A Jump-Type SDE Approach to Real-Valued Self-Similar Markov Processes

TL;DR

This paper introduces a jump-type SDE framework to classify symmetric real-valued self-similar Markov processes that only decrease via jumps and continuously leave zero, extending classical positive process classifications.

Contribution

It provides a novel jump-type SDE approach for classifying symmetric self-similar Markov processes that do not necessarily have zero as a trap, expanding existing theories.

Findings

Classified symmetric real-valued self-similar Markov processes using jump-type SDEs.

Developed a pseudo excursion construction for these processes.

Ensured solutions spend zero time at zero to handle singularities.

Abstract

In his 1972 paper, John Lamperti characterized all positive self-similar Markov processes as time-changes of exponentials of Levy processes. In the past decade the problem of classifying all non-negative self-similar Markov processes that do not necessarily have zero as a trap has been solved gradually via connections to ladder height processes and excursion theory. Motivated by a recent article of Chaumont, Rivero, Panti we classify via jump-type SDEs the symmetric real-valued self-similar Markov processes that only decrease the absolute value by jumps and leave zero continuously. Our construction of these self-similar processes involves a pseudo excursion construction and singular stochastic calculus arguments ensuring that solutions to the SDEs spend zero time at zero to avoid problems caused by a "bang-bang" drift.