La topologie fine a été introduite pour fournir un cadre intrinsèque à la théorie du potentiel. Cependant les ouverts fins ne possèdent pas certaines propriétés dont celle de Lindeberg. Cette considération nous conduit à introduire des topologies moins finies appelées $p$-topologies $(p\in {\mathbf{R}}_{+}^{*}$). Nous démontrons pour ces $p$-topologies un critère analogue à celui établi par N. Wiener, pour les ouverts fins. Puis nous nous intéressons à la théorie des équations différentielles stochastiques sur les $p$-ouverts.

In the present work, we consider spectrally positive Lévy processes $(X_t,t\ge 0)$ not drifting to $+\infty$ and we are interested in conditioning these processes to reach arbitrarily large heights (in the sense of the height process associated with $X$) before hitting $0$. This way we obtain a new conditioning of Lévy processes to stay positive. The (honest) law ${\mathbb{P}}_x^{☆}$ of this conditioned process (starting at $xgt;0$) is defined as a Doob $h$-transform via a martingale. For Lévy processes with infinite variation paths, this martingale...

Let ${\tau }_D\left(Z\right)$ be the first exit time of iterated Brownian motion from a domain $D\subset {ℝ}^n$ started at $z\in D$ and let $P_z[{\tau }_D\left(Z\right)>t]$ be its distribution. In this paper we establish the exact asymptotics of $P_z[{\tau }_D\left(Z\right)>t]$ over bounded domains as an improvement of the results in DeBlassie (2004) [DeBlassie, Ann. Appl. Prob.14 (2004) 1529–1558] and Nane (2006) [Nane, Stochastic Processes Appl.116 (2006) 905–916], for $z\in D$
 ${\displaystyle \underset{t\to \infty }{lim}{t}^{-1/2}exp\left(\frac{3}{2}{\pi }^{2/3}{\lambda }_D^{2/3}{t}^{1/3}\right)P_z[{\tau }_D\left(Z\right)>t]=C\left(z\right),}$
where $C\left(z\right)=\left({\lambda }_D{2}^{7/2}\right)/\sqrt{3\pi }{\left(\psi \left(z\right){\int }_D\psi \left(y\right)\mathrm{d}y\right)}^2$. Here λD is the first eigenvalue of the Dirichlet Laplacian $\frac{1}{2}\Delta$ in D, and ψ is the eigenfunction corresponding...

Let $m\ge 3$ be an integer. The so-called$m$-ary search treeis a discrete time Markov chain which is very popular in theoretical computer science, modelling famous algorithms used in searching and sorting. This random process satisfies a well-known phase transition: when $m\le 26$, the asymptotic behavior of the process is Gaussian, but for $m\ge 27$ it is no longer Gaussian and a limit $W^{DT}$ of a complex-valued martingale arises. In this paper, we consider the multitype branching process which is the continuous time version...

We consider excited random walks (ERWs) on ℤ with a bounded number of i.i.d. cookies per site without the non-negativity assumption on the drifts induced by the cookies. Kosygina and Zerner [15] have shown that when the total expected drift per site, δ, is larger than 1 then ERW is transient to the right and, moreover, for δ>4 under the averaged measure it obeys the Central Limit Theorem. We show that when δ∈(2, 4] the limiting behavior of an appropriately centered and scaled excited random...

We find limit shapes for a family of multiplicative measures on the set of partitions, induced by exponential generating functions with expansive parameters, ak∼Ckp−1, k→∞, p>0, where C is a positive constant. The measures considered are associated with the generalized Maxwell–Boltzmann models in statistical mechanics, reversible coagulation–fragmentation processes and combinatorial structures, known as assemblies. We prove a central limit theorem for fluctuations of a properly scaled partition...