The Annals of Probability Articles (Project Euclid)

Uniform convergence of Vapnik–Chervonenkis classes under ergodic sampling

Thu, 05 Aug 2010 15:41 EDT

Terrence M. Adams, Andrew B. Nobel

Source: Ann. Probab., Volume 38, Number 4, 1345--1367.

Abstract:
We show that if $\mathcal{X}$ is a complete separable metric space and $\mathcal{C}$ is a countable family of Borel subsets of $\mathcal{X}$ with finite VC dimension, then, for every stationary ergodic process with values in $\mathcal{X}$ , the relative frequencies of sets $C\in\mathcal{C}$ converge uniformly to their limiting probabilities. Beyond ergodicity, no assumptions are imposed on the sampling process, and no regularity conditions are imposed on the elements of $\mathcal{C}$ . The result extends existing work of Vapnik and Chervonenkis, among others, who have studied uniform convergence for i.i.d. and strongly mixing processes. Our method of proof is new and direct: it does not rely on symmetrization techniques, probability inequalities or mixing conditions. The uniform convergence of relative frequencies for VC-major and VC-graph classes of functions under ergodic sampling is established as a corollary of the basic result for sets.

Painting a graph with competing random walks

Fri, 08 Mar 2013 08:56 EST

Jason Miller

Source: Ann. Probab., Volume 41, Number 2, 636--670.

Abstract:
Let $X_{1}$, $X_{2}$ be independent random walks on $\mathbf{Z} _{n}^{d}$, $d\geq3$, each starting from the uniform distribution. Initially, each site of $\mathbf{Z} _{n}^{d}$ is unmarked, and, whenever $X_{i}$ visits such a site, it is set irreversibly to $i$. The mean of $|\mathcal{A} _{i}|$, the cardinality of the set $\mathcal{A} _{i}$ of sites painted by $i$, once all of $\mathbf{Z} _{n}^{d}$ has been visited, is $\frac{1}{2}n^{d}$ by symmetry. We prove the following conjecture due to Pemantle and Peres: for each $d\geq3$ there exists a constant $\alpha_{d}$ such that $\lim_{n\to\infty}\operatorname{Var} (|\mathcal{A} _{i}|)/h_{d}(n)=\frac{1}{4}\alpha_{d}$ where $h_{3}(n)=n^{4}$, $h_{4}(n)=n^{4}(\log n)$ and $h_{d}(n)=n^{d}$ for $d\geq5$. We will also identify $\alpha_{d}$ explicitly and show that $\alpha_{d}\to1$ as $d\to\infty$. This is a special case of a more general theorem which gives the asymptotics of $\operatorname{Var} (|\mathcal{A} _{i}|)$ for a large class of transient, vertex transitive graphs; other examples include the hypercube and the Caley graph of the symmetric group generated by transpositions.

A sufficient condition for the continuity of permanental processes with applications to local times of Markov processes

Fri, 08 Mar 2013 08:56 EST

Michael B. Marcus, Jay Rosen

Source: Ann. Probab., Volume 41, Number 2, 671--698.

Abstract:
We provide a sufficient condition for the continuity of real valued permanental processes. When applied to the subclass of permanental processes which consists of squares of Gaussian processes, we obtain the sufficient condition for continuity which is also known to be necessary. Using an isomorphism theorem of Eisenbaum and Kaspi which relates Markov local times and permanental processes, we obtain a general sufficient condition for the joint continuity of local times.

Asymptotic support theorem for planar isotropic Brownian flows

Fri, 08 Mar 2013 08:56 EST

Moritz Biskamp

Source: Ann. Probab., Volume 41, Number 2, 699--721.

Abstract:
It has been shown by various authors that the diameter of a given nontrivial bounded connected set $\mathcal{X}$ grows linearly in time under the action of an isotropic Brownian flow (IBF), which has a nonnegative top-Lyapunov exponent. In case of a planar IBF with a positive top-Lyapunov exponent, the precise deterministic linear growth rate $K$ of the diameter is known to exist. In this paper we will extend this result to an asymptotic support theorem for the time-scaled trajectories of a planar IBF $\varphi$, which has a positive top-Lyapunov exponent, starting in a nontrivial compact connected set $\mathcal{X}\subseteq\mathbf{R}^{2}$; that is, we will show convergence in probability of the set of time-scaled trajectories in the Hausdorff distance to the set of Lipschitz continuous functions on $[0,1]$ starting in $0$ with Lipschitz constant $K$.

Random Dirichlet environment viewed from the particle in dimension $d\ge3$

Fri, 08 Mar 2013 08:56 EST

Christophe Sabot

Source: Ann. Probab., Volume 41, Number 2, 722--743.

Abstract:
We consider random walks in random Dirichlet environment (RWDE), which is a special type of random walks in random environment where the exit probabilities at each site are i.i.d. Dirichlet random variables. On ${\mathbb{Z}}^{d}$, RWDE are parameterized by a $2d$-tuple of positive reals called weights. In this paper, we characterize for $d\ge3$ the weights for which there exists an absolutely continuous invariant probability distribution for the process viewed from the particle. We can deduce from this result and from [ Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011) 1–8] a complete description of the ballistic regime for $d\ge3$.

Shy couplings, $\operatorname{CAT} ({0})$ spaces, and the lion and man

Fri, 08 Mar 2013 08:56 EST

Maury Bramson, Krzysztof Burdzy, Wilfrid Kendall

Source: Ann. Probab., Volume 41, Number 2, 744--784.

Abstract:
Two random processes $X$ and $Y$ on a metric space are said to be $\varepsilon$-shy coupled if there is positive probability of them staying at least a positive distance $\varepsilon$ apart from each other forever. Interest in the literature centres on nonexistence results subject to topological and geometric conditions; motivation arises from the desire to gain a better understanding of probabilistic coupling. Previous nonexistence results for co-adapted shy coupling of reflected Brownian motion required convexity conditions; we remove these conditions by showing the nonexistence of shy co-adapted couplings of reflecting Brownian motion in any bounded $\operatorname{CAT} ({0})$ domain with boundary satisfying uniform exterior sphere and interior cone conditions, for example, simply-connected bounded planar domains with $C^{2}$ boundary. The proof uses a Cameron–Martin–Girsanov argument, together with a continuity property of the Skorokhod transformation and properties of the intrinsic metric of the domain. To this end, a generalization of Gauss’ lemma is established that shows differentiability of the intrinsic distance function for closures of $\operatorname{CAT} ({0})$ domains with boundaries satisfying uniform exterior sphere and interior cone conditions. By this means, the shy coupling question is converted into a Lion and Man pursuit–evasion problem.

