In a previous work,we proved the validity ofthe replica-symmetric solution for the Hop field model in a nontrivial domain of parameters. This was done at the accuracy ofthe LLN In a somewhat larger domain, we obtain a description at the level ofthe CLT, or, in the terminology of physics, we calculate the fluctuations around the mean field. This is obtained by refining the tools we developed for a rigorous use of the cavity method and proving new a priori estimates about the “localization”of the Gibbs measure.

We consider a model of diffusion in random media with a two-way coupling (i.e., a model in which the randomness of the medium influences the diffusing particles and where the diffusing particles change the medium). In this particular model, particles are injected at the origin with a time-dependent rate and diffuse among random traps. Each trap has a finite (random) depth, so that when it has absorbed a finite (random) number of particles it is “saturated,” and it no longer acts as a trap. This model comes from a problem of nuclear waste management. However, a very similar model has been studied recently by Gravner and Quastel with different tools (hydrodynamic limits). We compute the asymptotic behavior of the probability of survival of a particle born at some given time, both in the annealed and quenched cases, and show that three different situations occur depending on the injection rate. For weak injection, the typical survival strategy of the particle is as in Sznitman and the asymptotic behavior of this survival probability behaves as if there was no saturation effect. For medium injection rate, the picture is closer to that of internal DLA, as given by Lawler, Bramson and Griffeath. For large injection rates, the picture is less understood except in dimension one.

Generalized internal diffusion limited aggregation is a stochastic growth model on the lattice in which a finite number of sites act as Poisson sources of particles which then perform symmetric random walks with an attractive zero-range interaction until they reach the first site which has been visited by fewer than $\alpha$ particles, at which point they stop. Sites on which particles are frozen constitute the occupied set. We prove that in appropriate regimes the particle density has a hydrodynamic limit which is the one-phase Stefan problem. This is then used to study the asymptotic behavior of the occupied set. In two dimensions when the walks are independent with one source at the origin and $\alpha=1$, we obtain in particular that the occupied set is asymptotically a disc of radius $K\sqrt{t}$, where $K$ is the solution of $\exp (-K^2 /4) = \pi K^2$, settling a conjecture of Lawler, Bramson and Griffeath.

Let $M$ be a randomly chosen $n \times n$ permutation matrix. For a fixed arc of the unit circle, let $X$ be the number of eigenvalues of $M$ which lie in the specified arc. We calculate the large $n$ asymptotics for the mean and variance of $X$, and show that $(X -E[X])/( \text{Var} (X))^ {1/2}$ is asymptotically normally distributed. In addition, we show that for several fixed arcs $I_1,\ldots,I_m$, the corresponding random variables are jointly normal in the large $n$ limit.

The genealogy of a cluster in the multitype voter model can be defined in terms of a family of dual coalescing random walks. We represent the genealogy of a cluster as a point process in a size-time plane and show that in high dimensions the genealogy of the cluster at the origin has a weak Poisson limit.The limiting point process is the same as for the genealogy of the size-biased Galton-Watson tree. Moreover, our results show that the branching mechanism and the spatial effects of the voter model can be separated on a macroscopic scale. Our proofs are based on a probabilistic construction of the genealogy of the cluster at the origin derived from Harris’ graphical representation of the voter model.

Under very mild conditions, we prove that the limiting behavior of the component counts in a decomposable logarithmic combinatorial structure conforms to a single, unified pattern, which includes functional central limit theorems, Erdös-Turán laws, Poisson–Dirichlet limits for the large components and Poisson approximation in total variation for the total number ofcomponents. Our approach is entirely probabilistic, and the conditions can readily be verified in practice.

We consider a sequence of probability measures $\nu_n$ obtained by conditioning a random vector $X =(X_1,\ldots,X_d)$ with nonnegative integer valued components on

$$ X_1 + \dots + X_d = n - 1 $$

and give several sufficient conditions on the distribution of $X$ for $\nu_n$ to be stochastically increasing in $n$. The problem is motivated by an interacting particle system on the homogeneous tree in which each vertex has $d +1$ neighbors. This system is a variant of the contact process and was studied recently by A.Puha. She showed that the critical value for this process is 1/4 if $d = 2$ and gave a conjectured expression for the critical value for all $d$. Our results confirm her conjecture, by showing that certain $\nu_n$’s defined in terms of $d$-ary Catalan numbers are stochastically increasing in $n$. The proof uses certain combinatorial identities satisfied by the $d$-ary Catalan numbers.

