Expediting Feller process with stochastic resetting,Physical Review E

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Expediting Feller process with stochastic resetting
Physical Review E ( IF 2.2 ) Pub Date : 2022-09-26 , DOI: 10.1103/physreve.106.034133
Somrita Ray ₁

Affiliation

We explore the effect of stochastic resetting on the first-passage properties of the Feller process. The Feller process can be envisioned as space-dependent diffusion, with diffusion coefficient

D (x) = x

, in a potential

U (x) = x (\frac{x}{2} - θ)

that owns a minimum at

θ

. This restricts the process to the positive side of the origin and therefore, Feller diffusion can successfully model a vast array of phenomena in biological and social sciences, where realization of negative values is forbidden. In our analytically tractable model system, a particle that undergoes Feller diffusion is subject to Poissonian resetting, i.e., taken back to its initial position at a constant rate

r

, after random time epochs. We addressed the two distinct cases that arise when the relative position of the absorbing boundary (

x_{a}

) with respect to the initial position of the particle (

x_{0}

) differ, i.e., for (a)

x_{0} < x_{a}

and (b)

x_{a} < x_{0}

. Utilizing the Fokker-Planck description of the system, we obtained closed-form expressions for the Laplace transform of the survival probability and hence derived the exact expressions of the mean first-passage time

〈 T_{r} 〉

. Performing a comprehensive analysis on the optimal resetting rate (

r^{★}

) that minimize

〈 T_{r} 〉

and the maximal speedup that

r^{★}

renders, we identify the phase space where Poissonian resetting facilitates first-passage for Feller diffusion. We observe that for

x_{0} < x_{a}

, resetting accelerates first-passage when

θ < θ_{c}

, where

θ_{c}

is a critical value of

θ

that decreases when

x_{a}

is moved away from the origin. In stark contrast, for

x_{a} < x_{0}

, resetting accelerates first-passage when

θ > θ_{c}

, where

θ_{c}

is a critical value of

θ

that increases when

x_{0}

is moved away from the origin. Our study opens up the possibility of a series of subsequent works with more case-specific models of Feller diffusion with resetting.

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