A CLT for empirical processes involving time-dependent data

Fri, 08 Mar 2013 08:56 EST

James Kuelbs, Thomas Kurtz, Joel Zinn

Source: Ann. Probab., Volume 41, Number 2, 785--816.

Abstract:
For stochastic processes $\{X_{t} : t\in E\}$, we establish sufficient conditions for the empirical process based on $\{I_{X_{t}\le y}-\operatorname{Pr} (X_{t}\le y) : t\in E,y\in\mathbb{R}\}$ to satisfy the CLT uniformly in $t\in E$, $y\in\mathbb{R}$. Corollaries of our main result include examples of classical processes where the CLT holds, and we also show that it fails for Brownian motion tied down at zero and $E=[0,1]$.

Convergence of clock processes in random environments and ageing in the $p$-spin SK model

Fri, 08 Mar 2013 08:56 EST

Anton Bovier, Véronique Gayrard

Source: Ann. Probab., Volume 41, Number 2, 817--847.

Abstract:
We derive a general criterion for the convergence of clock processes in random dynamics in random environments that is applicable in cases when correlations are not negligible, extending recent results by Gayrard [(2010), (2011), forthcoming], based on general criterion for convergence of sums of dependent random variables due to Durrett and Resnick [ Ann. Probab. 6 (1978) 829–846]. We demonstrate the power of this criterion by applying it to the case of random hopping time dynamics of the $p$-spin SK model. We prove that on a wide range of time scales, the clock process converges to a stable subordinator almost surely with respect to the environment. We also show that a time-time correlation function converges to the arcsine law for this subordinator, almost surely. This improves recent results of Ben Arous, Bovier and Černý [ Comm. Math. Phys. 282 (2008) 663–695] that obtained similar convergence results in law, with respect to the random environment.

Nonconcentration of return times

Fri, 08 Mar 2013 08:56 EST

Ori Gurel-Gurevich, Asaf Nachmias

Source: Ann. Probab., Volume 41, Number 2, 848--870.

Abstract:
We show that the distribution of the first return time $\tau$ to the origin, $v$, of a simple random walk on an infinite recurrent graph is heavy tailed and nonconcentrated. More precisely, if $d_{v}$ is the degree of $v$, then for any $t\geq1$ we have \[\mathbf{P} _{v}(\tau\ge t)\ge\frac{c}{d_{v}\sqrt{t}}\] and \[\mathbf{P} _{v}(\tau=t\mid\tau\geq t)\leq\frac{C\log(d_{v}t)}{t}\] for some universal constants $c>0$ and $C<\infty$. The first bound is attained for all $t$ when the underlying graph is $\mathbb{Z}$, and as for the second bound, we construct an example of a recurrent graph $G$ for which it is attained for infinitely many $t$’s. Furthermore, we show that in the comb product of that graph $G$ with $\mathbb{Z}$, two independent random walks collide infinitely many times almost surely. This answers negatively a question of Krishnapur and Peres [ Electron. Commun. Probab. 9 (2004) 72–81] who asked whether every comb product of two infinite recurrent graphs has the finite collision property.

Mixing of the symmetric exclusion processes in terms of the corresponding single-particle random walk

Fri, 08 Mar 2013 08:56 EST

Roberto Imbuzeiro Oliveira

Source: Ann. Probab., Volume 41, Number 2, 871--913.

Abstract:
We prove an upper bound for the $\varepsilon$-mixing time of the symmetric exclusion process on any graph $G$, with any feasible number of particles. Our estimate is proportional to ${\mathsf{T}}_{\mathsf{RW}(G)}\ln(|V|/\varepsilon )$, where $|V|$ is the number of vertices in $G$, and ${\mathsf{T}}_{\mathsf{RW}(G)}$ is the $1/4$-mixing time of the corresponding single-particle random walk. This bound implies new results for symmetric exclusion on expanders, percolation clusters, the giant component of the Erdös–Rényi random graph and Poisson point processes in $\mathbb{R}^{d}$. Our technical tools include a variant of Morris’s chameleon process.

Rosenthal-type inequalities for the maximum of partial sums of stationary processes and examples

Fri, 08 Mar 2013 08:56 EST

Florence Merlevède, Magda Peligrad

Source: Ann. Probab., Volume 41, Number 2, 914--960.

Abstract:
The aim of this paper is to propose new Rosenthal-type inequalities for moments of order higher than $2$ of the maximum of partial sums of stationary sequences including martingales and their generalizations. As in the recent results by Peligrad et al. [ Proc. Amer. Math. Soc. 135 (2007) 541–550] and Rio [ J. Theoret. Probab. 22 (2009) 146–163], the estimates of the moments are expressed in terms of the norms of projections of partial sums. The proofs of the results are essentially based on a new maximal inequality generalizing the Doob maximal inequality for martingales and dyadic induction. Various applications are also provided.

Free subexponentiality

Fri, 08 Mar 2013 08:56 EST

Rajat Subhra Hazra, Krishanu Maulik

Source: Ann. Probab., Volume 41, Number 2, 961--988.

Abstract:
In this article, we introduce the notion of free subexponentiality , which extends the notion of subexponentiality in the classical probability setup to the noncommutative probability spaces under freeness. We show that distributions with regularly varying tails belong to the class of free subexponential distributions. This also shows that the partial sums of free random elements having distributions with regularly varying tails are tail equivalent to their maximum in the sense of Ben Arous and Voiculescu [ Ann. Probab. 34 (2006) 2037–2059]. The analysis is based on the asymptotic relationship between the tail of the distribution and the real and the imaginary parts of the remainder terms in Laurent series expansion of Cauchy transform, as well as the relationship between the remainder terms in Laurent series expansions of Cauchy and Voiculescu transforms, when the distribution has regularly varying tails.

