Some new results for McKean's graphs with applications to Kac's equation

The paper under review considers the Kac equation, which is an analog of the Boltzmann equation for Maxwell molecules in dimension one, and thus describes the evolution of the velocity distribution in a sort of one-dimensional gas. This equation is known to admit an explicit solution, in terms of a series, known as the Wild sums. The authors investigate the speed of convergence of this series. McKean gave a probabilistic interpretation of the Wild sums in terms of graphs. Using this interpretation, E. A. Carlen, M. C. V. Carvalho and E. Gabetta [J. Funct. Anal. 220 (2005)] showed that the error made when the series is truncated at the n-th stage is bounded from above by a(n^(?+?)), for some constant a, and where ? is the least negative eigenvalue of the linearized collision operator. In the present paper, the authors show that ? can be removed from the previous estimate. The proofs rely on a fine study of the probability distribution of the depth of a leaf in a McKean random tree.