We study the homogenization property of systems of quasi-linear PDEs of parabolic type with periodic coefficients, highly oscillating drift and highly oscillating nonlinear term. To this end, we propose a probabilistic approach based on the theory of forward–backward stochastic differential equations and introduce the new concept of “auxiliary SDEs.”

We study a nonlinear stochastic partial differential equation whose solution is the conditional log-Laplace functional of a superprocess in a random environment. We establish its existence and uniqueness by smoothing out the nonlinear term and making use of the particle system representation developed by Kurtz and Xiong [Stochastic Process. Appl. 83 (1999) 103–126]. We also derive the Wong–Zakai type approximation for this equation. As an application, we give a direct proof of the moment formulas of Skoulakis and Adler [Ann. Appl. Probab. 11 (2001) 488–543].

We prove that if $(X,{\mathfrak{A}},P)$ is an arbitrary probability space with countably generated σ-algebra ${\mathfrak{A}}$, $(Y,{\mathfrak{B}},Q)$ is an arbitrary complete probability space with a lifting ρ and $\widehat {R}$ is a complete probability measure on ${\mathfrak{A}}{\,\widehat{\otimes}_{R}\,}{\mathfrak{B}}$ determined by a regular conditional probability $\{S_y:y∈Y\}$ on ${\mathfrak{A}}$ with respect to ${\mathfrak{B}}$, then there exist a lifting π on $(X\times Y,{\mathfrak{A}}{\,\widehat{\otimes}_{R}\,}{\mathfrak{B}},\widehat {R})$ and liftings $σ_y$ on $(X,\widehat {\mathfrak{A}}_{y},\widehat {S}_{y})$, $y∈Y$, such that, for every $E\in{\mathfrak{A}}{\,\widehat{\otimes}_{R}\,}{\mathfrak{B}}$ and every $y∈Y$, $$[\pi(E)]^{y}=\sigma_{y}\bigl([\pi(E)]^{y}\bigr).$$ Assuming the absolute continuity of $R$ with respect to $P⊗Q$, we prove the existence of a regular conditional probability $\{T_y:y∈Y\}$ and liftings ϖ on $(X\times Y,{\mathfrak{A}}{\,\widehat{\otimes}_{R}\,}{\mathfrak{B}},\widehat {R})$, ρ' on $(Y,\mathfrak{B},\widehat {Q})$ and $σ_y$ on $(X,\widehat {\mathfrak{A}}_{y},\widehat {S}_{y})$, $y∈Y$, such that, for every $E\in{\mathfrak{A}}{\,\widehat{\otimes}_{R}\,}{\mathfrak{B}}$ and every $y∈Y$, $$[\varpi(E)]^{y}=\sigma_{y}\bigl([\varpi(E)]^{y}\bigr)$$ and $$\varpi(A\times B)=\bigcup_{y\in\rho'(B)}\sigma_{y}(A)\times\{y\}\qquad\mbox{if }A\times B\in{\mathfrak{A}}\times{\mathfrak{B}}.$$ Both results are generalizations of Musiał, Strauss and Macheras [Fund. Math. 166 (2000) 281–303] to the case of measures which are not necessarily products of marginal measures. We prove also that liftings obtained in this paper always convert $\widehat {R}$-measurable stochastic processes into their $\widehat {R}$-measurable modifications.

We first prove the L^p-convergence (p≥1) and a Fernique-type exponential integrability of divergence functionals for all Cameron–Martin vector fields with respect to the pinned Wiener measure on loop spaces over a compact Riemannian manifold. We then prove that the Driver flow is a smooth transform on path spaces in the sense of the Malliavin calculus and has an ∞-quasi-continuous modification which can be quasi-surely well defined on path spaces. This leads us to construct the Driver flow on loop spaces through the corresponding flow on path spaces. Combining these two results with the Cruzeiro lemma [J. Funct. Anal. 54 (1983) 206–227] we give an alternative proof of the quasi-invariance of the pinned Wiener measure under Driver’s flow on loop spaces which was established earlier by Driver [Trans. Amer. Math. Soc. 342 (1994) 375–394] and Enchev and Stroock [Adv. Math. 119 (1996) 127–154] by Doob’s h-processes approach together with the short time estimates of the gradient and the Hessian of the logarithmic heat kernel on compact Riemannian manifolds. We also establish the L^p-convergence (p≥1) and a Fernique-type exponential integrability theorem for the stochastic anti-development of pinned Brownian motions on compact Riemannian manifold with an explicit exponential exponent. Our results generalize and sharpen some earlier results due to Gross [J. Funct. Anal. 102 (1991) 268–313] and Hsu [Math. Ann. 309 (1997) 331–339]. Our method does not need any heat kernel estimate and is based on quasi-sure analysis and Sobolev estimates on path spaces.