A super Ornstein–Uhlenbeck process interacting with its center of mass

Fri, 08 Mar 2013 08:56 EST

Hardeep Gill

Source: Ann. Probab., Volume 41, Number 2, 989--1029.

Abstract:
We construct a supercritical interacting measure-valued diffusion with representative particles that are attracted to, or repelled from, the center of mass. Using the historical stochastic calculus of Perkins, we modify a super Ornstein–Uhlenbeck process with attraction to its origin, and prove continuum analogues of results of Engländer [ Electron. J. Probab. 15 (2010) 1938–1970] for binary branching Brownian motion. It is shown, on the survival set, that in the attractive case the mass normalized interacting measure-valued process converges almost surely to the stationary distribution of the Ornstein–Uhlenbeck process, centered at the limiting value of its center of mass. In the same setting, it is proven that the normalized super Ornstein–Uhlenbeck process converges a.s. to a Gaussian random variable, which strengthens a theorem of Engländer and Winter [ Ann. Inst. Henri Poincaré Probab. Stat. 42 (2006) 171–185] in this particular case. In the repelling setting, we show that the center of mass converges a.s., provided the repulsion is not too strong and then give a conjecture. This contrasts with the center of mass of an ordinary super Ornstein–Uhlenbeck process with repulsion, which is shown to diverge a.s. A version of a result of Tribe [ Ann. Probab. 20 (1992) 286–311] is proven on the extinction set; that is, as it approaches the extinction time, the normalized process in both the attractive and repelling cases converges to a random point a.s.

Super-Brownian motion as the unique strong solution to an SPDE

Fri, 08 Mar 2013 08:56 EST

Jie Xiong

Source: Ann. Probab., Volume 41, Number 2, 1030--1054.

Abstract:
A stochastic partial differential equation (SPDE) is derived for super-Brownian motion regarded as a distribution function valued process. The strong uniqueness for the solution to this SPDE is obtained by an extended Yamada–Watanabe argument. Similar results are also proved for the Fleming–Viot process.

The complete characterization of a.s. convergence of orthogonal series

Fri, 08 Mar 2013 08:56 EST

Witold Bednorz

Source: Ann. Probab., Volume 41, Number 2, 1055--1071.

Abstract:
In this paper we prove the complete characterization of a.s. convergence of orthogonal series in terms of existence of a majorizing measure. It means that for a given $(a_{n})^{\infty}_{n=1}$, $a_{n}>0$, series $\sum^{\infty}_{n=1}a_{n}\varphi_{n}$ is a.e. convergent for each orthonormal sequence $(\varphi_{n})^{\infty}_{n=1}$ if and only if there exists a measure $m$ on \[T=\{0\}\cup\Biggl\{\sum^{m}_{n=1}a_{n}^{2},m\geq 1\Biggr\}\] such that \[\sup_{t\in T}\int^{\sqrt{D(T)}}_{0}(m(B(t,r^{2})))^{-{1}/{2}}\,dr<\infty,\] where $D(T)=\sup_{s,t\in T}|s-t|$ and $B(t,r)=\{s\in T\dvtx |s-t|\leq r\}$. The presented approach is based on weakly majorizing measures and a certain partitioning scheme.

Sub-Gaussian tail bounds for the width and height of conditioned Galton–Watson trees

Fri, 08 Mar 2013 08:56 EST

Louigi Addario-Berry, Luc Devroye, Svante Janson

Source: Ann. Probab., Volume 41, Number 2, 1072--1087.

Abstract:
We study the height and width of a Galton–Watson tree with offspring distribution $\xi$ satisfying $\mathbb{E} \xi=1$, $0<\operatorname{Var} \xi<\infty$, conditioned on having exactly $n$ nodes. Under this conditioning, we derive sub-Gaussian tail bounds for both the width (largest number of nodes in any level) and height (greatest level containing a node); the bounds are optimal up to constant factors in the exponent. Under the same conditioning, we also derive essentially optimal upper tail bounds for the number of nodes at level $k$, for $1\leq k\leq n$.

Limit theorems for weighted nonlinear transformations of Gaussian stationary processes with singular spectra

Fri, 08 Mar 2013 08:56 EST

Alexander V. Ivanov, Nikolai Leonenko, María D. Ruiz-Medina, Irina N. Savich

Source: Ann. Probab., Volume 41, Number 2, 1088--1114.

Abstract:
The limit Gaussian distribution of multivariate weighted functionals of nonlinear transformations of Gaussian stationary processes, having multiple singular spectra, is derived, under very general conditions on the weight function. This paper is motivated by its potential applications in nonlinear regression, and asymptotic inference on nonlinear functionals of Gaussian stationary processes with singular spectra.

From logarithmic to subdiffusive polynomial fluctuations for internal DLA and related growth models

Mon, 29 Apr 2013 09:19 EDT

Amine Asselah, Alexandre Gaudillière

Source: Ann. Probab., Volume 41, Number 3A, 1115--1159.

Abstract:
We consider a cluster growth model on ${\mathbb{Z}}^{d}$, called internal diffusion limited aggregation (internal DLA). In this model, random walks start at the origin, one at a time, and stop moving when reaching a site not occupied by previous walks. It is known that the asymptotic shape of the cluster is spherical. When dimension is 2 or more, we prove that fluctuations with respect to a sphere are at most a power of the logarithm of its radius in dimension $d\ge2$. In so doing, we introduce a closely related cluster growth model, that we call the flashing process , whose fluctuations are controlled easily and accurately. This process is coupled to internal DLA to yield the desired bound. Part of our proof adapts the approach of Lawler, Bramson and Griffeath, on another space scale, and uses a sharp estimate (written by Blachère in our Appendix) on the expected time spent by a random walk inside an annulus.

Sublogarithmic fluctuations for internal DLA

Mon, 29 Apr 2013 09:19 EDT

Amine Asselah, Alexandre Gaudillière

Source: Ann. Probab., Volume 41, Number 3A, 1160--1179.

