Filters

2006 - 200912010 - 201921

More filters

201922
Reset filters

Settings

End date: 2019 × Start date: 2019 × Type: Text × Type: report × Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik × Subject: large deviations ×

Search Results

Now showing 1 - 10 of 22
  • Thumbnail Image
    Item
    The parabolic Anderson model with acceleration and deceleration
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) König, Wolfgang; Schmidt, Sylvia
    We describe the large-time moment asymptotics for the parabolic Anderson model where the speed of the diffusion is coupled with time, inducing an acceleration or deceleration. We find a lower critical scale, below which the mass flow gets stuck. On this scale, a new interesting variational problem arises in the description of the asymptotics. Furthermore, we find an upper critical scale above which the potential enters the asymptotics only via some average, but not via its extreme values. We make out altogether five phases, three of which can be described by results that are qualitatively similar to those from the constant-speed parabolic Anderson model in earlier work by various authors. Our proofs consist of adaptations and refinements of their methods, as well as a variational convergence method borrowed from finite elements theory.
  • Thumbnail Image
    Item
    Brownian occupation measures, compactness and large deviations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Mukherjee, Chiranjib; Varadhan, S.R. Srinivasa
    In proving large deviation estimates, the lower bound for open sets and upper bound for compact sets are essentially local estimates. On the other hand, the upper bound for closed sets is global and compactness of space or an exponential tightness estimate is needed to establish it. In dealing with the occupation measure $L_t(A)=frac1tint_0^t1_A(W_s) d s$ of the $d$ dimensional Brownian motion, which is not positive recurrent, there is no possibility of exponential tightness. The space of probability distributions $mathcal M_1(R^d)$ can be compactified by replacing the usual topology of weak c onvergence by the vague toplogy, where the space is treated as the dual of continuous functions with compact support. This is essentially the one point compactification of $R^d$ by adding a point at $infty$ that results in the compactification of $mathcal M_1(R^d)$ by allowing some mass to escape to the point at $infty$. If one were to use only test functions that are continuous and vanish at $infty$ then the compactification results in the space of sub-probability distributions $mathcal M_le 1(R^d)$ by ignoring the mass at $infty$. The main drawback of this compactification is that it ignores the underlying translation invariance. More explicitly, we may be interested in the space of equivalence classes of orbits $widetildemathcal M_1=widetildemathcal M_1(R^d)$ under the action of the translation group $R^d$ on $mathcal M_1(R^d)$. There are problems for which it is natural to compactify this space of orbits. We will provide such a compactification, prove a large deviation principle there and give an application to a relevant problem.
  • Thumbnail Image
    Item
    On the relation between gradient flows and the large-deviation principle, with applications to Markov chains and diffusion
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Mielke, Alexander; Peletier, Mark A.; Renger, D.R. Michiel
    Motivated by the occurence in rate functions of time-dependent large-deviation principles, we study a class of non-negative functions L that induce a flow, given by L(pt, pt) = 0. We derive necessary and sufficient conditions for the unique existence of a generalized gradient structure for the induced flow, as well as explicit formulas for the corresponding driving entropy and dissipation functional. In particular, we show how these conditions can be given a probabilistic interpretation when L is associated to the large deviations of a microscopic particle system. Finally, we illustrate the theory for independent Brownian particles with drift, which leads to the entropy-Wasserstein gradient structure, and for independent Markovian particles on a finite state space, which leads to a previously unknown gradient structure.
  • Thumbnail Image
    Item
    Large deviations for the local times of a random walk among random conductances
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) König, Wolfgang; Salvi, Michele; Wolff, Tilman
    We derive an annealed large deviation principle for the normalised local times of a continuous-time random walk among random conductances in a finite domain in $Z^d$ in the spirit of Donsker-Varadhan citeDV75. We work in the interesting case that the conductances may assume arbitrarily small values. Thus, the underlying picture of the principle is a joint strategy of small values of the conductances and large holding times of the walk. The speed and the rate function of our principle are explicit in terms of the lower tails of the conductance distribution. As an application, we identify the logarithmic asymptotics of the lower tails of the principal eigenvalue of the randomly perturbed negative Laplace operator in the domain.
