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The latest articles from The Annals of Probability on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTWed, 16 Mar 2011 09:23 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Uniform convergence of Vapnik–Chervonenkis classes under ergodic sampling
http://projecteuclid.org/euclid.aop/1278593952
<strong>Terrence M. Adams</strong>, <strong>Andrew B. Nobel</strong><p><strong>Source: </strong>Ann. Probab., Volume 38, Number 4, 1345--1367.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.aop/1278593952_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTA fractional kinetic process describing the intermediate time behaviour of cellular flowshttps://projecteuclid.org/euclid.aop/1520586272<strong>Martin Hairer</strong>, <strong>Gautam Iyer</strong>, <strong>Leonid Koralov</strong>, <strong>Alexei Novikov</strong>, <strong>Zsolt Pajor-Gyulai</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 2, 897--955.</p><p><strong>Abstract:</strong><br/>
This paper studies the intermediate time behaviour of a small random perturbation of a periodic cellular flow. Our main result shows that on time scales shorter than the diffusive time scale, the limiting behaviour of trajectories that start close enough to cell boundaries is a fractional kinetic process: a Brownian motion time changed by the local time of an independent Brownian motion. Our proof uses the Freidlin–Wentzell framework, and the key step is to establish an analogous averaging principle on shorter time scales.
As a consequence of our main theorem, we obtain a homogenization result for the associated advection diffusion equation. We show that on intermediate time scales the effective equation is a fractional time PDE that arises in modelling anomalous diffusion.
</p>projecteuclid.org/euclid.aop/1520586272_20180309040445Fri, 09 Mar 2018 04:04 ESTQuasi-symmetries of determinantal point processeshttps://projecteuclid.org/euclid.aop/1520586273<strong>Alexander I. Bufetov</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 2, 956--1003.</p><p><strong>Abstract:</strong><br/>
The main result of this paper is that determinantal point processes on $\mathbb{R}$ corresponding to projection operators with integrable kernels are quasi-invariant, in the continuous case, under the group of diffeomorphisms with compact support (Theorem 1.4); in the discrete case, under the group of all finite permutations of the phase space (Theorem 1.6). The Radon–Nikodym derivative is computed explicitly and is given by a regularized multiplicative functional. Theorem 1.4 applies, in particular, to the sine-process, as well as to determinantal point processes with the Bessel and the Airy kernels; Theorem 1.6 to the discrete sine-process and the Gamma kernel process. The paper answers a question of Grigori Olshanski.
</p>projecteuclid.org/euclid.aop/1520586273_20180309040445Fri, 09 Mar 2018 04:04 ESTFirst-passage percolation on Cartesian power graphshttps://projecteuclid.org/euclid.aop/1520586274<strong>Anders Martinsson</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 2, 1004--1041.</p><p><strong>Abstract:</strong><br/>
We consider first-passage percolation on the class of “high-dimensional” graphs that can be written as an iterated Cartesian product $G\square G\square\dots\square G$ of some base graph $G$ as the number of factors tends to infinity. We propose a natural asymptotic lower bound on the first-passage time between $(v,v,\dots,v)$ and $(w,w,\dots,w)$ as $n$, the number of factors, tends to infinity, which we call the critical time $t^{*}_{G}(v,w)$. Our main result characterizes when this lower bound is sharp as $n\rightarrow\infty$. As a corollary, we are able to determine the limit of the so-called diagonal time-constant in $\mathbb{Z}^{n}$ as $n\rightarrow\infty$ for a large class of distributions of passage times.
</p>projecteuclid.org/euclid.aop/1520586274_20180309040445Fri, 09 Mar 2018 04:04 ESTSPDE limit of the global fluctuations in rank-based modelshttps://projecteuclid.org/euclid.aop/1520586275<strong>Praveen Kolli</strong>, <strong>Mykhaylo Shkolnikov</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 2, 1042--1069.</p><p><strong>Abstract:</strong><br/>
We consider systems of diffusion processes (“particles”) interacting through their ranks (also referred to as “rank-based models” in the mathematical finance literature). We show that, as the number of particles becomes large, the process of fluctuations of the empirical cumulative distribution functions converges to the solution of a linear parabolic SPDE with additive noise. The coefficients in the limiting SPDE are determined by the hydrodynamic limit of the particle system which, in turn, can be described by the porous medium PDE. The result opens the door to a thorough investigation of large equity markets and investment therein. In the course of the proof, we also derive quantitative propagation of chaos estimates for the particle system.
</p>projecteuclid.org/euclid.aop/1520586275_20180309040445Fri, 09 Mar 2018 04:04 ESTExponentially concave functions and a new information geometryhttps://projecteuclid.org/euclid.aop/1520586276<strong>Soumik Pal</strong>, <strong>Ting-Kam Leonard Wong</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 2, 1070--1113.</p><p><strong>Abstract:</strong><br/>
A function is exponentially concave if its exponential is concave. We consider exponentially concave functions on the unit simplex. In a previous paper, we showed that gradient maps of exponentially concave functions provide solutions to a Monge–Kantorovich optimal transport problem and give a better gradient approximation than those of ordinary concave functions. The approximation error, called L-divergence, is different from the usual Bregman divergence. Using tools of information geometry and optimal transport, we show that L-divergence induces a new information geometry on the simplex consisting of a Riemannian metric and a pair of dually coupled affine connections which defines two kinds of geodesics. We show that the induced geometry is dually projectively flat but not flat. Nevertheless, we prove an analogue of the celebrated generalized Pythagorean theorem from classical information geometry. On the other hand, we consider displacement interpolation under a Lagrangian integral action that is consistent with the optimal transport problem and show that the action minimizing curves are dual geodesics. The Pythagorean theorem is also shown to have an interesting application of determining the optimal trading frequency in stochastic portfolio theory.
</p>projecteuclid.org/euclid.aop/1520586276_20180309040445Fri, 09 Mar 2018 04:04 ESTOn the cycle structure of Mallows permutationshttps://projecteuclid.org/euclid.aop/1520586277<strong>Alexey Gladkich</strong>, <strong>Ron Peled</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 2, 1114--1169.</p><p><strong>Abstract:</strong><br/>
We study the length of cycles of random permutations drawn from the Mallows distribution. Under this distribution, the probability of a permutation $\pi\in\mathbb{S}_{n}$ is proportional to $q^{\operatorname{inv}(\pi)}$ where $q>0$ and $\operatorname{inv}(\pi)$ is the number of inversions in $\pi$.
We focus on the case that $q<1$ and show that the expected length of the cycle containing a given point is of order $\min\{(1-q)^{-2},n\}$. This marks the existence of two asymptotic regimes: with high probability, when $n$ tends to infinity with $(1-q)^{-2}\ll n$ then all cycles have size $o(n)$ whereas when $n$ tends to infinity with $(1-q)^{-2}\gg n$ then macroscopic cycles, of size proportional to $n$, emerge. In the second regime, we prove that the distribution of normalized cycle lengths follows the Poisson–Dirichlet law, as in a uniformly random permutation. The results bear formal similarity with a conjectured localization transition for random band matrices.
Further results are presented for the variance of the cycle lengths, the expected diameter of cycles and the expected number of cycles. The proofs rely on the exact sampling algorithm for the Mallows distribution and make use of a special diagonal exposure process for the graph of the permutation.
</p>projecteuclid.org/euclid.aop/1520586277_20180309040445Fri, 09 Mar 2018 04:04 ESTInterlacements and the wired uniform spanning foresthttps://projecteuclid.org/euclid.aop/1520586278<strong>Tom Hutchcroft</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 2, 1170--1200.</p><p><strong>Abstract:</strong><br/>
We extend the Aldous–Broder algorithm to generate the wired uniform spanning forests (WUSFs) of infinite, transient graphs. We do this by replacing the simple random walk in the classical algorithm with Sznitman’s random interlacement process. We then apply this algorithm to study the WUSF, showing that every component of the WUSF is one-ended almost surely in any graph satisfying a certain weak anchored isoperimetric condition, that the number of ‘excessive ends’ in the WUSF is nonrandom in any graph, and also that every component of the WUSF is one-ended almost surely in any transient unimodular random rooted graph. The first two of these results answer positively two questions of Lyons, Morris and Schramm [ Electron. J. Probab. 13 (2008) 1702–1725], while the third extends a recent result of the author.
