Mostrando recursos 1 - 20 de 1.344

  1. Multifractal analysis for the occupation measure of stable-like processes

    Seuret, Stéphane; Yang, Xiaochuan
    In this article, we investigate the local behavior of the occupation measure $\mu $ of a class of real-valued Markov processes $\mathcal{M} $, defined via a SDE. This (random) measure describes the time spent in each set $A\subset \mathbb{R} $ by the sample paths of $\mathcal{M} $. We compute the multifractal spectrum of $\mu $, which turns out to be random, depending on the trajectory. This remarkable property is in sharp contrast with the results previously obtained on occupation measures of other processes (such as Lévy processes), where the multifractal spectrum is usually deterministic, almost surely. In addition, the shape...

  2. Weak error for the Euler scheme approximation of diffusions with non-smooth coefficients

    Konakov, Valentin; Menozzi, Stéphane
    We study the weak error associated with the Euler scheme of non degenerate diffusion processes with non smooth bounded coefficients. Namely, we consider the cases of Hölder continuous coefficients as well as piecewise smooth drifts with smooth diffusion matrices.

  3. Weak error for the Euler scheme approximation of diffusions with non-smooth coefficients

    Konakov, Valentin; Menozzi, Stéphane
    We study the weak error associated with the Euler scheme of non degenerate diffusion processes with non smooth bounded coefficients. Namely, we consider the cases of Hölder continuous coefficients as well as piecewise smooth drifts with smooth diffusion matrices.

  4. Scaling limits for the critical Fortuin-Kasteleyn model on a random planar map II: local estimates and empty reduced word exponent

    Gwynne, Ewain; Sun, Xin
    We continue our study of the inventory accumulation introduced by Sheffield (2011), which encodes a random planar map decorated by a collection of loops sampled from the critical Fortuin-Kasteleyn (FK) model. We prove various local estimates for the inventory accumulation model, i.e., estimates for the precise number of symbols of a given type in a reduced word sampled from the model. Using our estimates, we obtain the scaling limit of the associated two-dimensional random walk conditioned on the event that it stays in the first quadrant for one unit of time and ends up at a particular position in the...

  5. Scaling limits for the critical Fortuin-Kasteleyn model on a random planar map II: local estimates and empty reduced word exponent

    Gwynne, Ewain; Sun, Xin
    We continue our study of the inventory accumulation introduced by Sheffield (2011), which encodes a random planar map decorated by a collection of loops sampled from the critical Fortuin-Kasteleyn (FK) model. We prove various local estimates for the inventory accumulation model, i.e., estimates for the precise number of symbols of a given type in a reduced word sampled from the model. Using our estimates, we obtain the scaling limit of the associated two-dimensional random walk conditioned on the event that it stays in the first quadrant for one unit of time and ends up at a particular position in the...

  6. Extremes and gaps in sampling from a GEM random discrete distribution

    Pitman, Jim; Yakubovich, Yuri
    We show that in a sample of size $n$ from a $\mathsf{GEM} (0,\theta )$ random discrete distribution, the gaps $G_{i:n}:= X_{n-i+1:n} - X_{n-i:n}$ between order statistics $X_{1:n} \le \cdots \le X_{n:n}$ of the sample, with the convention $G_{n:n} := X_{1:n} - 1$, are distributed like the first $n$ terms of an infinite sequence of independent geometric$(i/(i+\theta ))$ variables $G_i$. This extends a known result for the minimum $X_{1:n}$ to other gaps in the range of the sample, and implies that the maximum $X_{n:n}$ has the distribution of $1 + \sum _{i=1}^n G_i$, hence the known result that $X_{n:n}$ grows like...

  7. Extremes and gaps in sampling from a GEM random discrete distribution

    Pitman, Jim; Yakubovich, Yuri
    We show that in a sample of size $n$ from a $\mathsf{GEM} (0,\theta )$ random discrete distribution, the gaps $G_{i:n}:= X_{n-i+1:n} - X_{n-i:n}$ between order statistics $X_{1:n} \le \cdots \le X_{n:n}$ of the sample, with the convention $G_{n:n} := X_{1:n} - 1$, are distributed like the first $n$ terms of an infinite sequence of independent geometric$(i/(i+\theta ))$ variables $G_i$. This extends a known result for the minimum $X_{1:n}$ to other gaps in the range of the sample, and implies that the maximum $X_{n:n}$ has the distribution of $1 + \sum _{i=1}^n G_i$, hence the known result that $X_{n:n}$ grows like...

