Abstract :
In the case of Maxwellian molecules, the Wild summation formula gives an expression for
the solution of the spatially homogeneous Boltzmann equation in terms of its initial data F as a
sum f (v, t)= ∞
n=1 e
−t (1−e
−t )n−1Q
+
n (F )(v). Here, Q
+
n (F ) is an average over n-fold iterated
Wild convolutions of F. If M denotes the Maxwellian equilibrium corresponding to F, then it
is of interest to determine bounds on the rate at which Q
+
n (F ) − M
L1(R) tends to zero. In
the case of the Kac model, we prove that for every >0, if F has moments of every order and
finite Fisher information, there is a constant C so that for all n, Q
+
n (F ) −M
L1(R) Cn +
where is the least negative eigenvalue for the linearized collision operator. We show that is
the best possible exponent by relating this estimate to a sharp estimate for the rate of relaxation
of f (·, t) to M. A key role in the analysis is played by a decomposition of Q
+
n (F ) into a
smooth part and a small part. This depends in an essential way on a probabilistic construction
of McKean. It allows us to circumvent difficulties stemming from the fact that the evolution
does not improve the qualitative regularity of the initial data.
