Journal: Electronic Journal of Probability

Abbreviation

Electron. J. Probab.

Publisher

Institute of Mathematical Statistics

Journal Volumes

ISSN

1083-6489

Description

Filters

Reset filters

Search Results

Publications1 - 10 of 41
  • Logarithmic Components of the Vacant Set for Random Walk on a Discrete Torus
    Item type: Journal Article
    Windisch, David (2008)
    Electronic Journal of Probability
    This work continues the investigation, initiated in a recent work by Benjamini and Sznitman, of percolative properties of the set of points not visited by a random walk on the discrete torus (Z/NZ)d up to time uNd in high dimension d. If u>0 is chosen sufficiently small it has been shown that with overwhelming probability this vacant set contains a unique giant component containing segments of length c0logN for some constant c0>0, and this component occupies a non-degenerate fraction of the total volume as N tends to infinity. Within the same setup, we investigate here the complement of the giant component in the vacant set and show that some components consist of segments of logarithmic size. In particular, this shows that the choice of a sufficiently large constant c0>0 is crucial in the definition of the giant component.
  • Stability of backward stochastic differential equations: the general Lipschitz case
    Item type: Journal Article
    Papapantoleon, Antonis; Possamaï, Dylan; Saplaouras, Alexandros (2023)
    Electronic Journal of Probability
    In this paper, we obtain stability results for backward stochastic differential equations with jumps (BSDEs) in a very general framework. More specifically, we consider a convergent sequence of standard data, each associated to their own filtration, and we prove that the associated sequence of (unique) solutions is also convergent. The current result extends earlier contributions in the literature of stability of BSDEs and unifies several frameworks for numerical approximations of BSDEs and their implementations.
  • No percolation in low temperature spin glass 1.2
    Item type: Journal Article
    Berger, Noam; Tessler, Ran J. (2017)
    Electronic Journal of Probability
    We consider the Edwards-Anderson Ising Spin Glass model for temperatures T≥0. We define notions of Boltzmann-Gibbs measure for the Edwards-Anderson spin glass at a given temperature, and of unsatisfied (frustrated) edges, and recall the notion of ground states. We prove that for low positive temperatures, in almost every spin configuration the graph formed by the unsatisfied edges is made of finite connected components. Similarly, for zero temperature, we show that in almost every ground state the graph of unsatisfied edges is a forest all of whose components are finite. In other words, for low enough temperatures the unsatisfied edges do not percolate.
  • Percolation on uniform infinite planar maps
    Item type: Journal Article
    Ménard, Laurent; Nolin, Pierre (2014)
    Electronic Journal of Probability
    We construct the uniform infinite planar map (UIPM), obtained as the n→∞ local limit of planar maps with n edges, chosen uniformly at random. We then describe how the UIPM can be sampled using a "peeling" process, in a similar way as for uniform triangulations. This process allows us to prove that for bond and site percolation on the UIPM, the percolation thresholds are pbondc=1/2 and psitec=2/3 respectively. This method also works for other classes of random infinite planar maps, and we show in particular that for bond percolation on the uniform infinite planar quadrangulation, the percolation threshold is pbondc=1/3.
  • Numerical schemes for G-Expectations
    Item type: Journal Article
    Dolinsky, Yan (2012)
    Electronic Journal of Probability
    We consider a discrete time analog of G-expectations and we prove that in the case where the time step goes to zero the corresponding values converge to the original G-expectation. Furthermore we provide error estimates for the convergence rate. This paper is continuation of Dolinsky, Nutz, and Soner (2012). Our main tool is a strong approximation theorem which we derive for general discrete time martingales.
  • Robust filtering and propagation of uncertainty in hidden Markov models
    Item type: Journal Article
    Allan, Andrew (2021)
    Electronic Journal of Probability
    We consider the filtering of continuous-time finite-state hidden Markov models, where the rate and observation matrices depend on unknown time-dependent parameters, for which no prior or stochastic model is available. We quantify and analyze how the induced uncertainty may be propagated through time as we collect new observations, and used to simultaneously provide robust estimates of the hidden signal and to learn the unknown parameters, via techniques based on pathwise filtering and new results on the optimal control of rough differential equations.