  8. Mean-field behavior for nearest-neighbor percolation in $d>10$

    Fitzner, Robert; van der Hofstad, Remco
    We prove that nearest-neighbor percolation in dimensions $d\geq 11$ displays mean-field behavior by proving that the infrared bound holds, in turn implying the finiteness of the percolation triangle diagram. The finiteness of the triangle implies the existence and mean-field values of various critical exponents, such as $\gamma =1, \beta =1, \delta =2$. We also prove sharp $x$-space asymptotics for the two-point function and the existence of various arm exponents. Such results had previously been obtained in unpublished work by Hara and Slade for nearest-neighbor percolation in dimension $d\geq 19$, so that we bring the dimension above which mean-field behavior is...

  9. Mean-field behavior for nearest-neighbor percolation in $d>10$

    Fitzner, Robert; van der Hofstad, Remco
    We prove that nearest-neighbor percolation in dimensions $d\geq 11$ displays mean-field behavior by proving that the infrared bound holds, in turn implying the finiteness of the percolation triangle diagram. The finiteness of the triangle implies the existence and mean-field values of various critical exponents, such as $\gamma =1, \beta =1, \delta =2$. We also prove sharp $x$-space asymptotics for the two-point function and the existence of various arm exponents. Such results had previously been obtained in unpublished work by Hara and Slade for nearest-neighbor percolation in dimension $d\geq 19$, so that we bring the dimension above which mean-field behavior is...

  10. Fluctuations for mean-field interacting age-dependent Hawkes processes

    Chevallier, Julien
    The propagation of chaos and associated law of large numbers for mean-field interacting age-dependent Hawkes processes (when the number of processes $n$ goes to $+\infty $) being granted by the study performed in [9], the aim of the present paper is to prove the resulting functional central limit theorem. It involves the study of a measure-valued process describing the fluctuations (at scale $n^{-1/2}$) of the empirical measure of the ages around its limit value. This fluctuation process is proved to converge towards a limit process characterized by a limit system of stochastic differential equations driven by a Gaussian noise instead...

  11. Fluctuations for mean-field interacting age-dependent Hawkes processes

    Chevallier, Julien
    The propagation of chaos and associated law of large numbers for mean-field interacting age-dependent Hawkes processes (when the number of processes $n$ goes to $+\infty $) being granted by the study performed in [9], the aim of the present paper is to prove the resulting functional central limit theorem. It involves the study of a measure-valued process describing the fluctuations (at scale $n^{-1/2}$) of the empirical measure of the ages around its limit value. This fluctuation process is proved to converge towards a limit process characterized by a limit system of stochastic differential equations driven by a Gaussian noise instead...

  12. An Eyring–Kramers law for the stochastic Allen–Cahn equation in dimension two

    Berglund, Nils; Di Gesù, Giacomo; Weber, Hendrik
    We study spectral Galerkin approximations of an Allen–Cahn equation over the two-dimensional torus perturbed by weak space-time white noise of strength $\sqrt{\varepsilon } $. We introduce a Wick renormalisation of the equation in order to have a system that is well-defined as the regularisation is removed. We show sharp upper and lower bounds on the transition times from a neighbourhood of the stable configuration $-1$ to the stable configuration $1$ in the asymptotic regime $\varepsilon \to 0$. These estimates are uniform in the discretisation parameter $N$, suggesting an Eyring–Kramers formula for the limiting renormalised stochastic PDE. The effect of the...

  13. An Eyring–Kramers law for the stochastic Allen–Cahn equation in dimension two

    Berglund, Nils; Di Gesù, Giacomo; Weber, Hendrik
    We study spectral Galerkin approximations of an Allen–Cahn equation over the two-dimensional torus perturbed by weak space-time white noise of strength $\sqrt{\varepsilon } $. We introduce a Wick renormalisation of the equation in order to have a system that is well-defined as the regularisation is removed. We show sharp upper and lower bounds on the transition times from a neighbourhood of the stable configuration $-1$ to the stable configuration $1$ in the asymptotic regime $\varepsilon \to 0$. These estimates are uniform in the discretisation parameter $N$, suggesting an Eyring–Kramers formula for the limiting renormalised stochastic PDE. The effect of the...

