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
-
1
Q
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
∥
L
1
(
R
)
tends to zero. In the case of the Kac model, we prove that for every
ε
>
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
∥
L
1
(
R
)
⩽
Cn
Λ
+
ε
where
Λ
is the least negative eigenvalue for the linearized collision operator. We show that
Λ
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.