  • An FBSDE approach to the Skorokhod embedding problem for Gaussian processes with non-linear drift
    Item type: Journal Article
    Fromm, Alexander; Imkeller, Peter; Prömel, David J. (2015)
    Electronic Journal of Probability
    We solve the Skorokhod embedding problem for a class of Gaussian processes including Brownian motion with non-linear drift. Our approach relies on solving an associated strongly coupled system of Forward Backward Stochastic Differential Equation (FBSDE), and investigating the regularity of the obtained solution. For this purpose we extend the existence, uniqueness and regularity theory of so called decoupling fields for Markovian FBSDE to a setting in which the coefficients are only locally Lipschitz continuous.
  • Superreplication under volatility uncertainty for measurable claims
    Item type: Journal Article
    Neufeld, Ariel; Nutz, Marcel (2013)
    Electronic Journal of Probability
    We establish the duality-formula for the superreplication price in a setting of volatility uncertainty which includes the example of "random G-expectation". In contrast to previous results, the contingent claim is not assumed to be quasi-continuous.
  • Disconnection by level sets of the discrete Gaussian free field and entropic repulsion
    Item type: Journal Article
    Nitzschner, Maximilian (2018)
    Electronic Journal of Probability
    We derive asymptotic upper and lower bounds on the large deviation probability that the level set of the Gaussian free field on Zd, d≥3, below a level α, disconnects the discrete blow-up of a compact set A from the boundary of the discrete blow-up of a box that contains A, when the level set of the Gaussian free field above α is in a strongly percolative regime. These bounds substantially strengthen the results of [21], where A was a box and the convexity of A played an important role in the proof. We also derive an asymptotic upper bound on the probability that the average of the Gaussian free field well inside the discrete blow-up of A is above a certain level when disconnection occurs. The derivation of the upper bounds uses the solidification estimates for porous interfaces that were derived in the work [15] of A.-S. Sznitman and the author to treat a similar disconnection problem for the vacant set of random interlacements. If certain critical levels for the Gaussian free field coincide, an open question at the moment, the asymptotic upper and lower bounds that we obtain for the disconnection probability match in principal order, and conditioning on disconnection lowers the average of the Gaussian free field well inside the discrete blow-up of A, which can be understood as entropic repulsion.
  • Polynomial growth in degree-dependent first passage percolation on spatial random graphs
    Item type: Journal Article
    Komjáthy, Júlia; Lapinskas, John; Lengler, Johannes; et al. (2024)
    Electronic Journal of Probability
    In this paper we study a version of (non-Markovian) first passage percolation on graphs, where the transmission time between two connected vertices is non-iid, but increases by a penalty factor polynomial in their expected degrees. Based on the exponent of the penalty-polynomial, this makes it increasingly harder to transmit to and from high degree vertices. This choice is motivated by awareness or time-limitations. For the iid part of the transmission times we allow any nonnegative distribution with regularly varying behaviour at 0. For the underlying graph models we choose spatial random graphs that have power-law degree distributions, so that the effect of the penalisation becomes visible: (finite and infinite) Geometric Inhomogeneous Random Graphs, and Scale-Free Percolation. In these spatial models, the connection probability between two vertices depends on their spatial distance and on their expected degrees. We prove that upon increasing the penalty exponent, the transmission time between two far away vertices x, y sweeps through four universal phases even for a single underlying graph: explosive (tight transmission times), polylogarithmic, polynomial but sublinear(|x − y|η0+o(1)) for an explicit η0 < 1), and linear (Θ(|x − y|)) in their Euclidean distance. Further, none of these phases are restricted to phase boundaries, and those are non-trivial in the main model parameters: the tail of the degree-distribution, a long-range parameter, and the exponent of regular variation of the iid part of the transmission times. In this paper we present proofs of lower bounds for the latter two phases and the upper bound for the linear phase. These complement the matching upper bounds for the polynomial regime in our companion paper
Publications1 - 10 of 41