Continuous-time statistics and generalized relaxation equations

TL;DR

This paper explores the connection between generalized relaxation equations and semi-Markov processes through simple examples, extending to continuous-time random statistics like sum and maximum, highlighting their theoretical relationships.

Contribution

It establishes a general framework linking anomalous relaxation equations with semi-Markov processes and applies it to convolution-type statistics.

Findings

Relation between relaxation equations and semi-Markov processes clarified

Detailed analysis of sum and maximum statistics in continuous time

Framework applicable to a broad class of convolution-type statistics

Abstract

Using two simple examples, the continuous-time random walk as well as a two state Markov chain, the relation between generalized anomalous relaxation equations and semi-Markov processes is illustrated. This relation is then used to discuss continuous-time random statistics in a general setting, for statistics of convolution-type. Two examples are presented in some detail: the sum statistic and the maximum statistic.