Finally, we construct a counterexample showing that almost sure one-endedness of WUSF components is not preserved by rough isometries of the underlying graph, answering negatively a further question of Lyons, Morris and Schramm.
</p>projecteuclid.org/euclid.aop/1520586278_20180309040445Fri, 09 Mar 2018 04:04 ESTGaussian and non-Gaussian fluctuations for mesoscopic linear statistics in determinantal processeshttps://projecteuclid.org/euclid.aop/1523520017<strong>Kurt Johansson</strong>, <strong>Gaultier Lambert</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 3, 1201--1278.</p><p><strong>Abstract:</strong><br/>
We study mesoscopic linear statistics for a class of determinantal point processes which interpolate between Poisson and random matrix statistics. These processes are obtained by modifying the spectrum of the correlation kernel of the Gaussian Unitary Ensemble (GUE) eigenvalue process. An example of such a system comes from considering the distribution of noncolliding Brownian motions in a cylindrical geometry, or a grand canonical ensemble of free fermions in a quadratic well at positive temperature. When the scale of the modification of the spectrum of the correlation kernel, related to the size of the cylinder or the temperature, is different from the scale in the mesoscopic linear statistic, we obtain a central limit theorem (CLT) of either Poisson or GUE type. On the other hand, in the critical regime where the scales are the same, we observe a non-Gaussian process in the limit. Its distribution is characterized by explicit but complicated formulae for the cumulants of smooth linear statistics. These results rely on an asymptotic sine-kernel approximation of the GUE kernel which is valid at all mesoscopic scales, and a generalization of cumulant computations of Soshnikov for the sine process. Analogous determinantal processes on the circle are also considered with similar results.
</p>projecteuclid.org/euclid.aop/1523520017_20180412040031Thu, 12 Apr 2018 04:00 EDTOn global fluctuations for non-colliding processeshttps://projecteuclid.org/euclid.aop/1523520018<strong>Maurice Duits</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 3, 1279--1350.</p><p><strong>Abstract:</strong><br/>
We study the global fluctuations for a class of determinantal point processes coming from large systems of non-colliding processes and non-intersecting paths. Our main assumption is that the point processes are constructed by biorthogonal families that satisfy finite term recurrence relations. The central observation of the paper is that the fluctuations of multi-time or multi-layer linear statistics can be efficiently expressed in terms of the associated recurrence matrices. As a consequence, we prove that different models that share the same asymptotic behavior of the recurrence matrices, also share the same asymptotic behavior for the global fluctuations. An important special case is when the recurrence matrices have limits along the diagonals, in which case we prove Central Limit Theorems for the linear statistics. We then show that these results prove Gaussian Free Field fluctuations for the random surfaces associated to these systems. To illustrate the results, several examples will be discussed, including non-colliding processes for which the invariant measures are the classical orthogonal polynomial ensembles and random lozenge tilings of a hexagon.
</p>projecteuclid.org/euclid.aop/1523520018_20180412040031Thu, 12 Apr 2018 04:00 EDTMultivariate approximation in total variation, I: Equilibrium distributions of Markov jump processeshttps://projecteuclid.org/euclid.aop/1523520019<strong>A. D. Barbour</strong>, <strong>M. J. Luczak</strong>, <strong>A. Xia</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 3, 1351--1404.</p><p><strong>Abstract:</strong><br/>
For integer valued random variables, the translated Poisson distributions form a flexible family for approximation in total variation, in much the same way that the normal family is used for approximation in Kolmogorov distance. Using the Stein–Chen method, approximation can often be achieved with error bounds of the same order as those for the CLT. In this paper, an analogous theory, again based on Stein’s method, is developed in the multivariate context. The approximating family consists of the equilibrium distributions of a collection of Markov jump processes, whose analogues in one dimension are the immigration-death processes with Poisson distributions as equilibria. The method is illustrated by providing total variation error bounds for the approximation of the equilibrium distribution of one Markov jump process by that of another. In a companion paper, it is shown how to use the method for discrete normal approximation in $\mathbb{Z}^{d}$.
</p>projecteuclid.org/euclid.aop/1523520019_20180412040031Thu, 12 Apr 2018 04:00 EDTMultivariate approximation in total variation, II: Discrete normal approximationhttps://projecteuclid.org/euclid.aop/1523520020<strong>A. D. Barbour</strong>, <strong>M. J. Luczak</strong>, <strong>A. Xia</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 3, 1405--1440.</p><p><strong>Abstract:</strong><br/>
The paper applies the theory developed in Part I to the discrete normal approximation in total variation of random vectors in $\mathbb{Z}^{d}$. We illustrate the use of the method for sums of independent integer valued random vectors, and for random vectors exhibiting an exchangeable pair. We conclude with an application to random colourings of regular graphs.
</p>projecteuclid.org/euclid.aop/1523520020_20180412040031Thu, 12 Apr 2018 04:00 EDTA Gaussian small deviation inequality for convex functionshttps://projecteuclid.org/euclid.aop/1523520021<strong>Grigoris Paouris</strong>, <strong>Petros Valettas</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 3, 1441--1454.</p><p><strong>Abstract:</strong><br/>
Let $Z$ be an $n$-dimensional Gaussian vector and let $f:\mathbb{R}^{n}\to \mathbb{R}$ be a convex function. We prove that
\[\mathbb{P}(f(Z)\leq \mathbb{E}f(Z)-t\sqrt{\operatorname{Var}f(Z)})\leq\exp (-ct^{2}),\] for all $t>1$ where $c>0$ is an absolute constant. As an application we derive variance-sensitive small ball probabilities for Gaussian processes.
</p>projecteuclid.org/euclid.aop/1523520021_20180412040031Thu, 12 Apr 2018 04:00 EDTA variational approach to dissipative SPDEs with singular drifthttps://projecteuclid.org/euclid.aop/1523520022<strong>Carlo Marinelli</strong>, <strong>Luca Scarpa</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 3, 1455--1497.</p><p><strong>Abstract:</strong><br/>
We prove global well-posedness for a class of dissipative semilinear stochastic evolution equations with singular drift and multiplicative Wiener noise. In particular, the nonlinear term in the drift is the superposition operator associated to a maximal monotone graph everywhere defined on the real line, on which neither continuity nor growth assumptions are imposed. The hypotheses on the diffusion coefficient are also very general, in the sense that the noise does not need to take values in spaces of continuous, or bounded, functions in space and time. Our approach combines variational techniques with a priori estimates, both pathwise and in expectation, on solutions to regularized equations.
</p>projecteuclid.org/euclid.aop/1523520022_20180412040031Thu, 12 Apr 2018 04:00 EDTStrong solutions to stochastic differential equations with rough coefficientshttps://projecteuclid.org/euclid.aop/1523520023<strong>Nicolas Champagnat</strong>, <strong>Pierre-Emmanuel Jabin</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 3, 1498--1541.</p><p><strong>Abstract:</strong><br/>
We study strong existence and pathwise uniqueness for stochastic differential equations in $\mathbb{R}^{d}$ with rough coefficients, and without assuming uniform ellipticity for the diffusion matrix. Our approach relies on direct quantitative estimates on solutions to the SDE, assuming Sobolev bounds on the drift and diffusion coefficients, and $L^{p}$ bounds for the solution of the corresponding Fokker–Planck PDE, which can be proved separately. This allows a great flexibility regarding the method employed to obtain these last bounds. Hence we are able to obtain general criteria in various cases, including the uniformly elliptic case in any dimension, the one-dimensional case and the Langevin (kinetic) case.
