Fine Properties Of Continuous Stable Trees

Theγ-stable trees are random measured compact metric spaces that appear asthe scaling limit of Galton-Watson trees whose ofspring distribution lies in a γ-stabledomain. Here, necessarilyγranges in (1,2]. They form a specific class of Le′vy trees(introduced by Le Gall and Le Jan in^[1]) and the Brownian caseγ=2corresponds toAldous Continuum Random Tree （short for CRT）^[2].In this PhD, we study fine properties of the local-time measures and the massmeasure of stable trees. The mass measure that is the natural measure onγ-stabletrees: it is the most spread out measure on the tree and it is an excat packing measure,and in the Brownian case it is an exact Hausdorf measure (see^[3,4]).We first discuss the minimum and the maximum of the local time measure of ballswith radius r on the level set of the Brownian tree （the levels sets are the sets at a fixeddistance from the root）. This show that the maximum of the local time measure isasymptotically equivalent to1/2r log1/r. We prove that the minimum of the local timemeasure is between r²（log1/r）2and r²（log1/r）^-2.In the general stable casesγ∈(1,2], we consider the minimum of the mass mea-γsure of balls with radius r and we show that this quantity is of order γ/r^γ-1（log1/r）^-1/γ-1We think that no similar result holds true for the maximum of the mass measure of ballswith radius r, except in the Brownian case: whenγ=2, we prove that this quantity isof order r²log1/r.In addition, we compute the exact constant for the lower local density of the massmeasure （and the upper one for the CRT）, which continues previous results from^[3–5].