It is proved in this paper that a weak parabolic Harnack inequality for a Markov semigroup implies the existence of a coupling and a shift coupling for the corresponding process with equal chances of success. This implies equality of the tail and invariant $\sigma$-fields for the diffusion as well as equality of the class of bounded parabolic functions and the class of bounded harmonic functions.

Measure-valued branching random walks (superprocesses) on $p$-adics are introduced and investigated. The uniqueness and existence of solutions to associated linear and nonlinear heat-type (parabolic) equations are proved, provided some condition on the parameter of the random walks is satisfied. The solutions of these equations are shown to be locally constant if their initial values are. Moreover, the heat-type equations can be identified with a system of ordinary differential equations. Conditions for the measure-valued branching stable random walks to possess the property of quasi-self-similarity are given, as well as a sufficient and necessary condition for these processes to be locally extinct. The latter result is consistent with the Euclidean case in the sense that the critical value for measure-valued branching stable processes to be locally extinct is the Hausdorff dimension of the image of the underlying processes divided by the dimension of the state space.

We establish a duality relation for the moments of bounded solutions to a class of one-dimensional parabolic stochastic partial differential equations. The equations are driven by multiplicative space-time white noise, with a non-Lipschitz multiplicative functional. The dual process is a system of branching Brownian particles. The same method can be applied to show uniqueness in law for a class of non-Lipschitz finite dimensional stochastic differential equations.

Consider the stochastic partial differential equation $$u_t=u_{xx}+u^{\gamma W},$$ where $x\in\mathbf{I}\equiv[0,J], W=W(t,x)$ is 2-parameter white noise, and we assume that the initial function $u(0,x)$ is nonnegative and not identically 0. We impose Dirichlet boundary conditions on u in the interval I. We say that u blows up in finite time, with positive probability, if there is a random time $T < \infty$ such that $$ P(\lim_{t \uparrow T} \sup_{x} u(t, x) = \infty) > 0. $$ It was known that if $\gamma<3/2$, then with probability 1, $u$ does not blowup in finite time. It was also known that there is a positive probability of finite time blowup for $\gamma$ sufficiently large.

We show that if $\gamma>3/2$, then there is a positive probability that u blows up in finite time.

Stochastic integral representation of martingales has been undergoing a renaissance due to questions motivated by stochastic finance theory. In the Brownian case one usually has formulas (of differing degrees of exactness) for the predictable integrands. We extend some of these to Markov cases where one does not necessarily have stochastic integral representation of all martingales. Moreover we study various convergence questions that arise naturally from (for example)approximations of “price processes” via Euler schemes for solutions of stochastic differential equations. We obtain general results of the following type: let $U, U^n$ be random variables with decompositions

$$ U = \alpha + \int_{0}^{\infty} \xi_s dX_s + N_\infty, $$

$$ U^n = \alpha_n + \int_{0}^{\infty} \xi^n_s dX^n_s + N^n_\infty, $$

where $X, N, X^n, N^n$ are martingales. If $X^n \to X$ and $U^n \to U$, when and how does $\xi^n\rightarrow\xi?$

Let $X$ be a strongly symmetric recurrent Markov process with state space $S$ and let $L_t^x$ denote the local time of $X$ at $X \in S$. For a fixed element 0 in the state space S, let $$ \tau(t) := \inf \{s: L^0_s > t \}. $$

The 0-potential density, $u_{\{0\}}(x, y)$, of the process $X$ killed at $T_0 = \inf \{s:X_s =0\}$ is symmetric and positive definite. Let $\eta = \{\eta_x; x \in S \}$ be a mean-zero Gaussian process with covariance $$ E_\eta (\eta_x \eta_y ) = u_{\{0\}}(x, y). $$ The main result of this paper is the following generalization of the classical second Ray–Knight theorem: for any $b \in R$ and $t > 0$, $$ \{L^x_{\tau(t)} + \frac{1}{2} (\eta_x + b)^2; x \in S \} = \{ \frac{1}{2} ( \eta_x + \sqrt{2+ b^2})^2 ; x \in S \} \text{ in law}. $$ A version of this theorem is also given when $X$ is transient.