</p>projecteuclid.org/euclid.aop/1523520023_20180412040031Thu, 12 Apr 2018 04:00 EDTDimensions of random covering sets in Riemann manifoldshttps://projecteuclid.org/euclid.aop/1523520024<strong>De-Jun Feng</strong>, <strong>Esa Järvenpää</strong>, <strong>Maarit Järvenpää</strong>, <strong>Ville Suomala</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 3, 1542--1596.</p><p><strong>Abstract:</strong><br/>
Let ${\mathbf{M}}$, ${\mathbf{N}}$ and ${\mathbf{K}}$ be $d$-dimensional Riemann manifolds. Assume that ${\mathbf{A}}:=(A_{n})_{n\in{\mathbb{N}}}$ is a sequence of Lebesgue measurable subsets of ${\mathbf{M}}$ satisfying a necessary density condition and ${\mathbf{x}}:=(x_{n})_{n\in{\mathbb{N}}}$ is a sequence of independent random variables, which are distributed on ${\mathbf{K}}$ according to a measure, which is not purely singular with respect to the Riemann volume. We give a formula for the almost sure value of the Hausdorff dimension of random covering sets ${\mathbf{E}}({\mathbf{x}},{\mathbf{A}}):=\limsup_{n\to\infty}A_{n}(x_{n})\subset{\mathbf{N}}$. Here, $A_{n}(x_{n})$ is a diffeomorphic image of $A_{n}$ depending on $x_{n}$. We also verify that the packing dimensions of ${\mathbf{E}}({\mathbf{x}},{\mathbf{A}})$ equal $d$ almost surely.
</p>projecteuclid.org/euclid.aop/1523520024_20180412040031Thu, 12 Apr 2018 04:00 EDTOptimal surviving strategy for drifted Brownian motions with absorptionhttps://projecteuclid.org/euclid.aop/1523520025<strong>Wenpin Tang</strong>, <strong>Li-Cheng Tsai</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 3, 1597--1650.</p><p><strong>Abstract:</strong><br/>
We study the “Up the River” problem formulated by Aldous (2002), where a unit drift is distributed among a finite collection of Brownian particles on $\mathbb{R}_{+}$, which are annihilated once they reach the origin. Starting $K$ particles at $x=1$, we prove Aldous’ conjecture [Aldous (2002)] that the “push-the-laggard” strategy of distributing the drift asymptotically (as $K\to\infty$) maximizes the total number of surviving particles, with approximately $\frac{4}{\sqrt{\pi}}\sqrt{K}$ surviving particles. We further establish the hydrodynamic limit of the particle density, in terms of a two-phase partial differential equation (PDE) with a moving boundary, by utilizing certain integral identities and coupling techniques.
</p>projecteuclid.org/euclid.aop/1523520025_20180412040031Thu, 12 Apr 2018 04:00 EDTDiscretisations of rough stochastic PDEshttps://projecteuclid.org/euclid.aop/1523520026<strong>M. Hairer</strong>, <strong>K. Matetski</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 3, 1651--1709.</p><p><strong>Abstract:</strong><br/>
We develop a general framework for spatial discretisations of parabolic stochastic PDEs whose solutions are provided in the framework of the theory of regularity structures and which are functions in time. As an application, we show that the dynamical $\Phi^{4}_{3}$ model on the dyadic grid converges after renormalisation to its continuous counterpart. This result in particular implies that, as expected, the $\Phi^{4}_{3}$ measure with a sufficiently small coupling constant is invariant for this equation and that the lifetime of its solutions is almost surely infinite for almost every initial condition.
</p>projecteuclid.org/euclid.aop/1523520026_20180412040031Thu, 12 Apr 2018 04:00 EDTMultidimensional SDEs with singular drift and universal construction of the polymer measure with white noise potentialhttps://projecteuclid.org/euclid.aop/1523520027<strong>Giuseppe Cannizzaro</strong>, <strong>Khalil Chouk</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 3, 1710--1763.</p><p><strong>Abstract:</strong><br/>
We study the existence and uniqueness of solution for stochastic differential equations with distributional drift by giving a meaning to the Stroock–Varadhan martingale problem associated to such equations. The approach we exploit is the one of paracontrolled distributions introduced in ( Forum Math. Pi 3 (2015) e6). As a result, we make sense of the three-dimensional polymer measure with white noise potential.
</p>projecteuclid.org/euclid.aop/1523520027_20180412040031Thu, 12 Apr 2018 04:00 EDTChaining, interpolation and convexity II: The contraction principlehttps://projecteuclid.org/euclid.aop/1523520028<strong>Ramon van Handel</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 3, 1764--1805.</p><p><strong>Abstract:</strong><br/>
The generic chaining method provides a sharp description of the suprema of many random processes in terms of the geometry of their index sets. The chaining functionals that arise in this theory are however notoriously difficult to control in any given situation. In the first paper in this series, we introduced a particularly simple method for producing the requisite multiscale geometry by means of real interpolation. This method is easy to use, but does not always yield sharp bounds on chaining functionals. In the present paper, we show that a refinement of the interpolation method provides a canonical mechanism for controlling chaining functionals. The key innovation is a simple but powerful contraction principle that makes it possible to efficiently exploit interpolation. We illustrate the utility of this approach by developing new dimension-free bounds on the norms of random matrices and on chaining functionals in Banach lattices. As another application, we give a remarkably short interpolation proof of the majorizing measure theorem that entirely avoids the greedy construction that lies at the heart of earlier proofs.
</p>projecteuclid.org/euclid.aop/1523520028_20180412040031Thu, 12 Apr 2018 04:00 EDTLimit theorems for Markov walks conditioned to stay positive under a spectral gap assumptionhttps://projecteuclid.org/euclid.aop/1528876816<strong>Ion Grama</strong>, <strong>Ronan Lauvergnat</strong>, <strong>Émile Le Page</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 1807--1877.</p><p><strong>Abstract:</strong><br/>
Consider a Markov chain $(X_{n})_{n\geq0}$ with values in the state space $\mathbb{X}$. Let $f$ be a real function on $\mathbb{X}$ and set $S_{n}=\sum_{i=1}^{n}f(X_{i})$, $n\geq1$. Let $\mathbb{P}_{x}$ be the probability measure generated by the Markov chain starting at $X_{0}=x$. For a starting point $y\in\mathbb{R}$, denote by $\tau_{y}$ the first moment when the Markov walk $(y+S_{n})_{n\geq1}$ becomes nonpositive. Under the condition that $S_{n}$ has zero drift, we find the asymptotics of the probability $\mathbb{P}_{x}(\tau_{y}>n)$ and of the conditional law $\mathbb{P}_{x}(y+S_{n}\leq \cdot\sqrt{n}\mid\tau_{y}>n)$ as $n\to+\infty$.
</p>projecteuclid.org/euclid.aop/1528876816_20180613040038Wed, 13 Jun 2018 04:00 EDTThe fourth moment theorem on the Poisson spacehttps://projecteuclid.org/euclid.aop/1528876817<strong>Christian Döbler</strong>, <strong>Giovanni Peccati</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 1878--1916.</p><p><strong>Abstract:</strong><br/>
We prove a fourth moment bound without remainder for the normal approximation of random variables belonging to the Wiener chaos of a general Poisson random measure. Such a result—that has been elusive for several years—shows that the so-called ‘fourth moment phenomenon’, first discovered by Nualart and Peccati [ Ann. Probab. 33 (2005) 177–193] in the context of Gaussian fields, also systematically emerges in a Poisson framework. Our main findings are based on Stein’s method, Malliavin calculus and Mecke-type formulae, as well as on a methodological breakthrough, consisting in the use of carré-du-champ operators on the Poisson space for controlling residual terms associated with add-one cost operators. Our approach can be regarded as a successful application of Markov generator techniques to probabilistic approximations in a nondiffusive framework: as such, it represents a significant extension of the seminal contributions by Ledoux [ Ann. Probab. 40 (2012) 2439–2459] and Azmoodeh, Campese and Poly [ J. Funct. Anal. 266 (2014) 2341–2359]. To demonstrate the flexibility of our results, we also provide some novel bounds for the Gamma approximation of nonlinear functionals of a Poisson measure.