Let W be the number of points in (0,t] of a stationary finite-state Markov renewal point process. We derive a bound for the total variation distance between the distribution of W and a compound Poisson distribution. For any nonnegative random variable ζ, we construct a “strong memoryless time” $\hat{\zeta}$ such that ζ−t is exponentially distributed conditional on $\{\hat{\zeta}\leq t,\,\zeta>t\}$, for each t. This is used to embed the Markov renewal point process into another such process whose state space contains a frequently observed state which represents loss of memory in the original process. We then write W as the accumulated reward of an embedded renewal reward process, and use a compound Poisson approximation error bound for this quantity by Erhardsson. For a renewal process, the bound depends in a simple way on the first two moments of the interrenewal time distribution, and on two constants obtained from the Radon–Nikodym derivative of the interrenewal time distribution with respect to an exponential distribution. For a Poisson process, the bound is 0.

This paper introduces a notion of 2-orthogonality for a sequence of polynomials to give extended versions of the Meixner and Feinsilver characterization results based on orthogonal polynomials. These new versions subsume the Letac–Mora characterization of the real natural exponential families having cubic variance function.

The motivation of this work is the study of the error term e_t^ɛ(x,ω) in the averaging method for differential equations perturbed by a dynamical system. Results of convergence in distribution for $({\frac{e_{t}^{\varepsilon}(x,\cdot)}{\sqrt{\varepsilon}}})_{\varepsilon>0}$ have been established in Khas’minskii [Theory Probab. Appl. 11 (1966) 211–228], Kifer [Ergodic Theory Dynamical Systems 15 (1995) 1143–1172] and Pène [ESAIM Probab. Statist. 6 (2002) 33–88]. We are interested here in the question of the rate of convergence in distribution of the family of random variables $({\frac{e_{t}^{\varepsilon}(x,\cdot)}{\sqrt{\varepsilon}}})_{\varepsilon>0}$ when ɛ goes to 0 (t>0 and $x\in{\bf R}^{d}$ being fixed). We will make an assumption of multiple decorrelation property (satisfied in several situations). We start by establishing a simpler result: the rate of convergence in the central limit theorem for regular multidimensional functions. In this context, we prove a result of convergence in distribution with rate of convergence in O(n^−1/2+α) for all α>0 (for the Prokhorov metric). This result can be seen as an extension of the main result of Pène [Comm. Math. Phys. 225 (2002) 91–119] to the case of d-dimensional functions. In a second time, we use the same method to establish a result of convergence in distribution for $({\frac{e_{t}^{\varepsilon}(x,\cdot)}{\sqrt{\varepsilon}}})_{\varepsilon>0}$ with rate of convergence in O(ɛ^1/2−α) (for the Prokhorov metric). We close this paper with a discussion (in the Appendix) about the behavior of the quantity $\Vert\sup_{0\le t\le T_{0}}\vert e_{t}^{\varepsilon}(x,\cdot)\vert_{\infty}\Vert_{L^{p}}$ under less stringent hypotheses.

Let qⁿ be a continuous density function in n-dimensional Euclidean space. We think of qⁿ as the density function of some random sequence Xⁿ with values in $\Bbb{R}^{n}$. For I⊂[1,n], let X_I denote the collection of coordinates X_i, i∈I, and let $\overline X_{I}$ denote the collection of coordinates X_i, i∉I. We denote by $Q_{I}(x_{I}|\bar{x}_{I})$ the joint conditional density function of X_I, given $\overline X_{I}$. We prove measure concentration for qⁿ in the case when, for an appropriate class of sets I, (i) the conditional densities $Q_{I}(x_{I}|\bar{x}_{I})$, as functions of x_I, uniformly satisfy a logarithmic Sobolev inequality and (ii) these conditional densities also satisfy a contractivity condition related to Dobrushin and Shlosman’s strong mixing condition.

