We consider critical reversible nearest particle systems. We assume that the associated renewal measure has large moments as well as some regularity conditions. It is shown that such processes, started from a nontrivial ergodic translation invariant distribution, converge in distribution to the upper invariant measure.

We provide a probabilistic interpretation of a class of natural capacities on Euclidean space in terms of the level sets of a suitably chosen multiparameter additive Lévy process X. We also present several probabilistic applications of the aforementioned potential-theoretic connections. They include areas such as intersections of Lévy processes and level sets, as well as Hausdorff dimension computations.

Consider the stochastic nonlinear oscillator equation

\[ \ddot{x} = -x -x^3 + \varepsilon^2 \beta \dot{x} + \varepsilon \sigma x \dot{W}_t \]

with $\beta < 0$ and $\sigma \neq 0$. If $4\beta + \sigma^2 > 0$ then for small enough $\varepsilon > 0$ the system $(x,\dot{x})$ is positive recurrent in ${\bf R}^2 \setminus \{(0,0\}$. Now let $\overline{\lambda}(\varepsilon)$ denote the top Lyapunov exponent for the linearization of this equation along trajectories. The main result asserts that

\[ \overline{\lambda}(\varepsilon) = \varepsilon^{2/3}\overline{\lambda} + O(\varepsilon^{4/3}) \qquad\mbox{as}\ \varepsilon \to 0 \]

with $\overline{\lambda} > 0$. This result depends crucially on the fact that the system above is a small perturbation of a Hamiltonian system. The method of proof can be applied to a more general class of small perturbations of two-dimensional Hamiltonian systems. The techniques used include (i) an extension of results of Pinsky and Wihstutz for perturbations of nilpotent linear systems, and (ii) a stochastic averaging argument involving

Consider the one-dimensional solution $X=\{X_t\}_{t \in [0,T]}$ of a possibly degenerate stochastic differential equation driven by a (non compensated) Poisson measure. We denote by $\mathcal{M}$ a set of deterministic integer-valued measures associated with the considered Poisson measure. For $m\in \mathcal{M}$, we denote by $S(m)=\{S_t(m)\}_{t\in[0,T]}$ the skeleton associated with $X$. We assume some regularity conditions, which allow to define a sort of “derivative” $D S_t(m)$ of $S_t(m)$ with respect to $m$. Then we fix $t \in\,]0,T]$, $y\in \reel$, and we prove that as soon there exists $m\in \mathcal{M}$ such that $S_t(m)=y$, $DS_t(m) \ne 0$ and $\Delta S_t(m) =0$, the law of $X_t$ is bounded below by a nonnegative measure admitting a continuous density not vanishing at $y$. In the case where the law of $X_t$ admits a continuous density $p_t$, this means that $p_t(y)>0$. We finally apply the described method in order to prove that the solution to a Kac equation without cutoff does never vanish.

We prove that, under fairly general conditions, a properly rescaled determinantal random point field converges to a generalized Gaussian random process.

We give equivalences for conditions like $X(T(r))/r\rightarrow 1$ and $X(T^{*}(r))/\allowbreak r\rightarrow 1$, where the convergence is in probability or almost sure, both as $r\rightarrow 0$ and $r\rightarrow \infty$, where $X$ is a L\'{e}vy process and $T(r)$ and $T^{*}(r)$ are the first exit times of $X$ out of the strip $\{(t,y):t> 0,|y|\leq r\}$ and half-plane $\{(t,y):t> 0$, $y\leq r\}$, respectively. We also show, using a result of Kesten, that $X(T^{*}(r))/r\rightarrow 1$ a.s.\ as $r\to 0$ is equivalent to $X$ ``creeping'' across a level.

This paper studies questions of changes of variables in a class of hyperbolic stochastic partial differential equations in two variables driven by white noise. Two types of changes of variables are considered: naive changes of variables which do not involve a change of filtration, which affect the equation much as though it were deterministic, and changes of variables that do involve a change of filtration, such as time-reversals. In particular, if the process in reversed coordinates does satisfy an s.p.d.e., then we show how this s.p.d.e. is related to the original one. Time-reversals for the Brownian sheet and for equations with constant coefficients are considered in detail. A necessary and sufficient condition is provided under which the reversal of the solution to the simplest hyperbolic s.p.d.e. with certain random initial conditions again satisfies such an s.p.d.e. This yields a negative conclusion concerning the reversal in time of the solution to the stochastic wave equation (in one spatial dimension) driven by white noise.