</p>projecteuclid.org/euclid.aop/1528876817_20180613040038Wed, 13 Jun 2018 04:00 EDTAnchored expansion, speed and the Poisson–Voronoi tessellation in symmetric spaceshttps://projecteuclid.org/euclid.aop/1528876818<strong>Itai Benjamini</strong>, <strong>Elliot Paquette</strong>, <strong>Joshua Pfeffer</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 1917--1956.</p><p><strong>Abstract:</strong><br/>
We show that a random walk on a stationary random graph with positive anchored expansion and exponential volume growth has positive speed. We also show that two families of random triangulations of the hyperbolic plane, the hyperbolic Poisson–Voronoi tessellation and the hyperbolic Poisson–Delaunay triangulation, have $1$-skeletons with positive anchored expansion. As a consequence, we show that the simple random walks on these graphs have positive hyperbolic speed. Finally, we include a section of open problems and conjectures on the topics of stationary geometric random graphs and the hyperbolic Poisson–Voronoi tessellation.
</p>projecteuclid.org/euclid.aop/1528876818_20180613040038Wed, 13 Jun 2018 04:00 EDTOptimal bilinear control of nonlinear stochastic Schrödinger equations driven by linear multiplicative noisehttps://projecteuclid.org/euclid.aop/1528876819<strong>Viorel Barbu</strong>, <strong>Michael Röckner</strong>, <strong>Deng Zhang</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 1957--1999.</p><p><strong>Abstract:</strong><br/>
We analyze the bilinear optimal control problem of quantum mechanical systems with final observation governed by a stochastic nonlinear Schrödinger equation perturbed by a linear multiplicative Wiener process. The existence of an open-loop optimal control and first-order Lagrange optimality conditions are derived, via Skorohod’s representation theorem, Ekeland’s variational principle and the existence for the linearized dual backward stochastic equation. Moreover, our approach in particular applies to the deterministic case.
</p>projecteuclid.org/euclid.aop/1528876819_20180613040038Wed, 13 Jun 2018 04:00 EDTRandom partitions of the plane via Poissonian coloring and a self-similar process of coalescing planar partitionshttps://projecteuclid.org/euclid.aop/1528876820<strong>David Aldous</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 2000--2037.</p><p><strong>Abstract:</strong><br/>
Plant differently colored points in the plane; then let random points (“Poisson rain”) fall, and give each new point the color of the nearest existing point. Previous investigation and simulations strongly suggest that the colored regions converge (in some sense) to a random partition of the plane. We prove a weak version of this, showing that normalized empirical measures converge to Lebesgue measures on a random partition into measurable sets. Topological properties remain an open problem. In the course of the proof, which heavily exploits time-reversals, we encounter a novel self-similar process of coalescing planar partitions. In this process, sets $A(z)$ in the partition are associated with Poisson random points $z$, and the dynamics are as follows. Points are deleted randomly at rate $1$; when $z$ is deleted, its set $A(z)$ is adjoined to the set $A(z^{\prime})$ of the nearest other point $z^{\prime}$.
</p>projecteuclid.org/euclid.aop/1528876820_20180613040038Wed, 13 Jun 2018 04:00 EDTScaling limit of two-component interacting Brownian motionshttps://projecteuclid.org/euclid.aop/1528876821<strong>Insuk Seo</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 2038--2063.</p><p><strong>Abstract:</strong><br/>
This paper presents our study of the asymptotic behavior of a two-component system of Brownian motions undergoing certain form of singular interactions. In particular, the system is a combination of two different types of particles and the mechanical properties and the interaction parameters depend on the corresponding type of particles. We prove that the hydrodynamic limit of the empirical densities of two types is the solution of a partial differential equation known as the Maxwell–Stefan equation.
</p>projecteuclid.org/euclid.aop/1528876821_20180613040038Wed, 13 Jun 2018 04:00 EDTLarge excursions and conditioned laws for recursive sequences generated by random matriceshttps://projecteuclid.org/euclid.aop/1528876822<strong>Jeffrey F. Collamore</strong>, <strong>Sebastian Mentemeier</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 2064--2120.</p><p><strong>Abstract:</strong><br/>
We study the large exceedance probabilities and large exceedance paths of the recursive sequence $V_{n}=M_{n}V_{n-1}+Q_{n}$, where $\{(M_{n},Q_{n})\}$ is an i.i.d. sequence, and $M_{1}$ is a $d\times d$ random matrix and $Q_{1}$ is a random vector, both with nonnegative entries. We impose conditions which guarantee the existence of a unique stationary distribution for $\{V_{n}\}$ and a Cramér-type condition for $\{M_{n}\}$. Under these assumptions, we characterize the distribution of the first passage time $T_{u}^{A}:=\inf\{n:V_{n}\in uA\}$, where $A$ is a general subset of $\mathbb{R}^{d}$, exhibiting that $T_{u}^{A}/u^{\alpha}$ converges to an exponential law for a certain $\alpha>0$. In the process, we revisit and refine classical estimates for $\mathbb{P}(V\in uA)$, where $V$ possesses the stationary law of $\{V_{n}\}$. Namely, for $A\subset\mathbb{R}^{d}$, we show that $\mathbb{P}(V\in uA)\sim C_{A}u^{-\alpha}$ as $u\to\infty$, providing, most importantly, a new characterization of the constant $C_{A}$. As a simple consequence of these estimates, we also obtain an expression for the extremal index of $\{|V_{n}|\}$. Finally, we describe the large exceedance paths via two conditioned limit theorems showing, roughly, that $\{V_{n}\}$ follows an exponentially-shifted Markov random walk, which we identify. We thereby generalize results from the theory of classical random walk to multivariate recursive sequences.
</p>projecteuclid.org/euclid.aop/1528876822_20180613040038Wed, 13 Jun 2018 04:00 EDTPhase transition for the Once-reinforced random walk on $\mathbb{Z}^{d}$-like treeshttps://projecteuclid.org/euclid.aop/1528876823<strong>Daniel Kious</strong>, <strong>Vladas Sidoravicius</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 2121--2133.</p><p><strong>Abstract:</strong><br/>
In this short paper, we consider the Once-reinforced random walk with reinforcement parameter $a$ on trees with bounded degree which are transient for the simple random walk. On each of these trees, we prove that there exists an explicit critical parameter $a_{0}$ such that the Once-reinforced random walk is almost surely recurrent if $a>a_{0}$ and almost surely transient if $a<a_{0}$. This provides the first examples of phase transition for the Once-reinforced random walk.
</p>projecteuclid.org/euclid.aop/1528876823_20180613040038Wed, 13 Jun 2018 04:00 EDTThe Brownian limit of separable permutationshttps://projecteuclid.org/euclid.aop/1528876824<strong>Frédérique Bassino</strong>, <strong>Mathilde Bouvel</strong>, <strong>Valentin Féray</strong>, <strong>Lucas Gerin</strong>, <strong>Adeline Pierrot</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 2134--2189.</p><p><strong>Abstract:</strong><br/>
We study uniform random permutations in an important class of pattern-avoiding permutations: the separable permutations. We describe the asymptotics of the number of occurrences of any fixed given pattern in such a random permutation in terms of the Brownian excursion. In the recent terminology of permutons, our work can be interpreted as the convergence of uniform random separable permutations towards a “Brownian separable permuton”.
</p>projecteuclid.org/euclid.aop/1528876824_20180613040038Wed, 13 Jun 2018 04:00 EDTCritical density of activated random walks on transitive graphshttps://projecteuclid.org/euclid.aop/1528876825<strong>Alexandre Stauffer</strong>, <strong>Lorenzo Taggi</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 2190--2220.</p><p><strong>Abstract:</strong><br/>
We consider the activated random walk model on general vertex-transitive graphs. A central question in this model is whether the critical density $\mu_{c}$ for sustained activity is strictly between 0 and 1. It was known that $\mu_{c}>0$ on $\mathbb{Z}^{d}$, $d\geq1$, and that $\mu_{c}<1$ on $\mathbb{Z}$ for small enough sleeping rate. We show that $\mu_{c}\to0$ as $\lambda\to0$ in all vertex-transitive transient graphs, implying that $\mu_{c}<1$ for small enough sleeping rate. We also show that $\mu_{c}<1$ for any sleeping rate in any vertex-transitive graph in which simple random walk has positive speed. Furthermore, we prove that $\mu_{c}>0$ in any vertex-transitive amenable graph, and that $\mu_{c}\in(0,1)$ for any sleeping rate on regular trees.