The framework of Stein’s method for Poisson process approximation is presented from the point of view of Palm theory, which is used to construct Stein identities and define local dependence. A general result (Theorem 2.3) in Poisson process approximation is proved by taking the local approach. It is obtained without reference to any particular metric, thereby allowing wider applicability. A Wasserstein pseudometric is introduced for measuring the accuracy of point process approximation. The pseudometric provides a generalization of many metrics used so far, including the total variation distance for random variables and the Wasserstein metric for processes as in Barbour and Brown [Stochastic Process. Appl. 43 (1992) 9–31]. Also, through the pseudometric, approximation for certain point processes on a given carrier space is carried out by lifting it to one on a larger space, extending an idea of Arratia, Goldstein and Gordon [Statist. Sci. 5 (1990) 403–434]. The error bound in the general result is similar in form to that for Poisson approximation. As it yields the Stein factor 1/λ as in Poisson approximation, it provides good approximation, particularly in cases where λ is large. The general result is applied to a number of problems including Poisson process modeling of rare words in a DNA sequence.

Let f_n denote a kernel density estimator of a continuous density f in d dimensions, bounded and positive. Let Ψ(t) be a positive continuous function such that ‖Ψf^β‖_∞<∞ for some 0<β<1/2. Under natural smoothness conditions, necessary and sufficient conditions for the sequence ${\sqrt{\frac{nh_{n}^{d}}{2|\log h_{n}^{d}|}}\|\Psi(t)(f_{n}(t)-Ef_{n}(t))\|_{\infty}}$ to be stochastically bounded and to converge a.s. to a constant are obtained. Also, the case of larger values of β is studied where a similar sequence with a different norming converges a.s. either to 0 or to +∞, depending on convergence or divergence of a certain integral involving the tail probabilities of Ψ(X). The results apply as well to some discontinuous not strictly positive densities.

We prove that the Hausdorff dimension of the trace of SLE₆ is almost surely 7/4 and give a more direct derivation of the result (due to Lawler–Schramm–Werner) that the dimension of its boundary is 4/3. We also prove that, for all κ<8, the SLE_κ trace has cut-points.

Consider a Markov chain on an infinite tree T=(V,E) rooted at ρ. In such a chain, once the initial root state σ({ρ}) is chosen, each vertex iteratively chooses its state from the one of its parent by an application of a Markov transition rule (and all such applications are independent). Let μ_j denote the resulting measure for σ({ρ})=j. The resulting measure μ_j is defined on configurations $\sigma=(\sigma(x))_{x\in V}\in \mathcal {A}^{V}$, where $\mathcal {A}$ is some finite set. Let μ_jⁿ denote the restriction of μ to the sigma-algebra generated by the variables σ(x), where x is at distance exactly n from ρ. Letting $\alpha_{n}=\max_{i,j\in \mathcal {A}}d_{\mathrm{TV}}(\mu_{i}^{n},\mu_{j}^{n})$, where d_TV denotes total variation distance, we say that the reconstruction problem is solvable if lim inf _n→∞α_n>0. Reconstruction solvability roughly means that the nth level of the tree contains a nonvanishing amount of information on the root of the tree as n→∞.

In this paper we study the problem of robust reconstruction. Let ν be a nondegenerate distribution on $\mathcal {A}$" and ɛ>0. Let σ be chosen according to μ_jⁿ and σ' be obtained from σ by letting for each node independently, σ(v)=σ'(v) with probability 1−ɛ and σ'(v) be an independent sample from ν otherwise. We denote by μ_jⁿ[ν,ɛ] the resulting measure on σ'. The measure μ_jⁿ[ν,ɛ] is a perturbation of the measure μ_jⁿ. Letting $\alpha_{n}(\nu,\varepsilon )=\max_{i,j\in \mathcal {A}}d_{\mathrm{TV}}(\mu_{i}^{n}[\nu,\varepsilon ],\mu_{j}^{n}[\nu,\varepsilon ])$, we say that the reconstruction problem is ν-robust-solvable if lim inf _n→∞α_n(ν,ɛ)>0 for all 0<ɛ<1. Roughly speaking, the reconstruction problem is robust-solvable if for any noise-rate and for all n, the nth level of the tree contains a nonvanishing amount of information on the root of the tree.