Abstract:
We consider internal diffusion limited aggregation in dimension larger than or equal to two. This is a random cluster growth model, where random walks start at the origin of the $d$-dimensional lattice, one at a time, and stop moving when reaching a site that is not occupied by previous walks. It is known that the asymptotic shape of the cluster is a sphere. When the dimension is two or more, we have shown in a previous paper that the inner (resp., outer) fluctuations of its radius is at most of order $\log(\mathrm{radius})$ [resp., $\log^{2}(\mathrm{radius})$]. Using the same approach, we improve the upper bound on the inner fluctuation to $\sqrt{\log(\mathrm{radius})}$ when $d$ is larger than or equal to three. The inner fluctuation is then used to obtain a similar upper bound on the outer fluctuation.

Invariant monotone coupling need not exist

Mon, 29 Apr 2013 09:19 EDT

Péter Mester

Source: Ann. Probab., Volume 41, Number 3A, 1180--1190.

Abstract:
We show by example that there is a Cayley graph, having two invariant random subgraphs $X$ and $Y$, such that there exists a monotone coupling between them in the sense that $X\subset Y$, although no such coupling can be invariant. Here, “invariant” means that the distribution is invariant under group multiplications.

On the law of the supremum of Lévy processes

Mon, 29 Apr 2013 09:19 EDT

L. Chaumont

Source: Ann. Probab., Volume 41, Number 3A, 1191--1217.

Abstract:
We show that the law of the overall supremum $\overline{X}_{t}=\sup_{s\let}X_{s}$ of a Lévy process $X$, before the deterministic time $t$ is equivalent to the average occupation measure $\mu_{t}^{+}(dx)=\int_{0}^{t}\mathbb{P} (X_{s}\in dx)\,ds$, whenever 0 is regular for both open halflines $(-\infty,0)$ and $(0,\infty)$. In this case, $\mathbb{P} (\overline{X}_{t}\in dx)$ is absolutely continuous for some (and hence for all) $t>0$ if and only if the resolvent measure of $X$ is absolutely continuous. We also study the cases where 0 is not regular for both halflines. Then we give absolute continuity criterions for the laws of $(g_{t},\overline{X}_{t})$ and $(g_{t},\overline{X}_{t},X_{t})$, where $g_{t}$ is the time at which the supremum occurs before $t$. The proofs of these results use an expression of the joint law $\mathbb{P} (g_{t}\in ds,X_{t}\in dx,\overline{X}_{t}\in dy)$ in terms of the entrance law of the excursion measure of the reflected process at the supremum and that of the reflected process at the infimum. As an application, this law is made (partly) explicit in some particular instances.

Sharp metastability threshold for an anisotropic bootstrap percolation model

Mon, 29 Apr 2013 09:19 EDT

H. Duminil-Copin, A. C. D. Van Enter

Source: Ann. Probab., Volume 41, Number 3A, 1218--1242.

Abstract:
Bootstrap percolation models have been extensively studied during the two past decades. In this article, we study the following “anisotropic” bootstrap percolation model: the neighborhood of a point $(m,n)$ is the set \[\{(m+2,n),(m+1,n),(m,n+1),(m-1,n),(m-2,n),(m,n-1)\}.\] At time 0, sites are occupied with probability $p$. At each time step, sites that are occupied remain occupied, while sites that are not occupied become occupied if and only if three of more sites in their neighborhood are occupied. We prove that it exhibits a sharp metastability threshold. This is the first mathematical proof of a sharp threshold for an anisotropic bootstrap percolation model.

Crossover distributions at the edge of the rarefaction fan

Mon, 29 Apr 2013 09:19 EDT

Ivan Corwin, Jeremy Quastel

Source: Ann. Probab., Volume 41, Number 3A, 1243--1314.

Abstract:
We consider the weakly asymmetric limit of simple exclusion process with drift to the left, starting from step Bernoulli initial data with $\rho_{-}<\rho_{+}$ so that macroscopically one has a rarefaction fan. We study the fluctuations of the process observed along slopes in the fan, which are given by the Hopf–Cole solution of the Kardar–Parisi–Zhang (KPZ) equation, with appropriate initial data. For slopes strictly inside the fan, the initial data is a Dirac delta function and the one point distribution functions have been computed in [ Comm. Pure Appl. Math. 64 (2011) 466–537] and [ Nuclear Phys. B 834 (2010) 523–542]. At the edge of the rarefaction fan, the initial data is one-sided Brownian . We obtain a new family of crossover distributions giving the exact one-point distributions of this process, which converge, as $T\nearrow\infty$ to those of the Airy $\mathcal{A}_{2\to \mathrm{BM}}$ process. As an application, we prove moment and large deviation estimates for the equilibrium Hopf–Cole solution of KPZ. These bounds rely on the apparently new observation that the FKG inequality holds for the stochastic heat equation. Finally, via a Feynman–Kac path integral, the KPZ equation also governs the free energy of the continuum directed polymer, and thus our formula may also be interpreted in those terms.

Spin glass models from the point of view of spin distributions

Mon, 29 Apr 2013 09:19 EDT

Dmitry Panchenko

Source: Ann. Probab., Volume 41, Number 3A, 1315--1361.

Abstract:
In many spin glass models, due to the symmetry among sites, any limiting joint distribution of spins under the annealed Gibbs measure admits the Aldous–Hoover representation encoded by a function $\sigma : [0,1]^{4}\to\{-1,+1\}$, and one can think of this function as a generic functional order parameter of the model. In a class of diluted models, and in the Sherrington–Kirkpatrick model, we introduce novel perturbations of the Hamiltonian that yield certain invariance and self-consistency equations for this generic functional order parameter and we use these invariance properties to obtain representations for the free energy in terms of $\sigma$. In the setting of the Sherrington–Kirkpatrick model, the self-consistency equations imply that the joint distribution of spins is determined by the joint distributions of the overlaps, and we give an explicit formula for $\sigma$ under the Parisi ultrametricity hypothesis. In addition, we discuss some connections with the Ghirlanda–Guerra identities and stochastic stability and describe the expected Parisi ansatz in the diluted models in terms of $\sigma$.

Convergence in law of the minimum of a branching random walk

Mon, 29 Apr 2013 09:19 EDT

Elie Aïdékon

Source: Ann. Probab., Volume 41, Number 3A, 1362--1426.

