Submitted by Maike Costa (maiksebas@gmail.com) on 2016-03-28T13:00:07Z No. of bitstreams: 1 arquivo total.pdf: 809257 bytes, checksum: 7406ef37d18bbaf1d9cdd5649f5cff19 (MD5) / Made available in DSpace on 2016-03-28T13:00:07Z (GMT). No. of bitstreams: 1 arquivo total.pdf: 809257 bytes, checksum: 7406ef37d18bbaf1d9cdd5649f5cff19 (MD5) Previous issue date: 2015-12-11 / Let X = fX 0;X0 = 0g be a mean zero -stable random walk on Z with inhomogeneous jump rates f 􀀀1 i ; i 2 Zg, with 2 (1; 2] and f i : i 2 Zg is a family of independent random walk variables with common marginal distribution in the basis of attraction of an -stable law with 2 (0; 2]. In this paper we derive results about the long time behavior of this process, we obtain the scaling limit. To this end, rst we will approach probability on metric spaces, speci cally treat the D space of the functions that are right-continuous and have left-hand limits. We will also expose some results dealing with stable laws that are directly related to the above problem. / Seja X = fX 0;X0 = 0g um passeio aleat orio de m edia zero -est avel sobre Z com taxas de saltos n~ao homog^eneas f 􀀀1 i ; i 2 Zg, com 2 (1; 2] e f i : i 2 Zg uma fam lia de vari aveis aleat orias independentes com distribui c~ao marginal comum na bacia de atra c~ao de uma lei -est avel com 2 (0; 2]. Neste trabalho, obtemos resultados sobre o comportamento a longo prazo deste processo obtendo seu limite de escala. Para isso, faremos previamente um estudo sobre probabilidade em espa cos m etricos, mais especi camente sobre o espa co D das fun coes cont nuas a direita com limite a esquerda. Tamb em iremos expor alguns resultados que tratam de leis est aveis que est~ao relacionadas diretamente ao problema supracitado.

L’objectif en arrière-plan est de montrer que plusieurs modèles de marches aléatoires en milieux aléatoires (MAMA) sont reliés à un modèle-jouet appelé le modèle de piège de Bouchaud. Le domaine des MAMA est très vaste, mais nous nous intéressons particulièrement à une classe de modèle où la marche est réversible et directionnellement transiente. En particulier, nous verrons pourquoi on pense que ces modèles se ressemblent et quel genre de similarités on s’attend à obtenir, une fois qu’on aura présenté le modèle de Bouchaud. Nous verrons aussi quelques techniques de base utilisés de ce domaine, telles que les temps de régénérations. Comme contribution, nous allons démontrer un théorème central limite pour la marche aléatoire β-biaisée sur un arbre de Galton-Watson. / This Master thesis is part of a larger project of linking the behaviours of a certain type of random walks in random environments (RWRE) with those of a toy model called the Bouchaud’s trap model. The domain of RWRE is very wide but our interest will be on a particular kind of models which are reversible and directionally transient. More specifically, we will see why those models have similar behaviours and what kind of results we could expect once we have reviewed the Bouchaud’s trap model. We will also present some basic technic used in this field, such as regeneration times. As a contribution, we will demonstrate a central limit theorem for the β-biased random walk on a Galton-Watson tree.