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 Clément Laurent Mathematics , 2010, Abstract: Let $(X_t,t\geq 0)$ be a random walk on $\mathbb{Z}^d$. Let $l_T(x)= \int_0^T \delta_x(X_s)ds$ the local time at the state $x$ and $I_T= \sum\limits_{x\in\mathbb{Z}^d} l_T(x)^q$ the q-fold self-intersection local time (SILT). In \cite{Castell} Castell proves a large deviations principle for the SILT of the simple random walk in the critical case $q(d-2)=d$. In the supercritical case $q(d-2)>d$, Chen and M\"orters obtain in \cite{ChenMorters} a large deviations principle for the intersection of $q$ independent random walks, and Asselah obtains in \cite{Asselah5} a large deviations principle for the SILT with $q=2$. We extend these results to an $\alpha$-stable process (i.e. $\alpha\in]0,2]$) in the case where $q(d-\alpha)\geq d$.
 Mathematics , 2009, Abstract: We study the object formally defined as \gamma\big([0,t]^{2}\big)=\int\int_{[0,t]^{2}} | X_{s}- X_{r}|^{-\sigma} dr ds-E\int\int_{[0,t]^{2}} | X_{s}- X_{r}|^{-\sigma} dr ds, where $X_{t}$ is the symmetric stable processes of index $0<\beta\le 2$ in $R^{d}$. When $\beta\le\sigma<\displaystyle\min \Big\{{3\over 2}\beta, d\Big\}$, this has to be defined as a limit, in the spirit of renormalized self-intersection local time. We obtain results about the large deviations and laws of the iterated logarithm for $\gamma$. This is applied to obtain results about stable processes in random potentials.
 Clément Laurent Mathematics , 2010, Abstract: Let $(X_t,t\geq 0)$ be a random walk on $\mathbb{Z}^d$. Let $l_t(x)= \int_0^t \delta_x(X_s)ds$ be the local time at site $x$ and $I_t= \sum\limits_{x\in\mathbb{Z}^d} l_t(x)^p$ the p-fold self-intersection local time (SILT). Becker and K\"onig have recently proved a large deviations principle for $I_t$ for all $(p,d)\in\mathbb{R}^d\times\mathbb{Z}^d$ such that $p(d-2)<2$. We extend these results to a broader scale of deviations and to the whole subcritical domain $p(d-2)  Fabienne Castell Mathematics , 2008, DOI: 10.1214/09-AOP499 Abstract: Let$(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$independent simple random walks.  Mathematics , 2008, Abstract: In this paper we apply Clark-Ocone formula to deduce an explicit integral representation for the renormalized self-intersection local time of the$d$% -dimensional fractional Brownian motion with Hurst parameter$H\in (0,1)$. As a consequence, we derive the existence of some exponential moments for this random variable.  Jay S. Rosen Mathematics , 2009, DOI: 10.1214/09-AIHP338 Abstract: We study$\gamma_{k}(x_2,...,x_k;t)$, the k-fold renormalized self-intersection local time for Brownian motion in$R^1$. Our main result says that$\gamma_{k}(x_2,...,x_k;t)$is continuously differentiable in the spatial variables, with probability 1.  Mathematics , 2004, Abstract: Let \beta_k(n) be the number of self-intersections of order k, appropriately renormalized, for a mean zero random walk X_n in Z^2 with 2+\delta moments. On a suitable probability space we can construct X_n and a planar Brownian motion W_t such that for each k\geq 2, |\beta_k(n)-\gamma_k(n)|=O(n^{-a}), a.s. for some a>0 where \gamma_k(n) is the renormalized self-intersection local time of order k at time 1 for the Brownian motion W_{nt}/\sqrt n.  Amine Asselah Mathematics , 2007, Abstract: We obtain a large deviations principle for the self-intersection local times for a symmetric random walk in dimension d>4. As an application, we obtain moderate deviations for random walk in random sceneries in some region of parameters.  Amine Asselah Mathematics , 2006, Abstract: We obtain large deviations estimates for the self-intersection local times for a symmetric random walk in dimension 3. Also, we show that the main contribution to making the self-intersection large, in a time period of length$n$, comes from sites visited less than some power of$\log(n)\$. This is opposite to the situation in dimensions larger or equal to 5. Finally, we present two applications of our estimates: (i) to moderate deviations estimates for the range of a random walk, and (ii) to moderate deviations for random walk in random sceneries.