This work is concerned with the structural analysis of stationary $\alpha$-stable random fields. Three distinct classes of such random fields are characterized and it is shown that every stationary $\alpha$-stable random field can be uniquely decomposed into the sum of three independent components belonging to these classes. Various examples of stationary $\alpha$-stable random fields are discussed in this context.

For a random walk with negative drift we study the exceedance probability (ruin probability) of a high threshold. The steps of this walk (claim sizes) constitute a stationary ergodic stable process. We study how ruin occurs in this situation and evaluate the asymptotic behavior of the ruin probability for a large variety of stationary ergodic stable processes. Our findings show that the order of magnitude of the ruin probability varies significantly from one model to another. In particular, ruin becomes much more likely when the claim sizes exhibit long-range dependence. The proofs exploit large deviation techniques for sums of dependent stable random variables and the series representation of a stable process as a function of a Poisson process.

It has been suggested that stochastic flows might be used to model the spread of passive tracers in a turbulent fluid.We define a stochastic flow by the equations $$\begin{eqnarray} \phi_0 (x) &=& x, \\ d\phi_t (x) &=& F (dt, \phi_t (x)),\end{eqnarray}$$ where $F(t, x)$ is a field of semimartingales on $x \in \mathbb{R}^d$ for $d \geq 2$ whose local characteristics are bounded and Lipschitz. The particles are points in a bounded set $\mathscr{X}$, and we ask how far the substance has spread in a time $T$. That is, we define $$\Phi^*_T = \sup_{x \in \mathscr{X}} \sup_{0 \leq t \leq T} || \phi_t (x) ||,$$ and seek to bound $P\{\Phi_T^*>z\}$.

Without drift, when $F(\cdot, x)$ are required to be martingales, although single points move on the order of $\sqrt{T}$, it is easy to construct examples in which the supremum $\Phi_T^*$ still grows linearly in time—that is, $\lim \inf_{T \to \infty}\Phi_T^* / T > 0$ almost surely. We show that this is an upper bound for the growth; that is, we compute a finite constant $K_0$, depending on the bounds for the local characteristics, such that $$ \limsup_{T \to \infty} \frac{\Phi^*_T}{T} \leq K_0 \text{ almost surely.}$$ A linear bound on growth holds even when the field itself includes a drift term.

Let $ X, X_1, X_2 \ldots$ be i.i.d. random variables with mean 0 and positive, finite variance $\sigma^2$, and set $S_n = X_1 + \cdots + X_n, n \geq 1$. Continuing earlier work related to strong laws, we prove the following analogs for the law of the iterated logarithm: $$\lim_{\varepsilon\searrow\sigma\sqrt{2}}\sqrt{\varepsilon^2−2\sigma^2}\sum_{n\ge3}\frac{1}{n}P(|S_n|\ge\varepsilon\sqrt{n\log\log n}+a_n)=\sigma\sqrt{2}$$ whenever $a_n = O(\sqrt{n}(\log \log n)^{-\gamma})$ for some $\gamma \geq 1/2$ (assuming slightly more than finite variance), and $$\lim_{\varepsilon\searrow 0}\varepsilon^2\sum_{n\ge3}\frac{1}{n \log n}P(|S_n|\ge\varepsilon\sqrt{n\log\log n})=\sigma^{2}.$$