</p>projecteuclid.org/euclid.aop/1528876825_20180613040038Wed, 13 Jun 2018 04:00 EDTIndistinguishability of the components of random spanning forestshttps://projecteuclid.org/euclid.aop/1528876826<strong>Ádám Timár</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 2221--2242.</p><p><strong>Abstract:</strong><br/>
We prove that the infinite components of the Free Uniform Spanning Forest (FUSF) of a Cayley graph are indistinguishable by any invariant property, given that the forest is different from its wired counterpart. Similar result is obtained for the Free Minimal Spanning Forest (FMSF). We also show that with the above assumptions there can only be 0, 1 or infinitely many components, which solves the problem for the FUSF of Caylay graphs completely. These answer questions by Benjamini, Lyons, Peres and Schramm for Cayley graphs, which have been open up to now. Our methods apply to a more general class of percolations, those satisfying “weak insertion tolerance”, and work beyond Cayley graphs, in the more general setting of unimodular random graphs.
</p>projecteuclid.org/euclid.aop/1528876826_20180613040038Wed, 13 Jun 2018 04:00 EDTWeak symmetric integrals with respect to the fractional Brownian motionhttps://projecteuclid.org/euclid.aop/1528876827<strong>Giulia Binotto</strong>, <strong>Ivan Nourdin</strong>, <strong>David Nualart</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 2243--2267.</p><p><strong>Abstract:</strong><br/>
The aim of this paper is to establish the weak convergence, in the topology of the Skorohod space, of the $\nu$-symmetric Riemann sums for functionals of the fractional Brownian motion when the Hurst parameter takes the critical value $H=(4\ell+2)^{-1}$, where $\ell=\ell(\nu)\geq1$ is the largest natural number satisfying $\int_{0}^{1}\alpha^{2j}\nu(d\alpha)=\frac{1}{2j+1}$ for all $j=0,\ldots,\ell-1$. As a consequence, we derive a change-of-variable formula in distribution, where the correction term is a stochastic integral with respect to a Brownian motion that is independent of the fractional Brownian motion.
</p>projecteuclid.org/euclid.aop/1528876827_20180613040038Wed, 13 Jun 2018 04:00 EDTOn the spectral radius of a random matrix: An upper bound without fourth momenthttps://projecteuclid.org/euclid.aop/1528876828<strong>Charles Bordenave</strong>, <strong>Pietro Caputo</strong>, <strong>Djalil Chafaï</strong>, <strong>Konstantin Tikhomirov</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 2268--2286.</p><p><strong>Abstract:</strong><br/>
Consider a square matrix with independent and identically distributed entries of zero mean and unit variance. It is well known that if the entries have a finite fourth moment, then, in high dimension, with high probability, the spectral radius is close to the square root of the dimension. We conjecture that this holds true under the sole assumption of zero mean and unit variance. In other words, that there are no outliers in the circular law. In this work, we establish the conjecture in the case of symmetrically distributed entries with a finite moment of order larger than two. The proof uses the method of moments combined with a novel truncation technique for cycle weights that might be of independent interest.
</p>projecteuclid.org/euclid.aop/1528876828_20180613040038Wed, 13 Jun 2018 04:00 EDTStochastic Airy semigroup through tridiagonal matriceshttps://projecteuclid.org/euclid.aop/1528876829<strong>Vadim Gorin</strong>, <strong>Mykhaylo Shkolnikov</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 2287--2344.</p><p><strong>Abstract:</strong><br/>
We determine the operator limit for large powers of random symmetric tridiagonal matrices as the size of the matrix grows. The result provides a novel expression in terms of functionals of Brownian motions for the Laplace transform of the Airy$_{\beta}$ process, which describes the largest eigenvalues in the $\beta$ ensembles of random matrix theory. Another consequence is a Feynman–Kac formula for the stochastic Airy operator of Edelman–Sutton and Ramirez–Rider–Virag.
As a side result, we find that the difference between the area underneath a standard Brownian excursion and one half of the integral of its squared local times is a Gaussian random variable.
</p>projecteuclid.org/euclid.aop/1528876829_20180613040038Wed, 13 Jun 2018 04:00 EDTOn the mixing time of Kac’s walk and other high-dimensional Gibbs samplers with constraintshttps://projecteuclid.org/euclid.aop/1528876830<strong>Natesh S. Pillai</strong>, <strong>Aaron Smith</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 2345--2399.</p><p><strong>Abstract:</strong><br/>
Determining the total variation mixing time of Kac’s random walk on the special orthogonal group $\mathrm{SO}(n)$ has been a long-standing open problem. In this paper, we construct a novel non-Markovian coupling for bounding this mixing time. The analysis of our coupling entails controlling the smallest singular value of a certain random matrix with highly dependent entries. The dependence of the entries in our matrix makes it not amenable to existing techniques in random matrix theory. To circumvent this difficulty, we extend some recent bounds on the smallest singular values of matrices with independent entries to our setting. These bounds imply that the mixing time of Kac’s walk on the group $\mathrm{SO}(n)$ is between $C_{1}n^{2}$ and $C_{2}n^{4}\log(n)$ for some explicit constants $0<C_{1},C_{2}<\infty$, substantially improving on the bound of $O(n^{5}\log(n)^{2})$ in the preprint of Jiang [Jiang (2012)]. Our methods may also be applied to other high dimensional Gibbs samplers with constraints, and thus are of independent interest. In addition to giving analytical bounds on the mixing time, our approach allows us to compute rigorous estimates of the mixing time by simulating the eigenvalues of a random matrix.
</p>projecteuclid.org/euclid.aop/1528876830_20180613040038Wed, 13 Jun 2018 04:00 EDTErrata to “Distance covariance in metric spaces”https://projecteuclid.org/euclid.aop/1528876831<strong>Russell Lyons</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 4, 2400--2405.</p><p><strong>Abstract:</strong><br/>
We correct several statements and proofs in our paper, Ann. Probab. 41 , no. 5 (2013), 3284–3305.
</p>projecteuclid.org/euclid.aop/1528876831_20180613040038Wed, 13 Jun 2018 04:00 EDTRoots of random polynomials with coefficients of polynomial growthhttps://projecteuclid.org/euclid.aop/1535097632<strong>Yen Do</strong>, <strong>Oanh Nguyen</strong>, <strong>Van Vu</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 5, 2407--2494.</p><p><strong>Abstract:</strong><br/>
In this paper, we prove optimal local universality for roots of random polynomials with arbitrary coefficients of polynomial growth. As an application, we derive, for the first time, sharp estimates for the number of real roots of these polynomials, even when the coefficients are not explicit. Our results also hold for series; in particular, we prove local universality for random hyperbolic series.
</p>projecteuclid.org/euclid.aop/1535097632_20180824040102Fri, 24 Aug 2018 04:01 EDTWell-posedness and regularity for quasilinear degenerate parabolic-hyperbolic SPDEhttps://projecteuclid.org/euclid.aop/1535097633<strong>Benjamin Gess</strong>, <strong>Martina Hofmanová</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 5, 2495--2544.</p><p><strong>Abstract:</strong><br/>
We study quasilinear degenerate parabolic-hyperbolic stochastic partial differential equations with general multiplicative noise within the framework of kinetic solutions. Our results are twofold: First, we establish new regularity results based on averaging techniques. Second, we prove the existence and uniqueness of solutions in a full $L^{1}$ setting requiring no growth assumptions on the nonlinearities. In addition, we prove a comparison result and an $L^{1}$-contraction property for the solutions, generalizing the results obtained in [ Ann. Probab. 44 (2016) 1916–1955].
