Study of Direct and Inverse First-exit Problems for Drifted Brownian Motion with Poissonian Resetting

Mario Abundo ()
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Mario Abundo: Università “Tor Vergata”

Methodology and Computing in Applied Probability, 2025, vol. 27, issue 4, 1-34

Abstract: Abstract We address some direct and inverse problems, for the first-exit time (FET) $$\tau$$ of a drifted Brownian motion with Poissonian resetting $$\mathcal {X}(t)$$ from an interval (0, b) and the first-exit area (FEA) A, namely the area swept out by $$\mathcal {X}(t)$$ till the time $$\tau$$ ; this type of diffusion process $$\mathcal {X}(t)$$ is characterized by the fact that a reset to the position $$x_R$$ can occur according to a homogeneous Poisson process with rate $$r>0.$$ When the initial position $$\mathcal {X}(0)= \eta \in (0,b)$$ is deterministic and fixed, the direct FET problem consists in investigating the statistical properties of the FET $$\tau ,$$ whilst the direct FEA problem studies the probability distribution of the FEA A. The inverse FET problem regards the case when $$\eta$$ is randomly distributed in (0, b) (while r and $$x_R$$ are fixed); if F(t) is a given distribution function on the time t axis, the inverse FET problem consists in finding the density g of $$\eta ,$$ if it exists, such that $$P[\tau \le t ] = F(t), \ t>0.$$ Several explicit examples of solutions to the inverse FET problem are provided.

Keywords: Diffusion with resetting; First-passage time; First-passage place; 60J60; 60H05; 60H10 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s11009-025-10216-z

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