  • Thumbnail Image
    Item
    Large deviations for cluster size distributions in a continuous classical many-body system
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Jansen, Sabine; König, Wolfgang; Metzger, Bernd
    An interesting problem in statistical physics is the condensation of classical particles in droplets or clusters when the pair-interaction is given by a stable Lennard-Jones-type potential. We study two aspects of this problem. We start by deriving a large deviations principle for the cluster size distribution for any inverse temperature $betain(0,infty)$ and particle density $rhoin(0,rho_rmcp)$ in the thermodynamic limit. Here $rho_rmcp >0$ is the close packing density. While in general the rate function is an abstract object, our second main result is the $Gamma$-convergence of the rate function towards an explicit limiting rate function in the low-temperature dilute limit $betatoinfty$, $rho downarrow 0$ such that $-beta^-1logrhoto nu$ for some $nuin(0,infty)$. The limiting rate function and its minimisers appeared in recent work, where the temperature and the particle density were coupled with the particle number. In the de-coupled limit considered here, we prove that just one cluster size is dominant, depending on the parameter $nu$. Under additional assumptions on the potential, the $Gamma$-convergence along curves can be strengthened to uniform bounds, valid in a low-temperature, low-density rectangle.
  • Thumbnail Image
    Item
    Brownian occupation measures, compactness and large deviations: Pair interaction
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Mukherjee, Chiranjib
    Continuing with the study of compactness and large deviations initiated in citeMV14, we turn to the analysis of Gibbs measures defined on two independent Brownian paths in $R^d$ interacting through a mutual self-attraction. This is expressed by the Hamiltonian $intint_R^2d V(x-y) mu(d x)nu(d y)$ with two probability measures $mu$ and $nu$ representing the occupation measures of two independent Brownian motions. Due to the mixed product of two independent measures, the crucial shift-invariance requirement of citeMV14 is slightly lost. However, such a mixed product of measures inspires a compactification of the quotient space of orbits of product measures, which is structurally slightly different from the one introduced in citeMV14. The orbits of the product of independent occupation measures are embedded in such a compactfication and a strong large deviation principle for these objects enables us to prove the desired asymptotic localization properties of the joint behavior of two independent paths under the Gibbs transformation. As a second application, we study the spatially smoothened parabolic Anderson model in $R^d$ with white noise potential and provide a direct computation of the annealed Lyapunov exponents of the smoothened solutions when the smoothing parameter goes to $0$.
  • Thumbnail Image
    Item
    Moment asymptotics for branching random walks in random environment
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Gün, Onur; König, Wolfgang; Sekulov´c, Ozren
    We consider the long-time behaviour of a branching random walk in random environment on the lattice Zd. The migration of particles proceeds according to simple random walk in continuous time, while the medium is given as a random potential of spatially dependent killing/branching rates. The main objects of our interest are the annealed moments m_np , i.e., the p-th moments over the medium of the n-th moment over the migration and killing/branching, of the local and global population sizes. For n = 1, this is well-understood citeGM98, as m_1 is closely connected with the parabolic Anderson model. For some special distributions, citeA00 extended this to ngeq2, but only as to the first term of the asymptotics, using (a recursive version of) a Feynman-Kac formula for m_n. In this work we derive also the second term of the asymptotics, for a much larger class of distributions. In particular, we show that m_n^p m_1^np are asymptotically equal, up to an error e^o(t). The cornerstone of our method is a direct Feynman-Kac-type formula for mn, which we establish using the spine techniques developed in citeHR1.1
  • Thumbnail Image
    Item
    Mean-field interaction of Brownian occupation measures. II: A rigorous construction of the Pekar process
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Bolthausen, Erwin; König, Wolfgang; Mukherjee, Chiranjib
