Mostrando recursos 1 - 20 de 111

  1. Stochastic control for a class of nonlinear kernels and applications

    Possamaï, Dylan; Tan, Xiaolu; Zhou, Chao
    We consider a stochastic control problem for a class of nonlinear kernels. More precisely, our problem of interest consists in the optimization, over a set of possibly nondominated probability measures, of solutions of backward stochastic differential equations (BSDEs). Since BSDEs are nonlinear generalizations of the traditional (linear) expectations, this problem can be understood as stochastic control of a family of nonlinear expectations, or equivalently of nonlinear kernels. Our first main contribution is to prove a dynamic programming principle for this control problem in an abstract setting, which we then use to provide a semimartingale characterization of the value function. We...

  2. A class of globally solvable Markovian quadratic BSDE systems and applications

    Xing, Hao; Žitković, Gordan
    We establish existence and uniqueness for a wide class of Markovian systems of backward stochastic differential equations (BSDE) with quadratic nonlinearities. This class is characterized by an abstract structural assumption on the generator, an a priori local-boundedness property, and a locally-Hölder-continuous terminal condition. We present easily verifiable sufficient conditions for these assumptions and treat several applications, including stochastic equilibria in incomplete financial markets, stochastic differential games and martingales on Riemannian manifolds.

  3. Random walks on the random graph

    Berestycki, Nathanaël; Lubetzky, Eyal; Peres, Yuval; Sly, Allan
    We study random walks on the giant component of the Erdős–Rényi random graph $\mathcal{G}(n,p)$ where $p=\lambda/n$ for $\lambda>1$ fixed. The mixing time from a worst starting point was shown by Fountoulakis and Reed, and independently by Benjamini, Kozma and Wormald, to have order $\log^{2}n$. We prove that starting from a uniform vertex (equivalently, from a fixed vertex conditioned to belong to the giant) both accelerates mixing to $O(\log n)$ and concentrates it (the cutoff phenomenon occurs): the typical mixing is at $(\nu\mathbf{d})^{-1}\log n\pm(\log n)^{1/2+o(1)}$, where $\nu$ and $\mathbf{d}$ are the speed of random walk and dimension of harmonic measure on...

  4. Lattice approximation to the dynamical $\Phi_{3}^{4}$ model

    Zhu, Rongchan; Zhu, Xiangchan
    We study the lattice approximations to the dynamical $\Phi^{4}_{3}$ model by paracontrolled distributions proposed in [Forum Math. Pi 3 (2015) e6]. We prove that the solutions to the lattice systems converge to the solution to the dynamical $\Phi_{3}^{4}$ model in probability, locally uniformly in time. Since the dynamical $\Phi_{3}^{4}$ model is not well defined in the classical sense and renormalisation has to be performed in order to define the nonlinear term, a corresponding suitable drift term is added in the stochastic equations for the lattice systems.

  5. An $L^{p}$ theory of sparse graph convergence II: LD convergence, quotients and right convergence

    Borgs, Christian; Chayes, Jennifer T.; Cohn, Henry; Zhao, Yufei
    We extend the $L^{p}$ theory of sparse graph limits, which was introduced in a companion paper, by analyzing different notions of convergence. Under suitable restrictions on node weights, we prove the equivalence of metric convergence, quotient convergence, microcanonical ground state energy convergence, microcanonical free energy convergence and large deviation convergence. Our theorems extend the broad applicability of dense graph convergence to all sparse graphs with unbounded average degree, while the proofs require new techniques based on uniform upper regularity. Examples to which our theory applies include stochastic block models, power law graphs and sparse versions of $W$-random graphs.

  6. Quenched invariance principle for random walks with time-dependent ergodic degenerate weights

    Andres, Sebastian; Chiarini, Alberto; Deuschel, Jean-Dominique; Slowik, Martin
    We study a continuous-time random walk, $X$, on $\mathbb{Z}^{d}$ in an environment of dynamic random conductances taking values in $(0,\infty)$. We assume that the law of the conductances is ergodic with respect to space–time shifts. We prove a quenched invariance principle for the Markov process $X$ under some moment conditions on the environment. The key result on the sublinearity of the corrector is obtained by Moser’s iteration scheme.

