TY - JOUR

T1 - Conditioned backward and forward times of diffusion with stochastic resetting

T2 - A renewal theory approach

AU - Masó-Puigdellosas, Axel

AU - Campos, Daniel

AU - Méndez, Vicenç

N1 - Publisher Copyright:
© 2022 American Physical Society.

PY - 2022/9/16

Y1 - 2022/9/16

N2 - 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 φ(t)∼t-1-α, significant changes in the properties of the conditioned backward and forward times happen at half-integer values of α due to the composition between the long-time scaling of diffusion P(x,t)∼1/t and the reset time PDF.

AB - 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 φ(t)∼t-1-α, significant changes in the properties of the conditioned backward and forward times happen at half-integer values of α due to the composition between the long-time scaling of diffusion P(x,t)∼1/t and the reset time PDF.

UR - http://www.scopus.com/inward/record.url?scp=85138996532&partnerID=8YFLogxK

U2 - 10.1103/physreve.106.034126

DO - 10.1103/physreve.106.034126

M3 - Article

C2 - 36266817

AN - SCOPUS:85138996532

VL - 106

JO - Physical Review E

JF - Physical Review E

SN - 2470-0045

IS - 3

M1 - 034126

ER -