Let $\Phi$ be a symmetric function, nondecreasing on $[0,\infty)$ and satisfying a $\Delta_2$ growth condition, $(X_1, Y_1), (X_2, Y_2),\ldots,( X_n,Y_n)$ be independent random vectors such that (for each $1\le i\le n)$ either $Y_i =X_i$ or $Y_i$ is independent of all the other variates, and the marginal distributions of $\{X_i\}$ and $\{Y_j\}$ are otherwise arbitrary. Let $\{f_ {ij}(x, y)\}_{1\le i,j\le n}$ be any array of real valued measurable functions.We present a method of obtaining the order of magnitude of $$E\Phi\left( \sum_{1 \le i, j \le n}f_{ij}(X_i,Y_j)\right).$$ The proof employs a double symmetrization, introducing independent copies $\{\tilde{X}_i,\tilde{Y}_j\}$ of $\{X_i,Y_j\}$, and moving from summands of the form $f _{ij}(X_i, Y_j)$ to what we call $f_{ij}^{(s)}(X_i,Y_j,\tilde{X}_i,\tilde{Y}_j)$. Substitution of fixed constants $\tilde{x}_i$ and $\tilde{y}_ j$ for $\tilde{X}_ i$ and $\tilde{Y}_ j$ results in $f_{ij}^{(s)}(X_i,Y_j,\tilde{x}_i,\tilde{y}_j)$, which equals $f_{ij}(X_i,Y_j)$ adjusted by a sum of quantities of first order separately in ${X_i}$ and ${Y_j}$. Introducing further explicit first-order adjustments, call them $g_{1ij}(X_i,\mathbf{\tilde{x}},\mathbf{\tilde{y}})$ and $g_{2ij}(Y_j,\mathbf{\tilde{x}},\mathbf{\tilde{y}})$, it is proved that $$\begin{equation} E\Phi\left( \sum_{1 \le i, j \le n}\left(f^{(s)}_{ij}(X_i,Y_j,\tilde{x}_i,\tilde{y}_j)−g_{1ij}(X_i,\mathbf{\tilde{x}},\mathbf{\tilde{y}})−g_{2ij}(Y_j,\mathbf{\tilde{x}},\mathbf{\tilde{y}})\right)\right)\\ \le_\alpha E\Phi\left(\sqrt{\sum_{1 \le i, j \le n}\left(f^{(s)}_{ij}(X_i,Y_j,\tilde{x}_i,\tilde{y}_j)\right)^2}\right)\approx_\alpha \Phi\left(\mathbf{f}^{(s)},\mathbf{X},\mathbf{Y},\mathbf{\tilde{x}},\mathbf{\tilde{y}}\right) \end{equation}$$ where the latter is an explicitly computable quantity. For any $\mathbf{\tilde{x}}^0$ and $\mathbf{\tilde{y}}^0$ which come within a factor of two of minimizing $\Phi(\mathbf{f}^{(s)}, \mathbf{X},\mathbf{Y},\mathbf{\tilde{x}},\mathbf{\tilde{y}})$ it is shown that $$ \begin{equation} \left.\begin{aligned} E\Phi &\left( \sum_{1 \le i, j \le n}f_{ij}(X_i,Y_j)\right)\\ \approx_\alpha & \max\left\{\Phi(\mathbf{f}^{(s)},\mathbf{X},\mathbf{Y},\mathbf{\tilde{x}}^0,\mathbf{\tilde{y}}^0),E\Phi \left(\sum_{1 \le i, j \le n}\Big(f_{i,j}(X_i,\tilde{y}^0_j)+f_{ij}(\tilde{x}^0_i,Y_j)\\ −f_{ij}(\tilde{x}^0_i,\tilde{y}^0_j)+g_{1ij}(X_i, \tilde{x}^0_i,\tilde{y}^0_j)+g_{2ij}(Y_j,\tilde{x}^0_i,\tilde{y}^0_j)\Big)\right)\right\}, \end{aligned}\right. \end{equation} $$ which is computable (approximable)in terms of the underlying random variables. These results extend to the expectation of $\Phi$ of a sum of functions of $k$-components.

Under some mild regularity on the normalizing sequence, we obtain necessary and sufficient conditions for the strong law of large numbers for (symmetrized) U-statistics.We also obtain necessary and sufficient conditions for the a.s. convergence of series of an analogous form.