Large deviations for intersection local times in critical dimension

Let $(X_t,t\geq0)$ be a continuous time simple random walk on $\mathbb{Z}^d$ ($d\geq3$), and let $l_T(x)$ be the time spent by $(X_t,t\geq0)$ on the site $x$ up to time $T$. We prove a large deviations principle for the $q$-fold self-intersection local time $I_T=\sum_{x\in\mathbb{Z}^d}l_T(x)^q$ in the critical case $q=\frac{d}{d-2}$. When $q$ is integer, we obtain similar results for the intersection local times of $q$ independent simple random walks.Comment: Published in at http://dx.doi.org/10.1214/09-AOP499 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

A note on random walk in random scenery

We consider a d-dimensional random walk in random scenery X(n), where the scenery consists of i.i.d. with exponential moments but a tail decay of the form exp(-c t^a) with a<d/2. We study the probability, when averaged over both randomness, that {X(n)>ny}. We show that this probability is of order exp(-(ny)^b) with b=a/(a+1).Comment: 13 page

Persistence exponent for random processes in Brownian scenery

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

Intertwining wavelets or Multiresolution analysis on graphs through random forests

We propose a new method for performing multiscale analysis of functions defined on the vertices of a finite connected weighted graph. Our approach relies on a random spanning forest to downsample the set of vertices, and on approximate solutions of Markov intertwining relation to provide a subgraph structure and a filter bank leading to a wavelet basis of the set of functions. Our construction involves two parameters q and q'. The first one controls the mean number of kept vertices in the downsampling, while the second one is a tuning parameter between space localization and frequency localization. We provide an explicit reconstruction formula, bounds on the reconstruction operator norm and on the error in the intertwining relation, and a Jackson-like inequality. These bounds lead to recommend a way to choose the parameters q and q'. We illustrate the method by numerical experiments.Comment: 39 pages, 12 figure

Self-Intersection Times for Random Walk, and Random Walk in Random Scenery

25 pages, and 1 figure.We consider Random Walk in Random Scenery , denoted $X_n$, where the random walk is symmetric on $Z^d$, with $d>4$, and the random field is made up of i.i.d random variables with a stretched exponential tail decay, with exponent $\alpha$ with $1n^{\beta}\}$ for $1/2<\beta<1$. To obtain such asymptotics, we establish large deviations estimates for the the self-intersection local times process

Exponential moments of self-intersection local times of stable random walks in subcritical dimensions

Let $(X_t, t \geq 0)$ be an $\alpha$-stable random walk with values in $\Z^d$. Let $l_t(x) = \int_0^t \delta_x(X_s) ds$ be its local time. For $p>1$, not necessarily integer, $I_t = \sum_x l_t^p(x)$ is the so-called $p$-fold self- intersection local time of the random walk. When $p(d -\alpha) < d$, we derive precise logarithmic asymptotics of the probability $P(I_t \geq r_t)$ for all scales r_t \gg \E(I_t). Our result extends previous works by Chen, Li and Rosen 2005, Becker and K\"onig 2010, and Laurent 2012