Abstract:
We consider the minimum of a super-critical branching random walk. Addario-Berry and Reed [ Ann. Probab. 37 (2009) 1044–1079] proved the tightness of the minimum centered around its mean value. We show that a convergence in law holds, giving the analog of a well-known result of Bramson [ Mem. Amer. Math. Soc. 44 (1983) iv+190] in the case of the branching Brownian motion.

Convergence to the equilibria for self-stabilizing processes in double-well landscape

Mon, 29 Apr 2013 09:19 EDT

Julian Tugaut

Source: Ann. Probab., Volume 41, Number 3A, 1427--1460.

Abstract:
We investigate the convergence of McKean–Vlasov diffusions in a nonconvex landscape. These processes are linked to nonlinear partial differential equations. According to our previous results, there are at least three stationary measures under simple assumptions. Hence, the convergence problem is not classical like in the convex case. By using the method in Benedetto et al. [ J. Statist. Phys. 91 (1998) 1261–1271] about the monotonicity of the free-energy, and combining this with a complete description of the set of the stationary measures, we prove the global convergence of the self-stabilizing processes.

Large deviations for the current and tagged particle in 1D nearest-neighbor symmetric simple exclusion

Mon, 29 Apr 2013 09:19 EDT

Sunder Sethuraman, S. R. S. Varadhan

Source: Ann. Probab., Volume 41, Number 3A, 1461--1512.

Abstract:
Laws of large numbers, starting from certain nonequilibrium measures, have been shown for the integrated current across a bond, and a tagged particle in one-dimensional symmetric nearest-neighbor simple exclusion [ Ann. Inst. Henri Poincaré Probab. Stat. 42 (2006) 567–577]. In this article, we prove corresponding large deviation principles and evaluate the rate functions, showing different growth behaviors near and far from their zeroes which connect with results in [ J. Stat. Phys. 136 (2009) 1–15].

Multi-point Green’s functions for SLE and an estimate of Beffara

Mon, 29 Apr 2013 09:19 EDT

Gregory F. Lawler, Brent M. Werness

Source: Ann. Probab., Volume 41, Number 3A, 1513--1555.

Abstract:
In this paper we define and prove of the existence of the multi-point Green’s function for $\mbox{SLE}$—a normalized limit of the probability that an $\mbox{SLE}_{\kappa}$ curve passes near to a pair of marked points in the interior of a domain. When $\kappa<8$ this probability is nontrivial, and an expression can be written in terms two-sided radial $\mbox{SLE}$. One of the main components to our proof is a refinement of a bound first provided by Beffara [ Ann. Probab. 36 (2008) 1421–1452]. This work contains a proof of this bound independent from the original.

SLE curves and natural parametrization

Mon, 29 Apr 2013 09:19 EDT

Gregory F. Lawler, Wang Zhou

Source: Ann. Probab., Volume 41, Number 3A, 1556--1584.

Abstract:
Developing the theory of two-sided radial and chordal $\mathit{SLE}$, we prove that the natural parametrization on $\mathit{SLE}_{\kappa}$ curves is well defined for all $\kappa<8$. Our proof uses a two-interior-point local martingale.

A Lamperti-type representation of continuous-state branching processes with immigration

Mon, 29 Apr 2013 09:19 EDT

M. Emilia Caballero, José Luis Pérez Garmendia, Gerónimo Uribe Bravo

Source: Ann. Probab., Volume 41, Number 3A, 1585--1627.

Abstract:
Guided by the relationship between the breadth-first walk of a rooted tree and its sequence of generation sizes, we are able to include immigration in the Lamperti representation of continuous-state branching processes. We provide a representation of continuous-state branching processes with immigration by solving a random ordinary differential equation driven by a pair of independent Lévy processes. Stability of the solutions is studied and gives, in particular, limit theorems (of a type previously studied by Grimvall, Kawazu and Watanabe and by Li) and a simulation scheme for continuous-state branching processes with immigration. We further apply our stability analysis to extend Pitman’s limit theorem concerning Galton–Watson processes conditioned on total population size to more general offspring laws.

Distance between two skew Brownian motions as a S.D.E. with jumps and law of the hitting time

Mon, 29 Apr 2013 09:19 EDT

Arnaud Gloter, Miguel Martinez

Source: Ann. Probab., Volume 41, Number 3A, 1628--1655.

Abstract:
In this paper, we consider two skew Brownian motions, driven by the same Brownian motion, with different starting points and different skewness coefficients. We show that we can describe the evolution of the distance between the two processes with a stochastic differential equation. This S.D.E. possesses a jump component driven by the excursion process of one of the two skew Brownian motions. Using this representation, we show that the local time of two skew Brownian motions at their first hitting time is distributed as a simple function of a Beta random variable. This extends a result by Burdzy and Chen [ Ann. Probab. 29 (2001) 1693–1715], where the law of coalescence of two skew Brownian motions with the same skewness coefficient is computed.

On Stratonovich and Skorohod stochastic calculus for Gaussian processes

Mon, 29 Apr 2013 09:19 EDT

Yaozhong Hu, Maria Jolis, Samy Tindel

Source: Ann. Probab., Volume 41, Number 3A, 1656--1693.

Abstract:
In this article, we derive a Stratonovich and Skorohod-type change of variables formula for a multidimensional Gaussian process with low Hölder regularity $\gamma$ (typically $\gamma \le1/4$). To this aim, we combine tools from rough paths theory and stochastic analysis.

Stable limit laws for randomly biased walks on supercritical trees

Mon, 29 Apr 2013 09:19 EDT

Alan Hammond

Source: Ann. Probab., Volume 41, Number 3A, 1694--1766.

