau_0)$, but not in general on $[0,\tau_0]$, where $\tau_0$ is the first time that $\phi$ hits the set ${\cal V}$ of points $x \in \partial G$ such that $d(x)$ contains a line; (iii) the graph of the ESM $\bar \Gamma$ is closed on the space of cadlag trajectories (with respect to both the uniform and the $J_1$-Skorokhod topologies).

The paper then focuses on a class of multi-dimensional ESPs on polyhedral domains with a non-empty ${\cal V}$-set. Uniqueness and existence of solutions for this class of ESPs is established and existence and pathwise uniqueness of strong solutions to the associated stochastic differential equations with reflection is derived. The associated reflected diffusions are also shown to satisfy the corresponding submartingale problem. Lastly, it is proved that these reflected diffusions are semimartingales on $[0,\tau_0]$. One motivation for the study of this class of reflected diffusions is that they arise as approximations of queueing networks in heavy traffic that use the so-called generalised processor sharing discipline.

We study moderate deviations for the renormalized self-intersection local time of planar random walks. We also prove laws of the iterated logarithm for such local times.

We present a solution to some discounted optimal stopping problem for the maximum of a geometric Brownian motion on a finite time interval. The method of proof is based on reducing the initial optimal stopping problem with the continuation region determined by an increasing continuous boundary surface to a parabolic free-boundary problem. Using the change-of-variable formula with local time on surfaces we show that the optimal boundary can be characterized as a unique solution of a nonlinear integral equation. The result can be interpreted as pricing American fixed-strike lookback option in a diffusion model with finite time horizon.

Let $(S_0,S_1,\dots)$ be a supermartingale relative to a nondecreasing sequence of $\sigma$-algebras $(H_{\le0},H_{\le1},\dots)$, with $S_0\le0$ almost surely (a.s.) and differences $X_i:=S_i-S_{i-1}$. Suppose that for every $i=1,2,\dots$ there exist $H_{\le(i-1)}$-measurable r.v.'s $C_{i-1}$ and $D_{i-1}$ and a positive real number $s_i$ such that $C_{i-1}\le X_i\le D_{i-1}$ and $D_{i-1}-C_{i-1}\le 2 s_i$ a.s. Then for all real $t$ and natural $n$ and all functions $f$ satisfying certain convexity conditions $ E f(S_n)\le E f(sZ)$, where $f_t(x):=\max(0,x-t)^5$, $s:=\sqrt{s_1^2+\dots+s_n^2}$, and $Z\sim N(0,1)$. In particular, this implies $ P(S_n\ge x)\le c_{5,0}P(sZ\ge x)\quad\forall x\in R$, where $c_{5,0}=5!(e/5)^5=5.699\dots.$ Results for $\max_{0\le k\le n}S_k$ in place of $S_n$ and for concentration of measure also follow.

For discrete-time stochastic processes, there is a close connection between return (resp. waiting) times and entropy (resp. relative entropy). Such a connection cannot be straightforwardly extended to the continuous-time setting. Contrarily to the discrete-time case one needs a reference measure on path space and so the natural object is relative entropy rather than entropy. In this paper we elaborate on this in the case of continuous-time Markov processes with finite state space. A reference measure of special interest is the one associated to the time-reversed process. In that case relative entropy is interpreted as the entropy production rate. The main results of this paper are: almost-sure convergence to relative entropy of the logarithm of waiting-times ratios suitably normalized, and their fluctuation properties (central limit theorem and large deviation principle).

An annulus $\mathrm{SLE}_{\kappa}$ trace tends to a single point on the target circle, and the density function of the end point satisfies some differential equation. Some martingales or local martingales are found for annulus $\mathrm{SLE}_4$, $\mathrm{SLE}_{8}$ and $\mathrm{SLE}_{8/3}$. From the local martingale for annulus $\mathrm{SLE}_4$ we find a candidate of discrete lattice model that may have annulus $\mathrm{SLE}_4$ as its scaling limit. The local martingale for annulus $\mathrm{SLE}_{8/3}$ is similar to those for chordal and radial $\mathrm{SLE}_{8/3}$. But it seems that annulus $\mathrm{SLE}_{8/3}$ does not satisfy the restriction property

We study the last-passage growth model on the planar integer lattice with exponential weights. With boundary conditions that represent the equilibrium exclusion process as seen from a particle right after its jump we prove that the variance of the last-passage time in a characteristic direction is of order $t^{2/3}$. With more general boundary conditions that include the rarefaction fan case we show that the last-passage time fluctuations are still of order $t^{1/3}$, and also that the transversal fluctuations of the maximal path have order $t^{2/3}$. We adapt and then build on a recent study of Hammersley's process by Cator and Groeneboom, and also utilize the competition interface introduced by Ferrari, Martin and Pimentel. The arguments are entirely probabilistic, and no use is made of the combinatorics of Young tableaux or methods of asymptotic analysis.

Consider the following forest-fire model where trees are located on sites of $\mathbb{Z}$. A site can be vacant or be occupied by a tree. Each vacant site becomes occupied at rate $1$, independently of the other sites. Each site is hit by lightning with rate $\lambda$, which burns down the occupied cluster of that site instantaneously. As $\lambda \downarrow 0$ this process is believed to display self-organised critical behaviour.

