This article deals with the asymptotic behavior as $t \rightarrow +\infty $ of the survival function $\mathbb{P}[T > t]$, where T is the first passage time above a non negative level of a random process starting from zero. In many cases of physical significance, the behavior is of the type $\mathbb{P}[T > t] = t^{-\theta +o(1)}$ for a known or unknown positive parameter $\theta$ which is called the persistence exponent. The problem is well understood for random walks or Levy processes but becomes more difficult for integrals of such processes, which are more related to physics. We survey recent results and open problems in this field.

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Persistence Probabilities and Exponents

We study right tail large deviations of the logarithm of the partition function for directed lattice paths in i.i.d. random potentials. The main purpose is the derivation of explicit formulas for the 1+1-dimensional exactly solvable case with log-gamma distributed random weights. Along the way we establish some regularity results for this rate function for general distributions in arbitrary dimensions.

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Large deviation rate functions for the partition function in a log-gamma distributed random potential

We study right tail large deviations of the logarithm of the partition function for directed lattice paths in i.i.d. random potentials. The main purpose is the derivation of explicit formulas for the $1+1$-dimensional exactly solvable case with log-gamma distributed random weights. Along the way we establish some regularity results for this rate function for general distributions in arbitrary dimensions.

https://hal.archives-ouvertes.fr/hal-01324693/document

Persistence probabilities for stationary increment processes

We study the persistence probability for some discrete-time, time-reversible processes. In particular, we deduce the persistence exponent in a number of examples: first, we deal with random walks in random sceneries (RWRS) in any dimension with Gaussian scenery. Second, we deal with sums of stationary Gaussian sequences with correlations exhibiting long-range dependence. Apart from the persistence probability we deal with the position of the maximum and the time spent on the positive half-axis by the process.

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Persistence exponent for discrete-time, time-reversible processes

In this paper we consider the persistence properties of random processes in Brownian scenery, which are examples of non-Markovian andnon-Gaussian processes. More precisely we study the asymptotic behaviour for large $T$, of the probability $P[ \sup_{t\in[0,T]} \Delta_t \leq 1] $where $\Delta_t = \int_{\mathbb{R}} L_t(x) \, dW(x).$Here $W={W(x); x\in\mathbb{R}}$ is a two-sided standard real Brownian motion and ${L_t(x); x\in\mathbb{R},t\geq 0}$ isthe local time of some self-similar random process $Y$, independent from the process $W$. We thus generalize the results of \cite{BFFN} where the increments of $Y$ were assumed to be independent.

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Persistence exponent for random processes in Brownian scenery

We use a generalization of Hoeffding's inequality to show concentration results for the free energy of disordered pinning models, assuming only that the disorder has a finite exponential moment. We also prove some concentration in- equalities for directed polymers in random environment, which we use to establish a large deviations result for the end position of the polymer under the polymer measure.

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Concentration inequalities for disordered models

We use a generalization of Hoeffding's inequality to show concentration results for the free energy of disordered pinning models, assuming only that the disorder has a finite exponential moment. We also prove some concentration inequalities for directed polymers in random environment, which we use to establish a large deviations results for the end position of the polymer under the polymer measure.

In this paper we consider the persistence properties of random processes in Brownian scenery, which are examples of non-Markovian and non-Gaussian processes. More precisely we study the asymptotic behaviour for large $T$, of the probability $P[ \sup\_{t\in[0,T]} \Delta\_t \leq 1] $ where $\Delta\_t = \int\_{\mathbb{R}} L\_t(x) \, dW(x).$ Here $W={W(x); x\in\mathbb{R}}$ is a two-sided standard real Brownian motion and ${L\_t(x); x\in\mathbb{R},t\geq 0}$ is the local time of some self-similar random process $Y$, independent from the process $W$. We thus generalize the results of \cite{BFFN} where the increments of $Y$ were assumed to be independent.

Frederique Watbled

Papers

Persistence exponent for random processes in Brownian scenery

Concentration inequalities for disordered models

Concentration inequalities for disordered models

Persistence exponent for random processes in Brownian scenery