On moments-preserving cosine families and semigroups in $C[0,1]$

TL;DR

This paper constructs a unique cosine family in continuous functions on [0,1] that preserves the first two moments, characterizes its generator, and analyzes its symmetry and long-term behavior.

Contribution

It introduces a moments-preserving cosine family generated by a restricted Laplace operator using Kelvin's method, with detailed domain and symmetry properties.

Findings

Existence of a unique moments-preserving cosine family in C[0,1]

Characterization of the generator's domain and boundary conditions

Analysis of the semigroup's long-time behavior

Abstract

We use the newly developed Kelvin's method of images \cite{kosinusy,kelvin} to show existence of a unique cosine family generated by a restriction of the Laplace operator in $C[0,1]$, that preserves the first two moments. We characterize the domain of its generator by specifying its boundary conditions. Also, we show that it enjoys inherent symmetry properties, and in particular that it leaves the subspaces of odd and even functions invariant. Furthermore, we provide information on long-time behavior of the related semigroup.