This paper is mainly concerned with the cluster size distribution in steady-state. Drossel, Clar and Schwabl (1993) claimed that the cluster size distribution has a certain power law behaviour which holds for cluster sizes that are not too large compared to some explicit cluster size $s_{max}$. The latter can be written in terms of $\lambda$ approximately as $s_{max}\ln(s_{max}) = 1/ \lambda$. However, Van den Berg and Jarai (2005) showed that this claim is not correct for cluster sizes of order $s_{max}$, which left the question for which cluster sizes the power law behaviour does hold. Our main result is a rigorous proof of the power law behaviour up to cluster sizes of the order $s_{max}^{1/3}$. Further, it proves the existence of a stationary translation invariant distribution, which was always assumed but never shown rigorously in the literature.

The infinite divisibility of probability distributions on the space $P (R )$ of probability distributions on $R$ is defined and related fundamental results such as the Levy-Khintchin formula, representation of Ito type of infinitely divisible RPD, stable RPD and Levy processes on $P (R )$ are obtained. As an application we investigate limiting behaviors of a simple model of a particle motion in a random environment

We analyze the differences between the horizontal and the vertical component of the simple random walk on the 2-dimensional comb. In particular we evaluate by combinatorial methods the asymptotic behaviour of the expected value of the distance from the origin, the maximal deviation and the maximal span in $n$ steps, proving that for all these quantities the order is $n^{1/4}$ for the horizontal projection and $n^{1/2}$ for the vertical one (the exact constants are determined). Then we rescale the two projections of the random walk dividing by $n^{1/4}$ and $n^{1/2}$ the horizontal and vertical ones, respectively. The limit process is obtained. With similar techniques the walk dimension is determined, showing that the Einstein relation between the fractal, spectral and walk dimensions does not hold on the comb.

We investigate small deviation properties of Gaussian random fields in the space $L_q(R^N,\mu)$ where $\mu$ is an arbitrary finite compactly supported Borel measure. Of special interest are hereby "thin" measures $\mu$, i.e., those which are singular with respect to the $N$--dimensional Lebesgue measure; the so-called self-similar measures providing a class of typical examples. For a large class of random fields (including, among others, fractional Brownian motions), we describe the behavior of small deviation probabilities via numerical characteristics of $\mu$, called mixed entropy, characterizing size and regularity of $\mu$. For the particularly interesting case of self-similar measures $\mu$, the asymptotic behavior of the mixed entropy is evaluated explicitly. As a consequence, we get the asymptotic of the small deviation for $N$-parameter fractional Brownian motions with respect to $L_q(R^N,\mu)$-norms. While the upper estimates for the small deviation probabilities are proved by purely probabilistic methods, the lower bounds are established by analytic tools concerning Kolmogorov and entropy numbers of Holder operators.

Small world models are networks consisting of many local links and fewer long range `shortcuts', used to model networks with a high degree of local clustering but relatively small diameter. Here, we concern ourselves with the distribution of typical inter-point network distances. We establish approximations to the distribution of the graph distance in a discrete ring network with extra random links, and compare the results to those for simpler models, in which the extra links have zero length and the ring is continuous.

Let $X=(X_{i,j})_{m\times n}, m\ge n$, be a complex Gaussian random matrix with mean zero and variance $\frac 1n$, let $S=X^*X$ be a sample covariance matrix. In this paper we are mainly interested in the limiting behavior of eigenvalues when $\frac mn\rightarrow \gamma\ge 1$ as $n\rightarrow\infty$. Under certain conditions on $k$, we prove the central limit theorem holds true for the $k$-th largest eigenvalues $\lambda_{(k)}$ as $k$ tends to infinity as $n\rightarrow\infty$. The proof is largely based on the Costin-Lebowitz-Soshnikov argument and the asymptotic estimates for the expectation and variance of the number of eigenvalues in an interval. The standard technique for the RH problem is used to compute the exact formula and asymptotic properties for the mean density of eigenvalues. As a by-product, we obtain a convergence speed of the mean density of eigenvalues to the Marchenko-Pastur distribution density under the condition $|\frac mn-\gamma|=O(\frac 1n)$.

We establish integral tests and laws of the iterated logarithm for the lower envelope of positive self-similar Markov processes at 0 and $+\infty$. Our proofs are based on the Lamperti representation and time reversal arguments. These results extend laws of the iterated logarithm for Bessel processes due to Dvoretzky and Erdos (1951), Motoo (1958), and Rivero (2003).

Consider an infinite random matrix $H=(h_{ij})_{0 < i,j}$ picked from the Gaussian Unitary Ensemble (GUE). Denote its main minors by $H_i=(h_{rs})_{1\leq r,s\leq i}$ and let the $j$:th largest eigenvalue of $H_i$ be $\mu^i_j$. We show that the configuration of all these eigenvalues $(i,\mu_j^i)$ form a determinantal point process on $\mathbb{N}\times\mathbb{R}$.

Furthermore we show that this process can be obtained as the scaling limit in random tilings of the Aztec diamond close to the boundary. We also discuss the corresponding limit for random lozenge tilings of a hexagon.

An Erratum to this paper has been published in Electronic Journal of Probability, Volume 12 (2007), paper number 37.