</p>projecteuclid.org/euclid.aop/1535097633_20180824040102Fri, 24 Aug 2018 04:01 EDTThree favorite sites occurs infinitely often for one-dimensional simple random walkhttps://projecteuclid.org/euclid.aop/1535097634<strong>Jian Ding</strong>, <strong>Jianfei Shen</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 5, 2545--2561.</p><p><strong>Abstract:</strong><br/>
For a one-dimensional simple random walk $(S_{t})$, for each time $t$ we say a site $x$ is a favorite site if it has the maximal local time. In this paper, we show that with probability 1 three favorite sites occurs infinitely often. Our work is inspired by Tóth [ Ann. Probab. 29 (2001) 484–503], and disproves a conjecture of Erdős and Révész [In Mathematical Structure—Computational Mathematics—Mathematical Modelling 2 (1984) 152–157] and of Tóth [ Ann. Probab. 29 (2001) 484–503].
</p>projecteuclid.org/euclid.aop/1535097634_20180824040102Fri, 24 Aug 2018 04:01 EDTZigzag diagrams and Martin boundaryhttps://projecteuclid.org/euclid.aop/1535097635<strong>Pierre Tarrago</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 5, 2562--2620.</p><p><strong>Abstract:</strong><br/>
We investigate the asymptotic behavior of random paths on a graded graph which describes the subword order for words in two letters. This graph, denoted by $\mathcal{Z}$, has been introduced by Viennot, who also discovered a remarkable bijection between paths on $\mathcal{Z}$ and sequences of permutations. Later on, Gnedin and Olshanski used this bijection to describe the set of Gibbs measures on this graph. Both authors also conjectured that the Martin boundary of $\mathcal{Z}$ should coincide with its minimal boundary. We give here a proof of this conjecture by describing the distribution of a large random path conditioned on having a prescribed endpoint. We also relate paths on the graph $\mathcal{Z}$ with paths on the Young lattice, and we finally give a central limit theorem for the Plancherel measure on the set of paths in $\mathcal{Z}$.
</p>projecteuclid.org/euclid.aop/1535097635_20180824040102Fri, 24 Aug 2018 04:01 EDTParacontrolled distributions and the 3-dimensional stochastic quantization equationhttps://projecteuclid.org/euclid.aop/1535097636<strong>Rémi Catellier</strong>, <strong>Khalil Chouk</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 5, 2621--2679.</p><p><strong>Abstract:</strong><br/>
We prove the existence and uniqueness of a local in time solution to the periodic $\Phi^{4}_{3}$ model of stochastic quantisation using the method of paracontrolled distributions introduced recently by M. Gubinelli, P. Imkeller and N. Perkowski in [ Forum Math., Pi 3 (2015) e6].
</p>projecteuclid.org/euclid.aop/1535097636_20180824040102Fri, 24 Aug 2018 04:01 EDTStable random fields indexed by finitely generated free groupshttps://projecteuclid.org/euclid.aop/1535097637<strong>Sourav Sarkar</strong>, <strong>Parthanil Roy</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 5, 2680--2714.</p><p><strong>Abstract:</strong><br/>
In this work, we investigate the extremal behaviour of left-stationary symmetric $\alpha$-stable (S$\alpha$S) random fields indexed by finitely generated free groups. We begin by studying the rate of growth of a sequence of partial maxima obtained by varying the indexing parameter of the field over balls of increasing size. This leads to a phase-transition that depends on the ergodic properties of the underlying nonsingular action of the free group but is different from what happens in the case of S$\alpha$S random fields indexed by $\mathbb{Z}^{d}$. The presence of this new dichotomy is confirmed by the study of a stable random field induced by the canonical action of the free group on its Furstenberg–Poisson boundary with the measure being Patterson–Sullivan. This field is generated by a conservative action but its maxima sequence grows as fast as the i.i.d. case contrary to what happens in the case of $\mathbb{Z}^{d}$. When the action of the free group is dissipative, we also establish that the scaled extremal point process sequence converges weakly to a novel class of point processes that we have termed as randomly thinned cluster Poisson processes . This limit too is very different from that in the case of a lattice.
</p>projecteuclid.org/euclid.aop/1535097637_20180824040102Fri, 24 Aug 2018 04:01 EDTRecursive construction of continuum random treeshttps://projecteuclid.org/euclid.aop/1535097638<strong>Franz Rembart</strong>, <strong>Matthias Winkel</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 5, 2715--2748.</p><p><strong>Abstract:</strong><br/>
We introduce a general recursive method to construct continuum random trees (CRTs) from independent copies of a random string of beads, that is, any random interval equipped with a random discrete probability measure, and from related structures. We prove the existence of these CRTs as a new application of the fixpoint method for recursive distribution equations formalised in high generality by Aldous and Bandyopadhyay.
We apply this recursive method to show the convergence to CRTs of various tree growth processes. We note alternative constructions of existing self-similar CRTs in the sense of Haas, Miermont and Stephenson, and we give for the first time constructions of random compact $\mathbb{R}$-trees that describe the genealogies of Bertoin’s self-similar growth fragmentations. In forthcoming work, we develop further applications to embedding problems for CRTs, providing a binary embedding of the stable line-breaking construction that solves an open problem of Goldschmidt and Haas.
</p>projecteuclid.org/euclid.aop/1535097638_20180824040102Fri, 24 Aug 2018 04:01 EDTControlled equilibrium selection in stochastically perturbed dynamicshttps://projecteuclid.org/euclid.aop/1535097639<strong>Ari Arapostathis</strong>, <strong>Anup Biswas</strong>, <strong>Vivek S. Borkar</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 5, 2749--2799.</p><p><strong>Abstract:</strong><br/>
We consider a dynamical system with finitely many equilibria and perturbed by small noise, in addition to being controlled by an “expensive” control. The controlled process is optimal for an ergodic criterion with a running cost that consists of the sum of the control effort and a penalty function on the state space. We study the optimal stationary distribution of the controlled process as the variance of the noise becomes vanishingly small. It is shown that depending on the relative magnitudes of the noise variance and the “running cost” for control, one can identify three regimes, in each of which the optimal control forces the invariant distribution of the process to concentrate near equilibria that can be characterized according to the regime. We also obtain moment bounds for the optimal stationary distribution. Moreover, we show that in the vicinity of the points of concentration the density of optimal stationary distribution approximates the density of a Gaussian, and we explicitly solve for its covariance matrix.
</p>projecteuclid.org/euclid.aop/1535097639_20180824040102Fri, 24 Aug 2018 04:01 EDTA new look at duality for the symbiotic branching modelhttps://projecteuclid.org/euclid.aop/1535097640<strong>Matthias Hammer</strong>, <strong>Marcel Ortgiese</strong>, <strong>Florian Völlering</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 5, 2800--2862.</p><p><strong>Abstract:</strong><br/>
The symbiotic branching model is a spatial population model describing the dynamics of two interacting types that can only branch if both types are present. A classical result for the underlying stochastic partial differential equation identifies moments of the solution via a duality to a system of Brownian motions with dynamically changing colors. In this paper, we revisit this duality and give it a new interpretation. This new approach allows us to extend the duality to the limit as the branching rate $\gamma$ is sent to infinity. This limit is particularly interesting since it captures the large scale behavior of the system. As an application of the duality, we can explicitly identify the $\gamma=\infty$ limit when the driving noises are perfectly negatively correlated. The limit is a system of annihilating Brownian motions with a drift that depends on the initial imbalance between the types.
</p>projecteuclid.org/euclid.aop/1535097640_20180824040102Fri, 24 Aug 2018 04:01 EDTAlternating arm exponents for the critical planar Ising modelhttps://projecteuclid.org/euclid.aop/1535097641<strong>Hao Wu</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 5, 2863--2907.</p><p><strong>Abstract:</strong><br/>
We derive the alternating arm exponents of the critical Ising model. We obtain six different patterns of alternating boundary arm exponents which correspond to the boundary conditions $(\ominus\oplus)$, $(\ominus\operatorname{free})$ and $(\operatorname{free}\operatorname{free})$, and the alternating interior arm exponents.
