Multi-dimensional smoothing transformations: existence, regularity and stability of fixed points
The authors study properties of \(d\)-dimensional smoothing transformations, defined on the probability measures. Two types of results are provided for this multidimensional case that has been little studied in the literature. The first concerns existence, uniqueness and regularity of fixed points for smoothing transformations, and a representation of those as a scale mixture of Gaussians. The second result is a sharp characterization of the basin of attraction of these fixed points for transient solutions to the multidimensional Kac equation. A particular set of assumptions allows to prove uniqueness of the fixed point in the whole class of probability measures with finite second moments, and to characterize all the fixed points as scale mixtures of Gaussians. The main difference with the previous results in this direction is that sharp bounds on the domain of attraction are established, i.e., convergence to equilibrium is proved for arbitrary initial conditions of finite variance, without further hypotheses on higher moments or on the covariance of the initial measure. The results of the paper do not improve on what is already known about the long-time asymptotics of the ``physical'' kinetic equations. In contrast, the results are valid for the much larger class of Kac equations. Under additional hypotheses on higher moments of the initial condition, the obtained qualitative convergence result can even be quantified on the level of the Fourier transform.
multi-dimensional smoothing transformations
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