We study some probabilistic representations, based on branching processes, of a simple nonlinear differential equation, i.e. $u'=\lambda u(au^R-1)$. The first approach is basically the same used by Le Jan and Sznitman for 3-d Navier-Stokes equations, which need small initial data to work. In our much simpler setting we are able to make this precise, finding all the cases where their method fails to give the solution. The second approach is based on a resummed representation, which we can prove to give all the solutions of the problem, even those with large initial data.

The purpose of this work is to study some possible application of FKG inequality to the Brownian motion and to Stochastic Differential Equations. We introduce a special ordering on the Wiener space and prove the FKG inequality with respect to this ordering. Then we apply this result on the solutions $X_t$ of a stochastic differential equation with a positive coefficient $\sigma$, we prove that these solutions $X_t$ are increasing with respect to the ordering, and finally we deduce a correlation inequality between the solution of different stochastic equations.

A recent theorem by M. Manstavicius (2004) provided a link between a certain function of transition probabilities of a strong Markov process and the boundedness of the $p$-variation of its trajectories. Here one assumption of that theorem is relaxed and an example is constructed to show that the Markov property cannot be easily dispensed with.

We give upper bounds for the probability $P(|f(X)-Ef(X)| > x)$, where $X$ is a stable random variable with index close to 2 and $f$ is a Lipschitz function. While the optimal upper bound is known to be of order $1/x^\alpha$ for large $x$, we establish, for smaller $x$, an upper bound of order $\exp(-x^\alpha/2)$, which relates the result to the gaussian concentration.

We prove a weak version of the law of large numbers for multi-dimensional finite range random walks in certain mixing elliptic random environments. This already improves previously existing results, where a law of large numbers was known only under strong enough transience. We also prove that for such walks the zero-one law implies a law of large numbers.

We examine the percolation model on $\mathbb{Z}^d$ by an approach involving lattice animals and their surface-area-to-volume ratio. For $\beta \in [0,2(d-1))$, let $f(\beta)$ be the asymptotic exponential rate in the number of edges of the number of lattice animals containing the origin which have surface-area-to-volume ratio $\beta$. The function $f$ is bounded above by a function which may be written in an explicit form. For low values of $\beta$ ($\beta \leq 1/p_c - 1$), equality holds, as originally demonstrated by F. Delyon. For higher values ($\beta > 1/p_c - 1$), the inequality is strict.

We introduce two critical exponents, one of which describes how quickly $f$ falls away from the explicit form as $\beta$ rises from $1/p_c - 1$, and the second of which describes how large clusters appear in the marginally subcritical regime of the percolation model. We demonstrate that the pair of exponents must satisfy certain inequalities. Other such inequalities yield sufficient conditions for the absence of an infinite cluster at the critical value (c.f. [4]). The first exponent is related to one of a more conventional nature in the scaling theory of percolation, that of correlation size. In deriving this relation, we find that there are two possible behaviours, depending on the value of the first exponent, for the typical surface-area-to-volume ratio of an unusually large cluster in the marginally subcritical regime.

Let $X$ and $Y$ be Poisson point processes on the real numbers with rates $l_1$ and $l_2$ respectively. We show that if $l_1 > l_2$, then there exists a deterministic map $f$ such that $f(X)$ and $Y$ have the same distribution, the joint distribution of $(X, f(X))$ is translation-invariant, and which is monotone in the sense that for all intervals $I$, $f(X)(I) \leq X(I)$, almost surely.

We determine exactly when a certain randomly weighted, self–normalized sum converges in distribution, partially verifying a 1965 conjecture of Leo Breiman. We, then, apply our results to characterize the asymptotic distribution of relative sums and to provide a short proof of a 1973 conjecture of Logan, Mallows, Rice and Shepp on the asymptotic distribution of self–normalized sums in the case of symmetry.

We review the concept of topological recurrence for weak Feller Markov chains on compact state spaces and explore the implications of this concept for the ergodicity of the processes. We also prove some conditions for existence and uniqueness of invariant measures of certain types. Examples are given from the class of iterated function systems on the real line.

