The asymptotic behavior of a class of nonlinear semigroups in Hadamard spaces

TL;DR

This paper investigates the asymptotic behavior of nonlinear semigroups in Hadamard spaces, demonstrating their weak convergence to fixed points and linking this to the proximal point algorithm, with applications to heat flows in singular spaces.

Contribution

It establishes the weak convergence of nonlinear semigroups in Hadamard spaces and connects this to the proximal point algorithm, offering new insights into heat flows in singular spaces.

Findings

Weak convergence of nonlinear semigroups to fixed points

Proximal point algorithm shares the same asymptotic behavior

New approach to heat flows in singular spaces

Abstract

We study a nonlinear semigroup associated to a nonexpansive mapping on a Hadamard space and establish its weak convergence to a fixed point. A discrete-time counterpart of such a semigroup, the proximal point algorithm, turns out to have the same asymptotic behavior. This complements several results in the literature -- both classical and more recent ones. As an application, we obtain a new approach to heat flows in singular spaces for discrete, as well as continuous times.