  14. Ricci curvature bounds for weakly interacting Markov chains

    Erbar, Matthias; Henderson, Christopher; Menz, Georg; Tetali, Prasad
    We establish a general perturbative method to prove entropic Ricci curvature bounds for interacting stochastic particle systems. We apply this method to obtain curvature bounds in several examples, namely: Glauber dynamics for a class of spin systems including the Ising and Curie–Weiss models, a class of hard-core models and random walks on groups induced by a conjugacy invariant set of generators.

  15. Ricci curvature bounds for weakly interacting Markov chains

    Erbar, Matthias; Henderson, Christopher; Menz, Georg; Tetali, Prasad
    We establish a general perturbative method to prove entropic Ricci curvature bounds for interacting stochastic particle systems. We apply this method to obtain curvature bounds in several examples, namely: Glauber dynamics for a class of spin systems including the Ising and Curie–Weiss models, a class of hard-core models and random walks on groups induced by a conjugacy invariant set of generators.

  16. The Brownian net and selection in the spatial $\Lambda $-Fleming-Viot process

    Etheridge, Alison; Freeman, Nic; Straulino, Daniel
    We obtain the Brownian net of [24] as the scaling limit of the paths traced out by a system of continuous (one-dimensional) space and time branching and coalescing random walks. This demonstrates a certain universality of the net, which we have not seen explored elsewhere. The walks themselves arise in a natural way as the ancestral lineages relating individuals in a sample from a biological population evolving according to the spatial Lambda-Fleming-Viot process. Our scaling reveals the effect, in dimension one, of spatial structure on the spread of a selectively advantageous gene through such a population.

  17. The Brownian net and selection in the spatial $\Lambda $-Fleming-Viot process

    Etheridge, Alison; Freeman, Nic; Straulino, Daniel
    We obtain the Brownian net of [24] as the scaling limit of the paths traced out by a system of continuous (one-dimensional) space and time branching and coalescing random walks. This demonstrates a certain universality of the net, which we have not seen explored elsewhere. The walks themselves arise in a natural way as the ancestral lineages relating individuals in a sample from a biological population evolving according to the spatial Lambda-Fleming-Viot process. Our scaling reveals the effect, in dimension one, of spatial structure on the spread of a selectively advantageous gene through such a population.

  18. Rigorous results for a population model with selection II: genealogy of the population

    Schweinsberg, Jason
    We consider a model of a population of fixed size $N$ undergoing selection. Each individual acquires beneficial mutations at rate $\mu _N$, and each beneficial mutation increases the individual’s fitness by $s_N$. Each individual dies at rate one, and when a death occurs, an individual is chosen with probability proportional to the individual’s fitness to give birth. Under certain conditions on the parameters $\mu _N$ and $s_N$, we show that the genealogy of the population can be described by the Bolthausen-Sznitman coalescent. This result confirms predictions of Desai, Walczak, and Fisher (2013), and Neher and Hallatschek (2013).

  19. Rigorous results for a population model with selection II: genealogy of the population

    Schweinsberg, Jason
    We consider a model of a population of fixed size $N$ undergoing selection. Each individual acquires beneficial mutations at rate $\mu _N$, and each beneficial mutation increases the individual’s fitness by $s_N$. Each individual dies at rate one, and when a death occurs, an individual is chosen with probability proportional to the individual’s fitness to give birth. Under certain conditions on the parameters $\mu _N$ and $s_N$, we show that the genealogy of the population can be described by the Bolthausen-Sznitman coalescent. This result confirms predictions of Desai, Walczak, and Fisher (2013), and Neher and Hallatschek (2013).

  20. Rigorous results for a population model with selection I: evolution of the fitness distribution

    Schweinsberg, Jason
    We consider a model of a population of fixed size $N$ undergoing selection. Each individual acquires beneficial mutations at rate $\mu _N$, and each beneficial mutation increases the individual’s fitness by $s_N$. Each individual dies at rate one, and when a death occurs, an individual is chosen with probability proportional to the individual’s fitness to give birth. Under certain conditions on the parameters $\mu _N$ and $s_N$, we obtain rigorous results for the rate at which mutations accumulate in the population and the distribution of the fitnesses of individuals in the population at a given time. Our results confirm predictions...

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