</p>projecteuclid.org/euclid.aop/1535097641_20180824040102Fri, 24 Aug 2018 04:01 EDTGaussian mixtures: Entropy and geometric inequalitieshttps://projecteuclid.org/euclid.aop/1535097642<strong>Alexandros Eskenazis</strong>, <strong>Piotr Nayar</strong>, <strong>Tomasz Tkocz</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 5, 2908--2945.</p><p><strong>Abstract:</strong><br/>
A symmetric random variable is called a Gaussian mixture if it has the same distribution as the product of two independent random variables, one being positive and the other a standard Gaussian random variable. Examples of Gaussian mixtures include random variables with densities proportional to $e^{-|t|^{p}}$ and symmetric $p$-stable random variables, where $p\in(0,2]$. We obtain various sharp moment and entropy comparison estimates for weighted sums of independent Gaussian mixtures and investigate extensions of the B-inequality and the Gaussian correlation inequality in the context of Gaussian mixtures. We also obtain a correlation inequality for symmetric geodesically convex sets in the unit sphere equipped with the normalized surface area measure. We then apply these results to derive sharp constants in Khinchine inequalities for vectors uniformly distributed on the unit balls with respect to $p$-norms and provide short proofs to new and old comparison estimates for geometric parameters of sections and projections of such balls.
</p>projecteuclid.org/euclid.aop/1535097642_20180824040102Fri, 24 Aug 2018 04:01 EDTThe survival probability of a critical multi-type branching process in i.i.d. random environmenthttps://projecteuclid.org/euclid.aop/1535097643<strong>E. Le Page</strong>, <strong>M. Peigné</strong>, <strong>C. Pham</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 5, 2946--2972.</p><p><strong>Abstract:</strong><br/>
We study the asymptotic behaviour of the probability of non-extinction of a critical multi-type Galton–Watson process in i.i.d. random environments by using limit theorems for products of positive random matrices. Under suitable assumptions, the survival probability is proportional to $1/\sqrt{n}$.
</p>projecteuclid.org/euclid.aop/1535097643_20180824040102Fri, 24 Aug 2018 04:01 EDTAiry point process at the liquid-gas boundaryhttps://projecteuclid.org/euclid.aop/1535097644<strong>Vincent Beffara</strong>, <strong>Sunil Chhita</strong>, <strong>Kurt Johansson</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 5, 2973--3013.</p><p><strong>Abstract:</strong><br/>
Domino tilings of the two-periodic Aztec diamond feature all of the three possible types of phases of random tiling models. These phases are determined by the decay of correlations between dominoes and are generally known as solid, liquid and gas. The liquid-solid boundary is easy to define microscopically and is known in many models to be described by the Airy process in the limit of a large random tiling. The liquid-gas boundary has no obvious microscopic description. Using the height function, we define a random measure in the two-periodic Aztec diamond designed to detect the long range correlations visible at the liquid-gas boundary. We prove that this random measure converges to the extended Airy point process. This indicates that, in a sense, the liquid-gas boundary should also be described by the Airy process.
</p>projecteuclid.org/euclid.aop/1535097644_20180824040102Fri, 24 Aug 2018 04:01 EDTPfaffian Schur processes and last passage percolation in a half-quadranthttps://projecteuclid.org/euclid.aop/1537862428<strong>Jinho Baik</strong>, <strong>Guillaume Barraquand</strong>, <strong>Ivan Corwin</strong>, <strong>Toufic Suidan</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 6, 3015--3089.</p><p><strong>Abstract:</strong><br/>
We study last passage percolation in a half-quadrant, which we analyze within the framework of Pfaffian Schur processes. For the model with exponential weights, we prove that the fluctuations of the last passage time to a point on the diagonal are either GSE Tracy–Widom distributed, GOE Tracy–Widom distributed or Gaussian, depending on the size of weights along the diagonal. Away from the diagonal, the fluctuations of passage times follow the GUE Tracy–Widom distribution. We also obtain a two-dimensional crossover between the GUE, GOE and GSE distribution by studying the multipoint distribution of last passage times close to the diagonal when the size of the diagonal weights is simultaneously scaled close to the critical point. We expect that this crossover arises universally in KPZ growth models in half-space. Along the way, we introduce a method to deal with diverging correlation kernels of point processes where points collide in the scaling limit.
</p>projecteuclid.org/euclid.aop/1537862428_20180925040105Tue, 25 Sep 2018 04:01 EDTPathwise uniqueness of the stochastic heat equation with spatially inhomogeneous white noisehttps://projecteuclid.org/euclid.aop/1537862429<strong>Eyal Neuman</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 6, 3090--3187.</p><p><strong>Abstract:</strong><br/>
We study the solutions of the stochastic heat equation driven by spatially inhomogeneous multiplicative white noise based on a fractal measure. We prove pathwise uniqueness for solutions of this equation when the noise coefficient is Hölder continuous of index $\gamma>1-\frac{\eta}{2(\eta+1)}$. Here $\eta\in(0,1)$ is a constant that defines the spatial regularity of the noise.
</p>projecteuclid.org/euclid.aop/1537862429_20180925040105Tue, 25 Sep 2018 04:01 EDTA quantitative central limit theorem for the Euler–Poincaré characteristic of random spherical eigenfunctionshttps://projecteuclid.org/euclid.aop/1537862432<strong>Valentina Cammarota</strong>, <strong>Domenico Marinucci</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 6, 3188--3228.</p><p><strong>Abstract:</strong><br/>
We establish here a quantitative central limit theorem (in Wasserstein distance) for the Euler–Poincaré characteristic of excursion sets of random spherical eigenfunctions in dimension 2. Our proof is based upon a decomposition of the Euler–Poincaré characteristic into different Wiener-chaos components: we prove that its asymptotic behaviour is dominated by a single term, corresponding to the chaotic component of order two. As a consequence, we show how the asymptotic dependence on the threshold level $u$ is fully degenerate, that is, the Euler–Poincaré characteristic converges to a single random variable times a deterministic function of the threshold. This deterministic function has a zero at the origin, where the variance is thus asymptotically of smaller order. We discuss also a possible unifying framework for the Lipschitz–Killing curvatures of the excursion sets for Gaussian spherical harmonics.
</p>projecteuclid.org/euclid.aop/1537862432_20180925040105Tue, 25 Sep 2018 04:01 EDTRepresentations and isomorphism identities for infinitely divisible processeshttps://projecteuclid.org/euclid.aop/1537862433<strong>Jan Rosiński</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 6, 3229--3274.</p><p><strong>Abstract:</strong><br/>
We propose isomorphism-type identities for nonlinear functionals of general infinitely divisible processes. Such identities can be viewed as an analogy of the Cameron–Martin formula for Poissonian infinitely divisible processes but with random translations. The applicability of such tools relies on precise understanding of Lévy measures of infinitely divisible processes and their representations, which are studied here in full generality. We illustrate this approach on examples of squared Bessel processes, Feller diffusions, permanental processes, as well as Lévy processes.
</p>projecteuclid.org/euclid.aop/1537862433_20180925040105Tue, 25 Sep 2018 04:01 EDTCoupling in the Heisenberg group and its applications to gradient estimateshttps://projecteuclid.org/euclid.aop/1537862434<strong>Sayan Banerjee</strong>, <strong>Maria Gordina</strong>, <strong>Phanuel Mariano</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 6, 3275--3312.</p><p><strong>Abstract:</strong><br/>
We construct a non-Markovian coupling for hypoelliptic diffusions which are Brownian motions in the three-dimensional Heisenberg group. We then derive properties of this coupling such as estimates on the coupling rate, and upper and lower bounds on the total variation distance between the laws of the Brownian motions. Finally, we use these properties to prove gradient estimates for harmonic functions for the hypoelliptic Laplacian which is the generator of Brownian motion in the Heisenberg group.
