On partial convolution and mean comparison

Abstract

We prove mean comparison results from a different perspective, where we introduce the concept of partial convolutions. For a parabolic initial data problem on the whole domain of dimension n we consider data functions which live on a subspace of lower dimension k and coefficient functions which live on the whole space or some subspace of dimension l. The pair of natural numbers (k, l) is called a strong partial convolution pair if for some coordinate transformations the coefficients functions of the equation and the initial data functions live in complementary linear spaces of dimension k and l respectively. Here the qualification 'strong' indicates that this transformation exists without any further restriction concerning the intersection of the original linear subspaces involved, and that refined concepts are possible in this respect. For purely second order parabolic equations we show that (1, n) for any n ≥ 2 is a strong partial convolution pair. As a consequence new criteria for convexity preservation for classes of initial value problems are obtained. A further consequence is mean comparison for univariate convex functions of a considerable class of sums of locally continuous martingales. We leave the problem of determination of all partial convolution pairs as an open problem. In the literature it is observed that (n, n) is not a strong partial convolution pair.