Abstract:
We consider a random walk on a supercritical Galton–Watson tree with leaves, where the transition probabilities of the walk are determined by biases that are randomly assigned to the edges of the tree. The biases are chosen independently on distinct edges, each one according to a given law that satisfies a logarithmic nonlattice condition. We determine the condition under which the walk is sub-ballistic, and, in the sub-ballistic regime, we find a formula for the exponent $\gamma \in(0,1)$ such that the distance $\vert X(n)\vert$ moved by the walk in time $n$ is of the order of $n^{\gamma }$. We prove a stable limiting law for walker distance at late time, proving that the rescaled walk $n^{-\gamma }\vert X(n)\vert$ converges in distribution to an explicitly identified function of the stable law of index $\gamma $. This paper is a counterpart to Ben Arous et al. [ Ann. Probab. 40 (2012) 280–338], in which it is proved that, in the model where the biases on edges are taken to be a given constant, there is a logarithmic periodicity effect that prevents the existence of a stable limit law for scaled walker displacement. It is randomization of edge-biases that is responsible for the emergence of the stable limit in the present article, while also introducing further correlations into the model in comparison with the constant bias case. The derivation requires the development of a detailed understanding of trap geometry and the interplay between traps and backbone. The paper may be considered as a sequel to Ben Arous and Hammond [ Comm. Pure Appl. Math. 65 (2012) 1481–1527], since it makes use of a result on the regular tail of the total conductance of a randomly biased subcritical Galton–Watson tree.

Variational characterization of the critical curve for pinning of random polymers

Wed, 15 May 2013 09:12 EDT

Dimitris Cheliotis, Frank den Hollander

Source: Ann. Probab., Volume 41, Number 3B, 1767--1805.

Abstract:
In this paper we look at the pinning of a directed polymer by a one-dimensional linear interface carrying random charges. There are two phases, localized and delocalized, depending on the inverse temperature and on the disorder bias. Using quenched and annealed large deviation principles for the empirical process of words drawn from a random letter sequence according to a random renewal process [Birkner, Greven and den Hollander, Probab. Theory Related Fields 148 (2010) 403–456], we derive variational formulas for the quenched, respectively, annealed critical curve separating the two phases. These variational formulas are used to obtain a necessary and sufficient criterion, stated in terms of relative entropies, for the two critical curves to be different at a given inverse temperature, a property referred to as relevance of the disorder. This criterion in turn is used to show that the regimes of relevant and irrelevant disorder are separated by a unique inverse critical temperature. Subsequently, upper and lower bounds are derived for the inverse critical temperature, from which sufficient conditions under which it is strictly positive, respectively, finite are obtained. The former condition is believed to be necessary as well, a problem that we will address in a forthcoming paper. Random pinning has been studied extensively in the literature. The present paper opens up a window with a variational view. Our variational formulas for the quenched and the annealed critical curve are new and provide valuable insight into the nature of the phase transition. Our results on the inverse critical temperature drawn from these variational formulas are not new, but they offer an alternative approach, that is, flexible enough to be extended to other models of random polymers with disorder.

Gelation for Marcus–Lushnikov process

Wed, 15 May 2013 09:12 EDT

Fraydoun Rezakhanlou

Source: Ann. Probab., Volume 41, Number 3B, 1806--1830.

Abstract:
The Marcus–Lushnikov process is a simple mean field model of coagulating particles that converges to the homogeneous Smoluchowski equation in the large mass limit. If the coagulation rates grow sufficiently fast as the size of particles get large, giant particles emerge in finite time. This is known as gelation, and such particles are known as gels. Gelation comes in different flavors: simple, instantaneous and complete. In the case of an instantaneous gelation, giant particles are formed in a very short time. If all particles coagulate to form a single particle in a time interval that stays bounded as total mass gets large, then we have a complete gelation. In this article, we describe conditions which guarantee any of the three possible gelations with explicit bounds on the size of gels and the time of their creations.

Monotone stability of quadratic semimartingales with applications to unbounded general quadratic BSDEs

Wed, 15 May 2013 09:12 EDT

Pauline Barrieu, Nicole El Karoui

Source: Ann. Probab., Volume 41, Number 3B, 1831--1863.

Abstract:
In this paper, we study the stability and convergence of some general quadratic semimartingales. Motivated by financial applications, we study simultaneously the semimartingale and its opposite. Their characterization and integrability properties are obtained through some useful exponential submartingale inequalities. Then, a general stability result, including the strong convergence of the martingale parts in various spaces ranging from $\mathbb{H}^{1}$ to BMO, is derived under some mild integrability condition on the exponential of the terminal value of the semimartingale. This can be applied in particular to BSDE-like semimartingales. This strong convergence result is then used to prove the existence of solutions of general quadratic BSDEs under minimal exponential integrability assumptions, relying on a regularization in both linear-quadratic growth of the quadratic coefficient itself. On the contrary to most of the existing literature, it does not involve the seminal result of Kobylanski [ Ann. Probab. 28 (2010) 558–602] on bounded solutions.

Patterns in Sinai’s walk

Wed, 15 May 2013 09:12 EDT

Dimitris Cheliotis, Bálint Virág

Source: Ann. Probab., Volume 41, Number 3B, 1900--1937.

Abstract:
Sinai’s random walk in random environment shows interesting patterns on the exponential time scale. We characterize the patterns that appear on infinitely many time scales after appropriate rescaling (a functional law of iterated logarithm). The curious rate function captures the difference between one-sided and two-sided behavior.

Stochastic geometric wave equations with values in compact Riemannian homogeneous spaces

Wed, 15 May 2013 09:12 EDT

Zdzisław Brzeźniak, Martin Ondreját

Source: Ann. Probab., Volume 41, Number 3B, 1938--1977.

Abstract:
Let $M$ be a compact Riemannian homogeneous space (e.g., a Euclidean sphere). We prove existence of a global weak solution of the stochastic wave equation $\mathbf{D}_{t}\partial_{t}u=\sum_{k=1}^{d}\mathbf{D}_{x_{k}}\partial_{x_{k}}u+f_{u}(Du)+g_{u}(Du)\dot{W}$ in any dimension $d\ge1$, where $f$ and $g$ are continuous multilinear maps, and $W$ is a spatially homogeneous Wiener process on $\mathbb{R}^{d}$ with finite spectral measure. A nonstandard method of constructing weak solutions of SPDEs, that does not rely on martingale representation theorem, is employed.

Regularity of solutions to quantum master equations: A stochastic approach

Wed, 15 May 2013 09:12 EDT

Carlos M. Mora

Source: Ann. Probab., Volume 41, Number 3B, 1978--2012.

