Conditioned backward and forward times of diffusion with stochastic resetting: A renewal theory approach,Physical Review E

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Conditioned backward and forward times of diffusion with stochastic resetting: A renewal theory approach
Physical Review E ( IF 2.2 ) Pub Date : 2022-09-16 , DOI: 10.1103/physreve.106.034126
Axel Masó-Puigdellosas ₁ , Daniel Campos ₁ , Vicenç Méndez ₁

Affiliation

Stochastic resetting can be naturally understood as a renewal process governing the evolution of an underlying stochastic process. In this work, we formally derive well-known results of diffusion with resets from a renewal theory perspective. Parallel to the concepts from renewal theory, we introduce the conditioned backward

B

and forward

F

times being the times since the last and until the next reset, respectively, given that the current state of the system

X (t)

is known. These magnitudes are introduced with the paradigmatic case of diffusion under resetting, for which the backward and forward times are conditioned to the position of the walker. We find analytical expressions for the conditioned backward and forward time probability density functions (PDFs), and we compare them with numerical simulations. The general expressions allow us to study particular scenarios. For instance, for power-law reset time PDFs such that