Standard techniques imply that if T is the rooted B-ary tree (where each node has B children) and if B|λ₂(M)|²>1, where λ₂(M) is the second largest eigenvalue of M (in absolute value), then for all nondegenerate ν, the reconstruction problem is ν-robust-solvable. We prove a converse and show that the reconstruction problem is not ν-robust-solvable if B|λ₂(M)|²<1. This proves a conjecture by the second author and Y. Peres. We also consider other models of noise and general trees.

Vertex-reinforced random walk (VRRW), defined by Pemantle in 1988, is a random process that takes values in the vertex set of a graph G, which is more likely to visit vertices it has visited before. Pemantle and Volkov considered the case when the underlying graph is the one-dimensional integer lattice ℤ. They proved that the range is almost surely finite and that with positive probability the range contains exactly five points. They conjectured that this second event holds with probability 1. The proof of this conjecture is the main purpose of this paper.

We first give a characterization of the L¹-transportation cost-information inequality on a metric space and next find some appropriate sufficient condition to transportation cost-information inequalities for dependent sequences. Applications to random dynamical systems and diffusions are studied.

We prove that every two-player nonzero–sum stopping game in discrete time admits an ɛ-equilibrium in randomized strategies for every ɛ>0. We use a stochastic variation of Ramsey’s theorem, which enables us to reduce the problem to that of studying properties of ɛ-equilibria in a simple class of stochastic games with finite state space.

In the standard formulation of the occupancy problem one considers the distribution of r balls in n cells, with each ball assigned independently to a given cell with probability 1/n. Although closed form expressions can be given for the distribution of various interesting quantities (such as the fraction of cells that contain a given number of balls), these expressions are often of limited practical use. Approximations provide an attractive alternative, and in the present paper we consider a large deviation approximation as r and n tend to infinity. In order to analyze the problem we first consider a dynamical model, where the balls are placed in the cells sequentially and “time” corresponds to the number of balls that have already been thrown. A complete large deviation analysis of this “process level” problem is carried out, and the rate function for the original problem is then obtained via the contraction principle. The variational problem that characterizes this rate function is analyzed, and a fairly complete and explicit solution is obtained. The minimizing trajectories and minimal cost are identified up to two constants, and the constants are characterized as the unique solution to an elementary fixed point problem. These results are then used to solve a number of interesting problems, including an overflow problem and the partial coupon collector’s problem.

We consider the set M_n of all n-truncated power moment sequences of probability measures on [0,1]. We endow this set with the uniform probability. Picking randomly a point in M_n, we show that the upper canonical measure associated with this point satisfies a large deviation principle. Moderate deviation are also studied completing earlier results on asymptotic normality given by Chang, Kemperman and Studden [Ann. Probab. 21 (1993) 1295–1309]. Surprisingly, our large deviations results allow us to compute explicitly the (n+1)th moment range size of the set of all probability measures having the same n first moments. The main tool to obtain these results is the representation of M_n on canonical moments [see the book of Dette and Studden].

A class of random recursive sequences (Y_n) with slowly varying variances as arising for parameters of random trees or recursive algorithms leads after normalizations to degenerate limit equations of the form $X\stackrel {\mathcal{L}}{=}X$. For nondegenerate limit equations the contraction method is a main tool to establish convergence of the scaled sequence to the “unique” solution of the limit equation. In this paper we develop an extension of the contraction method which allows us to derive limit theorems for parameters of algorithms and data structures with degenerate limit equation. In particular, we establish some new tools and a general convergence scheme, which transfers information on mean and variance into a central limit law (with normal limit). We also obtain a convergence rate result. For the proof we use selfdecomposability properties of the limit normal distribution which allow us to mimic the recursive sequence by an accompanying sequence in normal variables.