</p>projecteuclid.org/euclid.aop/1537862434_20180925040105Tue, 25 Sep 2018 04:01 EDTFirst-passage times for random walks with nonidentically distributed incrementshttps://projecteuclid.org/euclid.aop/1537862435<strong>Denis Denisov</strong>, <strong>Alexander Sakhanenko</strong>, <strong>Vitali Wachtel</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 6, 3313--3350.</p><p><strong>Abstract:</strong><br/>
We consider random walks with independent but not necessarily identical distributed increments. Assuming that the increments satisfy the well-known Lindeberg condition, we investigate the asymptotic behaviour of first-passage times over moving boundaries. Furthermore, we prove that a properly rescaled random walk conditioned to stay above the boundary up to time $n$ converges, as $n\to\infty$, towards the Brownian meander.
</p>projecteuclid.org/euclid.aop/1537862435_20180925040105Tue, 25 Sep 2018 04:01 EDTCanonical supermartingale couplingshttps://projecteuclid.org/euclid.aop/1537862436<strong>Marcel Nutz</strong>, <strong>Florian Stebegg</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 6, 3351--3398.</p><p><strong>Abstract:</strong><br/>
Two probability distributions $\mu$ and $\nu$ in second stochastic order can be coupled by a supermartingale, and in fact by many. Is there a canonical choice? We construct and investigate two couplings which arise as optimizers for constrained Monge–Kantorovich optimal transport problems where only supermartingales are allowed as transports. Much like the Hoeffding–Fréchet coupling of classical transport and its symmetric counterpart, the antitone coupling, these can be characterized by order-theoretic minimality properties, as simultaneous optimal transports for certain classes of reward (or cost) functions, and through no-crossing conditions on their supports; however, our two couplings have asymmetric geometries. Remarkably, supermartingale optimal transport decomposes into classical and martingale transport in several ways.
</p>projecteuclid.org/euclid.aop/1537862436_20180925040105Tue, 25 Sep 2018 04:01 EDTA weak version of path-dependent functional Itô calculushttps://projecteuclid.org/euclid.aop/1537862437<strong>Dorival Leão</strong>, <strong>Alberto Ohashi</strong>, <strong>Alexandre B. Simas</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 6, 3399--3441.</p><p><strong>Abstract:</strong><br/>
We introduce a variational theory for processes adapted to the multidimensional Brownian motion filtration that provides a differential structure allowing to describe infinitesimal evolution of Wiener functionals at very small scales. The main novel idea is to compute the “sensitivities” of processes, namely derivatives of martingale components and a weak notion of infinitesimal generator, via a finite-dimensional approximation procedure based on controlled inter-arrival times and approximating martingales. The theory comes with convergence results that allow to interpret a large class of Wiener functionals beyond semimartingales as limiting objects of differential forms which can be computed path wisely over finite-dimensional spaces. The theory reveals that solutions of BSDEs are minimizers of energy functionals w.r.t. Brownian motion driving noise.
</p>projecteuclid.org/euclid.aop/1537862437_20180925040105Tue, 25 Sep 2018 04:01 EDTLower bounds for the smallest singular value of structured random matriceshttps://projecteuclid.org/euclid.aop/1537862438<strong>Nicholas Cook</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 6, 3442--3500.</p><p><strong>Abstract:</strong><br/>
We obtain lower tail estimates for the smallest singular value of random matrices with independent but nonidentically distributed entries. Specifically, we consider $n\times n$ matrices with complex entries of the form
\[M=A\circ X+B=(a_{ij}\xi_{ij}+b_{ij}),\] where $X=(\xi_{ij})$ has i.i.d. centered entries of unit variance and $A$ and $B$ are fixed matrices. In our main result, we obtain polynomial bounds on the smallest singular value of $M$ for the case that $A$ has bounded (possibly zero) entries, and $B=Z\sqrt{n}$ where $Z$ is a diagonal matrix with entries bounded away from zero. As a byproduct of our methods we can also handle general perturbations $B$ under additional hypotheses on $A$, which translate to connectivity hypotheses on an associated graph. In particular, we extend a result of Rudelson and Zeitouni for Gaussian matrices to allow for general entry distributions satisfying some moment hypotheses. Our proofs make use of tools which (to our knowledge) were previously unexploited in random matrix theory, in particular Szemerédi’s regularity lemma, and a version of the restricted invertibility theorem due to Spielman and Srivastava.
</p>projecteuclid.org/euclid.aop/1537862438_20180925040105Tue, 25 Sep 2018 04:01 EDTThe scaling limits of the Minimal Spanning Tree and Invasion Percolation in the planehttps://projecteuclid.org/euclid.aop/1537862439<strong>Christophe Garban</strong>, <strong>Gábor Pete</strong>, <strong>Oded Schramm</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 6, 3501--3557.</p><p><strong>Abstract:</strong><br/>
We prove that the Minimal Spanning Tree and the Invasion Percolation Tree on a version of the triangular lattice in the complex plane have unique scaling limits, which are invariant under rotations, scalings, and, in the case of the $\mathsf{MST}$, also under translations. However, they are not expected to be conformally invariant. We also prove some geometric properties of the limiting $\mathsf{MST}$. The topology of convergence is the space of spanning trees introduced by Aizenman et al. [ Random Structures Algorithms 15 (1999) 319–365], and the proof relies on the existence and conformal covariance of the scaling limit of the near-critical percolation ensemble, established in our earlier works.
</p>projecteuclid.org/euclid.aop/1537862439_20180925040105Tue, 25 Sep 2018 04:01 EDTQuenched central limit theorem for random walks in doubly stochastic random environmenthttps://projecteuclid.org/euclid.aop/1537862440<strong>Bálint Tóth</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 6, 3558--3577.</p><p><strong>Abstract:</strong><br/>
We prove the quenched version of the central limit theorem for the displacement of a random walk in doubly stochastic random environment, under the $H_{-1}$-condition, with slightly stronger, $\mathcal{L}^{2+\varepsilon}$ (rather than $\mathcal{L}^{2}$) integrability condition on the stream tensor. On the way we extend Nash’s moment bound to the nonreversible, divergence-free drift case, with unbounded ($\mathcal{L}^{2+\varepsilon}$) stream tensor. This paper is a sequel of [ Ann. Probab. 45 (2017) 4307–4347] and relies on technical results quoted from there.
</p>projecteuclid.org/euclid.aop/1537862440_20180925040105Tue, 25 Sep 2018 04:01 EDTThe KLS isoperimetric conjecture for generalized Orlicz ballshttps://projecteuclid.org/euclid.aop/1537862441<strong>Alexander V. Kolesnikov</strong>, <strong>Emanuel Milman</strong>. <p><strong>Source: </strong>The Annals of Probability, Volume 46, Number 6, 3578--3615.</p><p><strong>Abstract:</strong><br/>
What is the optimal way to cut a convex bounded domain $K$ in Euclidean space $(\mathbb{R}^{n},|\cdot|)$ into two halves of equal volume, so that the interface between the two halves has least surface area? A conjecture of Kannan, Lovász and Simonovits asserts that, if one does not mind gaining a universal numerical factor (independent of $n$) in the surface area, one might as well dissect $K$ using a hyperplane. This conjectured essential equivalence between the former nonlinear isoperimetric inequality and its latter linear relaxation, has been shown over the last two decades to be of fundamental importance to the understanding of volume-concentration and spectral properties of convex domains. In this work, we address the conjecture for the subclass of generalized Orlicz balls
\[K=\{x\in\mathbb{R}^{n};\sum_{i=1}^{n}V_{i}(x_{i})\leq E\},\] confirming its validity for certain levels $E\in\mathbb{R}$ under a mild technical assumption on the growth of the convex functions $V_{i}$ at infinity [without which we confirm the conjecture up to a $\log(1+n)$ factor]. In sharp contrast to previous approaches for tackling the KLS conjecture, we emphasize that no symmetry is required from $K$. This significantly enlarges the subclass of convex bodies for which the conjecture is confirmed.
</p>projecteuclid.org/euclid.aop/1537862441_20180925040105Tue, 25 Sep 2018 04:01 EDT