  7. Dimensional improvements of the logarithmic Sobolev, Talagrand and Brascamp–Lieb inequalities

    Bolley, François; Gentil, Ivan; Guillin, Arnaud
    In this work, we consider dimensional improvements of the logarithmic Sobolev, Talagrand and Brascamp–Lieb inequalities. For this, we use optimal transport methods and the Borell–Brascamp–Lieb inequality. These refinements can be written as a deficit in the classical inequalities. They have the right scale with respect to the dimension. They lead to sharpened concentration properties as well as refined contraction bounds, convergence to equilibrium and short time behavior for the laws of solutions to stochastic differential equations.

  8. Random planar maps and growth-fragmentations

    Bertoin, Jean; Curien, Nicolas; Kortchemski, Igor
    We are interested in the cycles obtained by slicing at all heights random Boltzmann triangulations with a simple boundary. We establish a functional invariance principle for the lengths of these cycles, appropriately rescaled, as the size of the boundary grows. The limiting process is described using a self-similar growth-fragmentation process with explicit parameters. To this end, we introduce a branching peeling exploration of Boltzmann triangulations, which allows us to identify a crucial martingale involving the perimeters of cycles at given heights. We also use a recent result concerning self-similar scaling limits of Markov chains on the nonnegative integers. A motivation...

  9. An upper bound on the number of self-avoiding polygons via joining

    Hammond, Alan
    For $d\geq2$ and $n\in\mathbb{N}$ even, let $p_{n}=p_{n}(d)$ denote the number of length $n$ self-avoiding polygons in $\mathbb{Z}^{d}$ up to translation. The polygon cardinality grows exponentially, and the growth rate $\lim_{n\in2\mathbb{N}}p_{n}^{1/n}\in(0,\infty)$ is called the connective constant and denoted by $\mu$. Madras [J. Stat. Phys. 78 (1995) 681–699] has shown that $p_{n}\mu^{-n}\leq Cn^{-1/2}$ in dimension $d=2$. Here, we establish that $p_{n}\mu^{-n}\leq n^{-3/2+o(1)}$ for a set of even $n$ of full density when $d=2$. We also consider a certain variant of self-avoiding walk and argue that, when $d\geq3$, an upper bound of $n^{-2+d^{-1}+o(1)}$ holds on a full density set for the counterpart in...

  10. Path-dependent equations and viscosity solutions in infinite dimension

    Cosso, Andrea; Federico, Salvatore; Gozzi, Fausto; Rosestolato, Mauro; Touzi, Nizar
    Path-dependent partial differential equations (PPDEs) are natural objects to study when one deals with non-Markovian models. Recently, after the introduction of the so-called pathwise (or functional or Dupire) calculus [see Dupire (2009)], in the case of finite-dimensional underlying space various papers have been devoted to studying the well-posedness of such kind of equations, both from the point of view of regular solutions [see, e.g., Dupire (2009) and Cont (2016) Stochastic Integration by Parts and Functional Itô Calculus 115–207, Birkhäuser] and viscosity solutions [see, e.g., Ekren et al. (2014) Ann. Probab. 42 204–236]. In this paper, motivated by the study of...

  11. Size biased couplings and the spectral gap for random regular graphs

    Cook, Nicholas; Goldstein, Larry; Johnson, Tobias
    Let $\lambda$ be the second largest eigenvalue in absolute value of a uniform random $d$-regular graph on $n$ vertices. It was famously conjectured by Alon and proved by Friedman that if $d$ is fixed independent of $n$, then $\lambda=2\sqrt{d-1}+o(1)$ with high probability. In the present work, we show that $\lambda=O(\sqrt{d})$ continues to hold with high probability as long as $d=O(n^{2/3})$, making progress toward a conjecture of Vu that the bound holds for all $1\le d\le n/2$. Prior to this work the best result was obtained by Broder, Frieze, Suen and Upfal (1999) using the configuration model, which hits a barrier...

