Non-contraction of heat flow on Minkowski spaces

TL;DR

This paper investigates the contractivity of gradient flows in normed and Finsler spaces, introducing a new convexity concept, and demonstrates that heat flow on Minkowski spaces (excluding inner product spaces) is not contractive under quadratic Wasserstein distance.

Contribution

It introduces a novel convexity notion linked to flow contractivity and proves non-contractivity of heat flow on non-inner product Minkowski spaces.

Findings

Heat flow on Minkowski spaces is not contractive with respect to quadratic Wasserstein distance.

Contractivity is equivalent to a new form of convexity different from traditional geodesic convexity.

The new convexity concept characterizes flow contractivity in Finsler manifolds.

Abstract

We study contractivity properties of gradient flows for functions on normed spaces or, more generally, on Finsler manifolds. Contractivity of the flows turns out to be equivalent to a new notion of convexity for the functions. This is different from the usual convexity along geodesics in non-Riemannian Finsler manifolds. As an application, we show that the heat flow on Minkowski normed spaces other than inner product spaces is not contractive with respect to the quadratic Wasserstein distance.