    We consider mean-field interactions corresponding to Gibbs measures on interacting Brownian paths in three dimensions. The interaction is self-attractive and is given by a singular Coulomb potential. The logarithmic asymptotics of the partition function for this model were identified in the 1980s by Donsker and Varadhan [DV83] in terms of the Pekar variational formula, which coincides with the behavior of the partition function corresponding to the polaron problem under strong coupling. Based on this, Spohn ([Sp87]) made a heuristic observation that the strong coupling behavior of the polaron path measure, on certain time scales, should resemble a process, named as the itPekar process, whose distribution could somehow be guessed from the limiting asymptotic behavior of the mean-field measures under interest, whose rigorous analysis remained open. The present paper is devoted to a precise analysis of these mean-field path measures and convergence of the normalized occupation measures towards an explicit mixture of the maximizers of the Pekar variational problem. This leads to a rigorous construction of the aforementioned Pekar process and hence, is a contribution to the understanding of the ``mean-field approximation" of the polaron problem on the level of path measures. The method of our proof is based on the compact large deviation theory developed in [MV14], its extension to the uniform strong metric for the singular Coulomb interaction carried out in [KM15], as well as an idea inspired by a itpartial path exchange argument appearing in [BS97]
  • Thumbnail Image
    Item
    Connection times in large ad hoc mobile networks
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Döring, Hanna; Faraud, Gabriel; König, Wolfgang
    We study connectivity properties in a probabilistic model for a large mobile ad-hoc network. We consider a large number of participants of the system moving randomly, independently and identically distributed in a large domain, with a space-dependent population density of finite, positive order and with a fixed time horizon. Messages are instantly transmitted according to a relay principle, i.e., they are iteratedly forwarded from participant to participant over distances 2R, with 2R the communication radius, until they reach the recipient. In mathematical terms, this is a dynamic continuum percolation model. We consider the connection time of two sample participants, the amount of time over which these two are connected with each other. In the above thermodynamic limit, we find that the connectivity induced by the system can be described in terms of the counterplay of a local, random, and a global, deterministic mechanism, and we give a formula for the limiting behaviour. A prime example of the movement schemes that we consider is the well-known random waypoint model (RWP). Here we describe the decay rate, in the limit of large time horizons, of the probability that the portion of the connection time is less than the expectation.
  • Thumbnail Image
    Item
    Large deviations for the local times of a random walk among random conductances in a growing box
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) König, Wolfgang; Wolff, Tilman
    We derive an annealed large deviation principle (LDP) for the normalised and rescaled local times of a continuous-time random walk among random conductances (RWRC) in a time-dependent, growing box in Zd. We work in the interesting case that the conductances are positive, but may assume arbitrarily small values. Thus, the underlying picture of the principle is a joint strategy of small conductance values and large holding times of the walk. The speed and the rate function of our principle are explicit in terms of the lower tails of the conductance distribution as well as the time-dependent size of the box. An interesting phase transition occurs if the thickness parameter of the conductance tails exceeds a certain threshold: for thicker tails, the random walk spreads out over the entire growing box, for thinner tails it stays confined to some bounded region. In fact, in the first case, the rate function turns out to be equal to the p-th power of the p-norm of the gradient of the square root for some p (2d d+2; 2). This extends the Donsker-Varadhan-Gärtner rate function for the local times of Brownian motion (with deterministic environment) from p = 2 to these values. As corollaries of our LDP, we derive the logarithmic asymptotics of the non-exit probability of the RWRC from the growing box, and the Lifshitz tails of the generator of the RWRC, the randomised Laplace operator. To contrast with the annealed, not uniformly elliptic case, we also provide an LDP in the quenched setting for conductances that are bounded and bounded away from zero. The main tool here is a spectral homogenisation result, based on a quenched invariance principle for the RWRC.