  12. Nonbacktracking spectrum of random graphs: Community detection and nonregular Ramanujan graphs

    Bordenave, Charles; Lelarge, Marc; Massoulié, Laurent
    A nonbacktracking walk on a graph is a directed path such that no edge is the inverse of its preceding edge. The nonbacktracking matrix of a graph is indexed by its directed edges and can be used to count nonbacktracking walks of a given length. It has been used recently in the context of community detection and has appeared previously in connection with the Ihara zeta function and in some generalizations of Ramanujan graphs. In this work, we study the largest eigenvalues of the nonbacktracking matrix of the Erdős–Rényi random graph and of the stochastic block model in the regime...

  13. Existence conditions of permanental and multivariate negative binomial distributions

    Eisenbaum, Nathalie; Maunoury, Franck
    Existence conditions of permanental distributions are deeply connected to existence conditions of multivariate negative binomial distributions. The aim of this paper is twofold. It answers several questions generated by recent works on this subject, but it also goes back to the roots of this field and fixes existing gaps in older papers concerning conditions of infinite divisibility for these distributions.

  14. Existence conditions of permanental and multivariate negative binomial distributions

    Eisenbaum, Nathalie; Maunoury, Franck
    Existence conditions of permanental distributions are deeply connected to existence conditions of multivariate negative binomial distributions. The aim of this paper is twofold. It answers several questions generated by recent works on this subject, but it also goes back to the roots of this field and fixes existing gaps in older papers concerning conditions of infinite divisibility for these distributions.

  15. Existence conditions of permanental and multivariate negative binomial distributions

    Eisenbaum, Nathalie; Maunoury, Franck
    Existence conditions of permanental distributions are deeply connected to existence conditions of multivariate negative binomial distributions. The aim of this paper is twofold. It answers several questions generated by recent works on this subject, but it also goes back to the roots of this field and fixes existing gaps in older papers concerning conditions of infinite divisibility for these distributions.

  16. Existence conditions of permanental and multivariate negative binomial distributions

    Eisenbaum, Nathalie; Maunoury, Franck
    Existence conditions of permanental distributions are deeply connected to existence conditions of multivariate negative binomial distributions. The aim of this paper is twofold. It answers several questions generated by recent works on this subject, but it also goes back to the roots of this field and fixes existing gaps in older papers concerning conditions of infinite divisibility for these distributions.

  17. The vacant set of two-dimensional critical random interlacement is infinite

    Comets, Francis; Popov, Serguei
    For the model of two-dimensional random interlacements in the critical regime (i.e., $\alpha=1$), we prove that the vacant set is a.s. infinite, thus solving an open problem from [Commun. Math. Phys. 343 (2016) 129–164]. Also, we prove that the entrance measure of simple random walk on annular domains has certain regularity properties; this result is useful when dealing with soft local times for excursion processes.

  18. The vacant set of two-dimensional critical random interlacement is infinite

    Comets, Francis; Popov, Serguei
    For the model of two-dimensional random interlacements in the critical regime (i.e., $\alpha=1$), we prove that the vacant set is a.s. infinite, thus solving an open problem from [Commun. Math. Phys. 343 (2016) 129–164]. Also, we prove that the entrance measure of simple random walk on annular domains has certain regularity properties; this result is useful when dealing with soft local times for excursion processes.

  19. The vacant set of two-dimensional critical random interlacement is infinite

    Comets, Francis; Popov, Serguei
    For the model of two-dimensional random interlacements in the critical regime (i.e., $\alpha=1$), we prove that the vacant set is a.s. infinite, thus solving an open problem from [Commun. Math. Phys. 343 (2016) 129–164]. Also, we prove that the entrance measure of simple random walk on annular domains has certain regularity properties; this result is useful when dealing with soft local times for excursion processes.

  20. The vacant set of two-dimensional critical random interlacement is infinite

    Comets, Francis; Popov, Serguei
    For the model of two-dimensional random interlacements in the critical regime (i.e., $\alpha=1$), we prove that the vacant set is a.s. infinite, thus solving an open problem from [Commun. Math. Phys. 343 (2016) 129–164]. Also, we prove that the entrance measure of simple random walk on annular domains has certain regularity properties; this result is useful when dealing with soft local times for excursion processes.

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