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What are the precise transition density asymptotics for Levy processes with stochastic resetting when the underlying Levy process lacks a finite first moment?

Related: Dynkin-Lamperti theorem for stable processes, Flajolet-Odlyzko Tauberian theorem for algebraic singularities, Blumenthal-Getoor index classification of Levy processes

The paper on Levy processes with stochastic resetting derives asymptotic formulas for transition densities, but these results appear to rely on moment conditions that exclude the most heavy-tailed Levy processes, specifically those whose jump distributions have infinite first moment such as stable processes with stability index below one. The open problem is to characterize whether a coherent asymptotic theory for transition densities can be developed in this regime, and if so, what qualitatively different behavior emerges compared to the finite-mean case. This matters because many empirical applications in finance and physics involve precisely these ultra-heavy-tailed processes where resetting competes with very large infrequent jumps in a non-trivial way.

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