Abstract:
Applying probabilistic techniques we study regularity properties of quantum master equations (QMEs) in the Lindblad form with unbounded coefficients; a density operator is regular if, roughly speaking, it describes a quantum state with finite energy. Using the linear stochastic Schrödinger equation we deduce that solutions of QMEs preserve the regularity of the initial states under a general nonexplosion condition. To this end, we develop the probabilistic representation of QMEs, and we prove the uniqueness of solutions for adjoint quantum master equations. By means of the nonlinear stochastic Schrödinger equation, we obtain the existence of regular stationary solutions for QMEs, under a Lyapunov-type condition.

Subcritical percolation with a line of defects

Wed, 15 May 2013 09:12 EDT

S. Friedli, D. Ioffe, Y. Velenik

Source: Ann. Probab., Volume 41, Number 3B, 2013--2046.

Abstract:
We consider the Bernoulli bond percolation process $\mathbb{P} _{p,p'}$ on the nearest-neighbor edges of $\mathbb{Z} ^{d}$, which are open independently with probability $p<p_{c}$, except for those lying on the first coordinate axis, for which this probability is $p'$. Define \[\xi_{p,p'}:=-\lim_{n\to\infty}n^{-1}\log\mathbb{P} _{p,p'}(0\leftrightarrow n\mathbf{e} _{1})\] and $\xi_{p}:=\xi_{p,p}$. We show that there exists $p_{c}'=p_{c}'(p,d)$ such that $\xi_{p,p'}=\xi_{p}$ if $p'<p_{c}'$ and $\xi_{p,p'}<\xi_{p}$ if $p'>p_{c}'$. Moreover, $p_{c}'(p,2)=p_{c}'(p,3)=p$, and $p_{c}'(p,d)>p$ for $d\geq 4$. We also analyze the behavior of $\xi_{p}-\xi_{p,p'}$ as $p'\downarrow p_{c}'$ in dimensions $d=2,3$. Finally, we prove that when $p'>p_{c}'$, the following purely exponential asymptotics holds: \[\mathbb{P} _{p,p'}(0\leftrightarrow n\mathbf{e} _{1})=\psi_{d}e^{-\xi_{p,p'}n}\bigl(1+o(1)\bigr)\] for some constant $\psi_{d}=\psi_{d}(p,p')$, uniformly for large values of $n$. This work gives the first results on the rigorous analysis of pinning-type problems, that go beyond the effective models and don’t rely on exact computations.

Suprema of Lévy processes

Wed, 15 May 2013 09:12 EDT

Mateusz Kwaśnicki, Jacek Małecki, Michał Ryznar

Source: Ann. Probab., Volume 41, Number 3B, 2047--2065.

Abstract:
In this paper we study the supremum functional $M_{t}=\sup_{0\le s\le t}X_{s}$, where $X_{t}$, $t\ge0$, is a one-dimensional Lévy process. Under very mild assumptions we provide a simple, uniform estimate of the cumulative distribution function of $M_{t}$. In the symmetric case we find an integral representation of the Laplace transform of the distribution of $M_{t}$ if the Lévy–Khintchin exponent of the process increases on $(0,\infty)$.

Symmetric random walks on $\mathrm{Homeo}^{+}(\mathbf{R})$

Wed, 15 May 2013 09:12 EDT

B. Deroin, V. Kleptsyn, A. Navas, K. Parwani

Source: Ann. Probab., Volume 41, Number 3B, 2066--2089.

Abstract:
We study symmetric random walks on finitely generated groups of orientation-preserving homeomorphisms of the real line. We establish an oscillation property for the induced Markov chain on the line that implies a weak form of recurrence. Except for a few special cases, which can be treated separately, we prove a property of “global stability at a finite distance”: roughly speaking, there exists a compact interval such that any two trajectories get closer and closer whenever one of them returns to the compact interval. The probabilistic techniques employed here lead to interesting results for the study of group actions on the line. For instance, we show that under a suitable change of the coordinates, the drift of every point becomes zero provided that the action is minimal. As a byproduct, we recover the fact that every finitely generated group of homeomorphisms of the real line is topologically conjugate to a group of (globally) Lipschitz homeomorphisms. Moreover, we show that such a conjugacy may be chosen in such a way that the displacement of each element is uniformly bounded.

Exchangeable sequences driven by an absolutely continuous random measure

Wed, 15 May 2013 09:12 EDT

Patrizia Berti, Luca Pratelli, Pietro Rigo

Source: Ann. Probab., Volume 41, Number 3B, 2090--2102.

Abstract:
Let $S$ be a Polish space and $(X_{n}:n\geq1)$ an exchangeable sequence of $S$-valued random variables. Let $\alpha_{n}(\cdot)=P(X_{n+1}\in\cdot\mid X_{1},\ldots,X_{n})$ be the predictive measure and $\alpha$ a random probability measure on $S$ such that $\alpha_{n}\stackrel{\mathrm{weak}}{\longrightarrow}\alpha$ a.s. Two (related) problems are addressed. One is to give conditions for $\alpha\ll\lambda$ a.s., where $\lambda$ is a (nonrandom) $\sigma$-finite Borel measure on $S$. Such conditions should concern the finite dimensional distributions $\mathcal{L}(X_{1},\ldots,X_{n})$, $n\geq1$, only. The other problem is to investigate whether $\Vert\alpha_{n}-\alpha\Vert\stackrel{\mathrm{a.s.}}{\longrightarrow}0$, where $\Vert\cdot\Vert$ is total variation norm. Various results are obtained. Some of them do not require exchangeability, but hold under the weaker assumption that $(X_{n})$ is conditionally identically distributed, in the sense of [ Ann. Probab. 32 (2004) 2029–2052].

Convergence of the largest singular value of a polynomial in independent Wigner matrices

Wed, 15 May 2013 09:12 EDT

Greg W. Anderson

Source: Ann. Probab., Volume 41, Number 3B, 2103--2181.

Abstract:
For polynomials in independent Wigner matrices, we prove convergence of the largest singular value to the operator norm of the corresponding polynomial in free semicircular variables, under fourth moment hypotheses. We actually prove a more general result of the form “no eigenvalues outside the support of the limiting eigenvalue distribution.” We build on ideas of Haagerup–Schultz–Thorbjørnsen on the one hand and Bai–Silverstein on the other. We refine the linearization trick so as to preserve self-adjointness and we develop a secondary trick bearing on the calculation of correction terms. Instead of Poincaré-type inequalities, we use a variety of matrix identities and $L^{p}$ estimates. The Schwinger–Dyson equation controls much of the analysis.

Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site potential

Wed, 15 May 2013 09:12 EDT

Georg Menz, Felix Otto

Source: Ann. Probab., Volume 41, Number 3B, 2182--2224.

Abstract:
We consider a noninteracting unbounded spin system with conservation of the mean spin. We derive a uniform logarithmic Sobolev inequality (LSI) provided the single-site potential is a bounded perturbation of a strictly convex function. The scaling of the LSI constant is optimal in the system size. The argument adapts the two-scale approach of Grunewald, Villani, Westdickenberg and the second author from the quadratic to the general case. Using an asymmetric Brascamp–Lieb-type inequality for covariances, we reduce the task of deriving a uniform LSI to the convexification of the coarse-grained Hamiltonian, which follows from a general local Cramér theorem.

On the chaotic character of the stochastic heat equation, before the onset of intermitttency

Wed, 15 May 2013 09:12 EDT

Daniel Conus, Mathew Joseph, Davar Khoshnevisan

Source: Ann. Probab., Volume 41, Number 3B, 2225--2260.

Abstract:
We consider a nonlinear stochastic heat equation $\partial_{t}u=\frac{1}{2}\partial_{xx}u+\sigma(u)\partial_{xt}W$, where $\partial_{xt}W$ denotes space–time white noise and $\sigma:\mathbf{R} \to\mathbf{R} $ is Lipschitz continuous. We establish that, at every fixed time $t>0$, the global behavior of the solution depends in a critical manner on the structure of the initial function $u_{0}$: under suitable conditions on $u_{0}$ and $\sigma$, $\sup_{x\in\mathbf{R} }u_{t}(x)$ is a.s. finite when $u_{0}$ has compact support, whereas with probability one, $\limsup_{|x|\to\infty}u_{t}(x)/({\log}|x|)^{1/6}>0$ when $u_{0}$ is bounded uniformly away from zero. This sensitivity to the initial data of the stochastic heat equation is a way to state that the solution to the stochastic heat equation is chaotic at fixed times, well before the onset of intermittency.

Limit theorems for iteration stable tessellations

Wed, 15 May 2013 09:12 EDT

Tomasz Schreiber, Christoph Thäle

Source: Ann. Probab., Volume 41, Number 3B, 2261--2278.

Abstract:
The intent of this paper is to describe the large scale asymptotic geometry of iteration stable (STIT) tessellations in $\mathbb{R}^{d}$, which form a rather new, rich and flexible class of random tessellations considered in stochastic geometry. For this purpose, martingale tools are combined with second-order formulas proved earlier to establish limit theorems for STIT tessellations. More precisely, a Gaussian functional central limit theorem for the surface increment process induced a by STIT tessellation relative to an initial time moment is shown. As second main result, a central limit theorem for the total edge length/facet surface is obtained, with a normal limit distribution in the planar case and, most interestingly, with a nonnormal limit showing up in all higher space dimensions.

Spectral statistics of Erdős–Rényi graphs I: Local semicircle law

Wed, 15 May 2013 09:12 EDT

László Erdős, Antti Knowles, Horng-Tzer Yau, Jun Yin

Source: Ann. Probab., Volume 41, Number 3B, 2279--2375.

Abstract:
We consider the ensemble of adjacency matrices of Erdős–Rényi random graphs, that is, graphs on $N$ vertices where every edge is chosen independently and with probability $p\equiv p(N)$. We rescale the matrix so that its bulk eigenvalues are of order one. We prove that, as long as $pN\to\infty$ (with a speed at least logarithmic in $N$), the density of eigenvalues of the Erdős–Rényi ensemble is given by the Wigner semicircle law for spectral windows of length larger than $N^{-1}$ (up to logarithmic corrections). As a consequence, all eigenvectors are proved to be completely delocalized in the sense that the $\ell^{\infty}$-norms of the $\ell^{2}$-normalized eigenvectors are at most of order $N^{-1/2}$ with a very high probability. The estimates in this paper will be used in the companion paper [Spectral statistics of Erdős–Rényi graphs II: Eigenvalue spacing and the extreme eigenvalues (2011) Preprint] to prove the universality of eigenvalue distributions both in the bulk and at the spectral edges under the further restriction that $pN\gg N^{2/3}$.

The solution of the perturbed Tanaka-equation is pathwise unique

Wed, 15 May 2013 09:12 EDT

Vilmos Prokaj

Source: Ann. Probab., Volume 41, Number 3B, 2376--2400.

Abstract:
The Tanaka equation $dX_{t}=\operatorname{sign}(X_{t})\,dB_{t}$ is an example of a stochastic differential equation (SDE) without strong solution. Hence pathwise uniqueness does not hold for this equation. In this note we prove that if we modify the right-hand side of the equation, roughly speaking, with a strong enough additive noise, independent of the Brownian motion $B$, then the solution of the obtained equation is pathwise unique.

On explosions in heavy-tailed branching random walks

Wed, 15 May 2013 12:58 EDT

Omid Amini, Luc Devroye, Simon Griffiths, Neil Olver

Source: Ann. Probab., Volume 41, Number 3B, 1864--1899.

Abstract:
Consider a branching random walk on $\mathbb{R}$, with offspring distribution $Z$ and nonnegative displacement distribution $W$. We say that explosion occurs if an infinite number of particles may be found within a finite distance of the origin. In this paper, we investigate this phenomenon when the offspring distribution $Z$ is heavy-tailed. Under an appropriate condition, we are able to characterize the pairs $(Z,W)$ for which explosion occurs, by demonstrating the equivalence of explosion with a seemingly much weaker event: that the sum over generations of the minimum displacement in each generation is finite. Furthermore, we demonstrate that our condition on the tail is best possible for this equivalence to occur. We also investigate, under additional smoothness assumptions, the behavior of $M_{n}$, the position of the particle in generation $n$ closest to the origin, when explosion does not occur (and hence $\lim_{n\rightarrow\infty}M_{n}=\infty$).