Mostrando recursos 1 - 20 de 1.787

  1. Harmonic moments and large deviations for a supercritical branching process in a random environment

    Grama, Ion; Liu, Quansheng; Miqueu, Eric
    Let $(Z_n)_{n\geq 0}$ be a supercritical branching process in an independent and identically distributed random environment $\xi =(\xi _n)_{n\geq 0}$. We study the asymptotic behavior of the harmonic moments $\mathbb{E} \left [Z_n^{-r} | Z_0=k \right ]$ of order $r>0$ as $n \to \infty $, when the process starts with $k$ initial individuals. We exhibit a phase transition with the critical value $r_k>0$ determined by the equation $\mathbb E p_1^k(\xi _0) = \mathbb E m_0^{-r_k},$ where $m_0=\sum _{j=0}^\infty j p_j (\xi _0)$, $(p_j(\xi _0))_{j\geq 0}$ being the offspring distribution given the environnement $\xi _0$. Contrary to the constant environment case (the...

  2. Invariant measures for stochastic functional differential equations

    Butkovsky, Oleg; Scheutzow, Michael
    We establish new general sufficient conditions for the existence of an invariant measure for stochastic functional differential equations and exponential or subexponential convergence to the equilibrium. The obtained conditions extend the Veretennikov–Khasminskii conditions for SDEs and are optimal in a certain sense.

  3. A tightness criterion for random fields, with application to the Ising model

    Furlan, Marco; Mourrat, Jean-Christophe
    We present a criterion for a family of random distributions to be tight in local Hölder and Besov spaces of possibly negative regularity on general domains. We then apply this criterion to find the sharp regularity of the magnetization field of the two-dimensional Ising model at criticality, answering a question of [8].

  4. An iterative technique for bounding derivatives of solutions of Stein equations

    Döbler, Christian; Gaunt, Robert E.; Vollmer, Sebastian J.
    We introduce a simple iterative technique for bounding derivatives of solutions of Stein equations $Lf=h-\mathbb{E} h(Z)$, where $L$ is a linear differential operator and $Z$ is the limit random variable. Given bounds on just the solutions or certain lower order derivatives of the solution, the technique allows one to deduce bounds for derivatives of any order, in terms of supremum norms of derivatives of the test function $h$. This approach can be readily applied to many Stein equations from the literature. We consider a number of applications; in particular, we derive new bounds for derivatives of any order of the...

  5. Local limits of Markov branching trees and their volume growth

    Pagnard, Camille
    We are interested in the local limits of families of random trees that satisfy the Markov branching property, which is fulfilled by a wide range of models. Loosely, this property entails that given the sizes of the sub-trees above the root, these sub-trees are independent and their distributions only depend upon their respective sizes. The laws of the elements of a Markov branching family are characterised by a sequence of probability distributions on the sets of integer partitions which describes how the sizes of the sub-trees above the root are distributed. ¶ We prove that under some natural assumption on...

  6. Local limits of Markov branching trees and their volume growth

    Pagnard, Camille
    We are interested in the local limits of families of random trees that satisfy the Markov branching property, which is fulfilled by a wide range of models. Loosely, this property entails that given the sizes of the sub-trees above the root, these sub-trees are independent and their distributions only depend upon their respective sizes. The laws of the elements of a Markov branching family are characterised by a sequence of probability distributions on the sets of integer partitions which describes how the sizes of the sub-trees above the root are distributed. ¶ We prove that under some natural assumption on...

  7. Local limits of Markov branching trees and their volume growth

    Pagnard, Camille
    We are interested in the local limits of families of random trees that satisfy the Markov branching property, which is fulfilled by a wide range of models. Loosely, this property entails that given the sizes of the sub-trees above the root, these sub-trees are independent and their distributions only depend upon their respective sizes. The laws of the elements of a Markov branching family are characterised by a sequence of probability distributions on the sets of integer partitions which describes how the sizes of the sub-trees above the root are distributed. ¶ We prove that under some natural assumption on...

  8. Long time asymptotics of unbounded additive functionals of Markov processes

    Gao, Fuqing
    Under hypercontractivity and $L_p$-integrability of transition density for some $p>1$, we use the perturbation theory of linear operators to obtain existence of long time asymptotics of exponentials of unbounded additive functionals of Markov processes and establish the moderate deviation principle for the functionals. For stochastic differential equations with multiplicative noise, we show the hypercontractivity and the integrability based on Wang’s Harnack inequality. As an application of our general results, we obtain the existence of these asymptotics and the moderate deviation principle of additive functionals with quadratic growth for the stochastic differential equations with multiplicative noise under some explicit conditions on...

  9. Long time asymptotics of unbounded additive functionals of Markov processes

    Gao, Fuqing
    Under hypercontractivity and $L_p$-integrability of transition density for some $p>1$, we use the perturbation theory of linear operators to obtain existence of long time asymptotics of exponentials of unbounded additive functionals of Markov processes and establish the moderate deviation principle for the functionals. For stochastic differential equations with multiplicative noise, we show the hypercontractivity and the integrability based on Wang’s Harnack inequality. As an application of our general results, we obtain the existence of these asymptotics and the moderate deviation principle of additive functionals with quadratic growth for the stochastic differential equations with multiplicative noise under some explicit conditions on...

  10. Long time asymptotics of unbounded additive functionals of Markov processes

    Gao, Fuqing
    Under hypercontractivity and $L_p$-integrability of transition density for some $p>1$, we use the perturbation theory of linear operators to obtain existence of long time asymptotics of exponentials of unbounded additive functionals of Markov processes and establish the moderate deviation principle for the functionals. For stochastic differential equations with multiplicative noise, we show the hypercontractivity and the integrability based on Wang’s Harnack inequality. As an application of our general results, we obtain the existence of these asymptotics and the moderate deviation principle of additive functionals with quadratic growth for the stochastic differential equations with multiplicative noise under some explicit conditions on...