Smirnov obtained the distribution $F$ for his $ \omega ^2$-test in the form of a certain series. $F$ is identical to the distribution of the the Brownian bridge in the $L^2$ norm. Smirnov, Kac and Shepp determined the Laplace--Stieltjes transform of $F$. Anderson and Darling expressed $F$ in terms of Bessel functions. In the present paper we compute the moments of $F$ and their asymptotics, obtain expansions of $F$ and its density $f$ in terms of the parabolic cylinder functions and Laguerre functions, and determine their asymptotics for the small and large values of the argument. A novel derivation of expansions of Smirnov and of Anderson and Darling is obtained.

Stochastic forward integrals for processes more general than semimartingales are shown to exist, generalized forms of Itô–Wentzell formula and covariation formula are proved, and one-dimensional stochastic equations driven by finite quadratic variation processes and semimartingales are solved. This generalized stochastic calculus is motivated by applications to uniqueness and dependence on parameters for stochastic equations with nonregular drift.

Distinct excursion intervals of a Brownian motion (that correspond to a fixed level) have no common endpoints. What is the situation for distinct excursion sets of a Brownian sheet? These sets are termed Brownian bubbles in the literature, and this paper examines how bubbles from fixed or random levels come into contact with each other, by examining whether or not the Brownian sheet restricted to a specific type of curve can have a point of increase. At random levels, we show that points of increase can occur along horizontal lines, while at fixed levels, such a point of increase can occur at the corner of a broken line segment with a right-angle. In addition, the Hausdorff dimension of the set of points with this last property is shown to be 1/2 a.s.

We study the Poisson boundary ($\equiv$ representation of bounded harmonic functions) of Markov operators on discrete state spaces that are invariant under the action of a transitive group of permutations. This automorphism group is locally compact, but not necessarily discrete or unimodular. The main technical tool is the entropy theory which we develop along the same lines as in the case of random walks on countable groups, while, however, the implementation is different and exploits discreteness of the state space on the one hand and the path space of the induced random walk on the nondiscrete group on the other. Various new examples are given as applications, including a description of the Poisson boundary for random walks on vertex-transitive graphs with infinitely many ends and on the Diestel-Leader graphs.

We prove the validity of the “replica-symmetric” solution of the Sherrington–Kirkpatrick model in a region that (probably) coincides with the region predicted by the physicists.

We analyze the existence and properties of right inverses $K$ for nonsymmetric Lévy processes $X$, extending recent work of Evans in the symmetric setting. First, both $X$ and $-X$ have right inverses if and only if $X$ is recurrent and has a nontrivial Gaussian component. Our main result is then a description of the excursion measure $n^Z$ of the strong Markov process $Z=X-L$ (reflected process) where $L_t=\inf\{x>0:K_x>t\}$. Specifically, $n^Z$ is essentially the restriction of $n^X$ to the ``excursions starting negative.'' Second, when only asking for right inverses of $X$, a certain ``strength of asymmetry'' is needed. Millar's notion of creeping turns out necessary but not sufficient for the existence of right inverses. We analyze this both in the bounded and unbounded variation case with a particular emphasis on results in terms of the Lévy–Khintchine characteristics.

Within the framework of information geometry, the interaction among units of a stochastic system is quantified in terms of the Kullback–Leibler divergence of the underlying joint probability distribution from an appropriate exponential family. In the present paper, the main example for such a family is given by the set of all factorizable random fields. Motivated by this example, the locally farthest points from an arbitrary exponential family $\mathcal{E}$ are studied. In the corresponding dynamical setting, such points can be generated by the structuring process with respect to $\mathcal{E}$ as a repelling set. The main results concern the low complexity of such distributions which can be controlled by the dimension of $\mathcal{E}$.

Let $\{X_n\}^{\infty}_0$ be a Markov chain with values in $[0,1]$ generated by the iteration of random logistic maps defined by $X_{n+1}=f_{C_{n+1}}(X_n)\equiv C_{n+1}X_n(1-X_n)$, $n=0,1,2,\ldots\,$, with $\{C_n\}^{\infty}_1$ being independent and identically distributed random variables with values in $[0,4]$ and independent of $X_0$. This paper provides a class of examples where $C_i$ take only two values $\lambda$ and $\mu$ such that there exist two distinct invariant probability distributions $\pi_0$ and $\pi_1$ supported by the open interval $(0,1)$. This settles a question raised by R. N. Bhattacharya.

The random-cluster model is a dependent percolation model that has applications in the study of Ising and Potts models. In this paper, several new results are obtained for the random-cluster model on nonamenable graphs with cluster parameter $q \geq 1$. Among these, the main ones are the absence of percolation for the free random-cluster measure at the critical value and examples of planar regular graphs with regular dual where $p_ \mathrm{c}^{\mathrm{free}} (q) > p_ \mathrm{u}^{\mathrm{wired}} (q)$ for $q$ large enough. The latter follows from considerations of isoperimetric constants, and we give the first nontrivial explicit calculations of such constants. Such considerations are also used to prove nonrobust phase transition for the Potts model on nonamenable regular graphs.