On the range of a two-dimensional conditioned simple random walk

Gantert, Nina; Popov, Serguei; Vachkovskaia, Marina

Annales Henri Lebesgue, Tome 2 (2019), pp. 349-368.

Nous considérons une marche aléatoire simple en dimension $2$ conditionnée à ne jamais atteindre l’origine. Ce processus est une chaîne de Markov, à savoir la transformation de Doob $h$ de la marche aléatoire simple par rapport au noyau potentiel. Il est connu que ce processus est transient et nous montrons qu’il est « presque récurrent » en ce sens que chaque ensemble infini est visité infiniment souvent, presque sûrement. Nous prouvons que, pour un « grand » ensemble, la proportion des sites visités par la marche aléatoire conditionnée est approximativement une variable aléatoire uniforme dans $[0,1]$. En outre, étant donné un ensemble $G\subset {ℝ}^2$ qui « n’entoure » pas l’origine, nous prouvons que p.s., il existe un nombre infini d’entiers naturels $k$ tels que $kG\cap {\mathbb{Z}}^2$ ne soit pas visité. Ces résultats suggèrent que l’amplitude de la marche simple conditionnée a un comportement « fractal ».

We consider the two-dimensional simple random walk conditioned on never hitting the origin. This process is a Markov chain, namely it is the Doob $h$-transform of the simple random walk with respect to the potential kernel. It is known to be transient and we show that it is “almost recurrent” in the sense that each infinite set is visited infinitely often, almost surely. We prove that, for a “large” set, the proportion of its sites visited by the conditioned walk is approximately a Uniform$[0,1]$ random variable. Also, given a set $G\subset {ℝ}^2$ that does not “surround” the origin, we prove that a.s. there is an infinite number of $k$’s such that $kG\cap {\mathbb{Z}}^2$ is unvisited. These results suggest that the range of the conditioned walk has “fractal” behavior.

Classification : 60J1060G50, 82C41
Mots clés : random interlacements, range, transience, simple random walk, Doob’s $h$-transform

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