  11. Long time asymptotics of unbounded additive functionals of Markov processes

    Gao, Fuqing
    Under hypercontractivity and $L_p$-integrability of transition density for some $p>1$, we use the perturbation theory of linear operators to obtain existence of long time asymptotics of exponentials of unbounded additive functionals of Markov processes and establish the moderate deviation principle for the functionals. For stochastic differential equations with multiplicative noise, we show the hypercontractivity and the integrability based on Wang’s Harnack inequality. As an application of our general results, we obtain the existence of these asymptotics and the moderate deviation principle of additive functionals with quadratic growth for the stochastic differential equations with multiplicative noise under some explicit conditions on...

  12. Long time asymptotics of unbounded additive functionals of Markov processes

    Gao, Fuqing
    Under hypercontractivity and $L_p$-integrability of transition density for some $p>1$, we use the perturbation theory of linear operators to obtain existence of long time asymptotics of exponentials of unbounded additive functionals of Markov processes and establish the moderate deviation principle for the functionals. For stochastic differential equations with multiplicative noise, we show the hypercontractivity and the integrability based on Wang’s Harnack inequality. As an application of our general results, we obtain the existence of these asymptotics and the moderate deviation principle of additive functionals with quadratic growth for the stochastic differential equations with multiplicative noise under some explicit conditions on...

  13. Renormalizability of Liouville quantum field theory at the Seiberg bound

    David, François; Kupiainen, Antti; Rhodes, Rémi; Vargas, Vincent
    Liouville Quantum Field Theory (LQFT) can be seen as a probabilistic theory of 2d Riemannian metrics $e^{\phi (z)}|dz|^2$, conjecturally describing scaling limits of discrete $2d$-random surfaces. The law of the random field $\phi $ in LQFT depends on weights $\alpha \in \mathbb{R} $ that in classical Riemannian geometry parametrize power law singularities in the metric. A rigorous construction of LQFT has been carried out in [3] in the case when the weights are below the so called Seiberg bound: $\alpha

  14. Renormalizability of Liouville quantum field theory at the Seiberg bound

    David, François; Kupiainen, Antti; Rhodes, Rémi; Vargas, Vincent
    Liouville Quantum Field Theory (LQFT) can be seen as a probabilistic theory of 2d Riemannian metrics $e^{\phi (z)}|dz|^2$, conjecturally describing scaling limits of discrete $2d$-random surfaces. The law of the random field $\phi $ in LQFT depends on weights $\alpha \in \mathbb{R} $ that in classical Riemannian geometry parametrize power law singularities in the metric. A rigorous construction of LQFT has been carried out in [3] in the case when the weights are below the so called Seiberg bound: $\alpha

  15. Renormalizability of Liouville quantum field theory at the Seiberg bound

    David, François; Kupiainen, Antti; Rhodes, Rémi; Vargas, Vincent
    Liouville Quantum Field Theory (LQFT) can be seen as a probabilistic theory of 2d Riemannian metrics $e^{\phi (z)}|dz|^2$, conjecturally describing scaling limits of discrete $2d$-random surfaces. The law of the random field $\phi $ in LQFT depends on weights $\alpha \in \mathbb{R} $ that in classical Riemannian geometry parametrize power law singularities in the metric. A rigorous construction of LQFT has been carried out in [3] in the case when the weights are below the so called Seiberg bound: $\alpha

  16. Renormalizability of Liouville quantum field theory at the Seiberg bound

    David, François; Kupiainen, Antti; Rhodes, Rémi; Vargas, Vincent
    Liouville Quantum Field Theory (LQFT) can be seen as a probabilistic theory of 2d Riemannian metrics $e^{\phi (z)}|dz|^2$, conjecturally describing scaling limits of discrete $2d$-random surfaces. The law of the random field $\phi $ in LQFT depends on weights $\alpha \in \mathbb{R} $ that in classical Riemannian geometry parametrize power law singularities in the metric. A rigorous construction of LQFT has been carried out in [3] in the case when the weights are below the so called Seiberg bound: $\alpha

  17. Renormalizability of Liouville quantum field theory at the Seiberg bound

    David, François; Kupiainen, Antti; Rhodes, Rémi; Vargas, Vincent
    Liouville Quantum Field Theory (LQFT) can be seen as a probabilistic theory of 2d Riemannian metrics $e^{\phi (z)}|dz|^2$, conjecturally describing scaling limits of discrete $2d$-random surfaces. The law of the random field $\phi $ in LQFT depends on weights $\alpha \in \mathbb{R} $ that in classical Riemannian geometry parametrize power law singularities in the metric. A rigorous construction of LQFT has been carried out in [3] in the case when the weights are below the so called Seiberg bound: $\alpha

  18. Asymptotic direction for random walks in mixing random environments

    Guerra, Enrique; Ramírez, Alejandro F.
    We prove that every random walk in a uniformly elliptic random environment satisfying the cone-mixing condition and a non-effective polynomial ballisticity condition with high enough degree has an asymptotic direction.

  19. Asymptotic direction for random walks in mixing random environments

    Guerra, Enrique; Ramírez, Alejandro F.
    We prove that every random walk in a uniformly elliptic random environment satisfying the cone-mixing condition and a non-effective polynomial ballisticity condition with high enough degree has an asymptotic direction.

  20. Asymptotic direction for random walks in mixing random environments

    Guerra, Enrique; Ramírez, Alejandro F.
    We prove that every random walk in a uniformly elliptic random environment satisfying the cone-mixing condition and a non-effective polynomial ballisticity condition with high enough degree has an asymptotic direction.

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