Consider a real valued stationary process $X=\{X_s : s\in \mathbb{R}\}$. For a fixed $t\in \mathbb{R}$ and a set $D$ in the state space of $X$, let $g_t$ and $d_t$ denote the starting and the ending time, respectively, of an excursion from and to $D$ (straddling $t$). Introduce also the occupation times $I^+_t$ and $I^-_t$ above and below, respectively, the observed level at time $t$ during such an excursion. In this note we show that the pairs $(I^+_t, I^-_t)$ and $(t-g_t, d_t-t)$ are identically distributed. This somewhat curious property is, in fact, seen to be a fairly simple consequence of the known general uniform sojourn law which implies that conditionally on $I^+_t + I^-_t = v$ the variable $I^+_t$ (and also $I^-_t$) is uniformly distributed on $$. We also particularize to the stationary diffusion case and show, e.g., that the distribution of $I^-_t+I^+_t$ is a mixture of gamma distributions.

We consider a last-passage directed percolation model in $Z_+^2$, with i.i.d. weights whose common distribution has a finite $(2+p)$th moment. We study the fluctuations of the passage time from the origin to the point $(n,n^a)$. We show that, for suitable $a$ (depending on $p$), this quantity, appropriately scaled, converges in distribution as $n\to\infty$ to the Tracy-Widom distribution, irrespective of the underlying weight distribution. The argument uses a coupling to a Brownian directed percolation problem and the strong approximation of Komlós, Major and Tusnády.

There are two types of particles interacting on a homogeneous tree of degree $d+1$. The particles of the first type colonize the empty space with exponential rate 1, but cannot take over the vertices that are occupied by the second type. The particles of the second type spread with exponential rate $\lambda$. They colonize the neighboring vertices that are either vacant or occupied by the representatives of the opposite type, and annihilate the particles of the type 1 as they reach them. There exists a critical value $\lambda_c=(2d-1)+\sqrt{(2d-1)^2-1}$ such that the first type survives with positive probability for $\lambda < \lambda_c$, and dies out with probability one for $\lambda > \lambda_c$. We also find the growth profile which characterizes the rate of growth of the type 1 in the space-time on the event of survival.

In this paper a method for proving homogenization of divergence form elliptic equations is extended to the non-divergence case. A new proof of homogenization is given when the coefficients in the equation are assumed to be stationary and ergodic. A rate of convergence theorem in homogenization is also obtained, under the assumption that the coefficients are i.i.d. and the elliptic equation can be solved by a convergent perturbation series.

This paper concerns the Markov process duality between the one-dimensional heat equation driven by Fisher-Wright white noise and slowly coalescing Brownian particles. A representation is found for the law of the solution $x \to U(t,x)$ to the stochastic PDE, at a fixed time, in terms of a labelled system of such particles.

Novikov and Shiryaev (2004) give explicit solutions to a class of optimal stopping problems for random walks based on other similar examples given in Darling et al. (1972). We give the analogue of their results when the random walk is replaced by a Lévy process. Further we show that the solutions show no contradiction with the conjecture given in Alili and Kyprianou (2004) that there is smooth pasting at the optimal boundary if and only if the boundary of the stopping reigion is irregular for the interior of the stopping region.

We conjecture that the Parisi functional in the SK model is convex in the functional order parameter and prove a partial result that shows the convexity along one-sided directions. A consequence of this result is the log-convexity of $L_m$ norm for a class or random variables.

Initially a car is placed with probability $p$ at each site of the two-dimensional integer lattice. Each car is equally likely to be East-facing or North-facing, and different sites receive independent assignments. At odd time steps, each North-facing car moves one unit North if there is a vacant site for it to move into. At even time steps, East-facing cars move East in the same way. We prove that when $p$ is sufficiently close to 1 traffic is jammed, in the sense that no car moves infinitely many times. The result extends to several variant settings, including a model with cars moving at random times, and higher dimensions.

The free energy of quenched disordered systems is bounded above by the free energy of the corresponding annealed system. This bound may be improved by applying the annealing procedure, which is just Jensen inequality, after having modified the Hamiltonian in a way that the quenched expressions are left unchanged. This procedure is often viewed as a partial annealing or as a constrained annealing, in the sense that the term that is added may be interpreted as a Lagrange multiplier on the disorder variables.

In this note we point out that, for a family of models, some of which have attracted much attention, the multipliers of the form of empirical averages of local functions cannot improve on the basic annealed bound from the viewpoint of characterizing the phase diagram. This class of multipliers is the one that is suitable for computations and it is often believed that in this class one can approximate arbitrarily well the quenched free energy.

Stochastic delay differential equations (SDDE) on a manifold $M$ depend intrinsically on a connection $\nabla$ in this space. The main geometric result in this notes concerns the horizontal lift of solutions of SDDE on a manifold $M$ to an SDDE in the frame bundle $BM$, hence the lifted equation should come together with the prolonged horizontal connection $\nabla^H$ on $BM$. We show that every horizontal semimartingale can be represented as a solution of an SDDE.

We use excursion theory and the ergodic theorem to present an extreme-value analysis of the classical law of the iterated logarithm (LIL) for Brownian motion. A simplified version of our method also proves, in a paragraph, the classical theorem of Darling and Erdős (1956).

Let $B$ be a standard Brownian motion and let $b_\gamma$ be a piecewise linear continuous boundary function. In this paper we obtain an exact asymptotic expansion of $P\{ B(t) < b_\gamma(t), \forall t\in [0,1]\} $ provided that the boundary function satisfies $\lim_{\gamma \to \infty} b_\gamma(t^*)= -\infty$ for some $t^*\in (0,1]$.

We use our maximum inequality for $p$-th order random variables ($p>1$) to prove a strong law of large numbers (SLLN) for sequences of $p$-th order random variables. In particular, in the case $p=2$ our result shows that $\sum f(k)/k < \infty$ is a sufficient condition for SLLN for $f$-quasi-stationary sequences. It was known that the above condition, under the additional assumption of monotonicity of $f$, implies SLLN (Erdos (1949), Gal and Koksma (1950), Gaposhkin (1977), Moricz (1977)). Besides getting rid of the monotonicity condition, the inequality enables us to extend thegeneral result to $p$-th order random variables, as well as to the case of Banach-space-valued random variables.

We study partition distributions in a population genetics model incorporating symmetric selection and mutation. They generalize Ewens distributions in the infinitely-many-neutral-alleles model, an explicit expression of which is known as the Ewens sampling formula. A sampling formula for the generalized model is obtained by means of calculus for Poisson and gamma processes.

We use stochastic analysis to study the random motion of a rigid body under a white noise perturbation. We obtain a formula for the angular velocity in an average sense and discuss the stability near a principle axis.

We consider a transient one-dimensional random walk $X_n$ in random environment having zero asymptotic speed. For a class of non-i.i.d. environments we show that $\log X_n / \log n$ converges in probability to a positive constant.

We define a Fractional Brownian Motion indexed by a sphere, or more generally by a compact rank one symmetric space, and prove that it exists if, and only if, $0 < H \leq 1/2$. We then prove that Fractional Brownian Motion indexed by an hyperbolic space exists if, and only if, $0 < H \leq 1/2$. At last, we prove that Fractional Brownian Motion indexed by a real tree exists when $0 < H \leq 1/2$.

Starting with a percolation model in $\mathbb{Z}^d$ in the subcritical regime, we consider a random walk described as follows: the probability of transition from $x$ to $y$ is proportional to some function $f$ of the size of the cluster of $y$. This function is supposed to be increasing, so that the random walk is attracted by bigger clusters. For $f(t)=e^{\beta t}$ we prove that there is a phase transition in $\beta$, i.e., the random walk is subdiffusive for large $\beta$ and is diffusive for small $\beta$.

We consider minimax trees with independent, identically distributed leaf values that have a continuous distribution function $F_V$ being strictly increasing on the range where $0 < F_V < 1$. It was shown by Pearl that the root value of such trees converges to a deterministic limit in probability without any scaling. We show that after normalization we have convergence in distribution to a nondegenerate limit random variable.

In a recent work, Boyd, Diaconis and Xiao introduced a semidefinite programming approach for computing the fastest mixing Markov chain on a graph of allowed transitions, given a target stationary distribution. In this paper, we show that standard mixing time analysis techniques—variational characterizations, conductance, canonical paths—can be used to give simple, nontrivial lower and upper bounds on the fastest mixing time. To test the applicability of this idea, we consider several detailed examples including the Glauber dynamics of the Ising model.

Christian Houdre has kindly pointed us to a paper by A. Fuks, A. Joffe and J. Teugels, where their Theorem 5.3 is our Proposition 3 in the case $0 < \alpha < 1$.