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Study of direct and inverse first-exit problems for drifted Brownian motion with Poissonian resetting

Mario Abundo

TL;DR

The paper studies direct and inverse first-exit time (FET) and first-exit area (FEA) problems for a drifted Brownian motion with Poissonian resetting in the interval $(0,b)$. It develops a generator-based ODE framework to characterize exit probabilities, Laplace transforms, and moments of the FET and FEA, with explicit elementary solutions available in the drifted BM with resetting case, and non-elementary cases handled via special functions. For the inverse problem, a Laplace-transform relation between the FET density and the initial-position density is derived, with symmetry constraints yielding uniqueness and explicit existence results demonstrated through several examples including uniform, Beta, and exponential-type densities. The results provide a systematic toolkit for direct and inverse exit problems in resetting diffusion models, with potential applications in biology, finance, and queueing theory, and suggest natural extensions to more general diffusions and random reset mechanisms.

Abstract

\noindent We address some direct and inverse problems, for the first-exit time (FET) $τ$ of a drifted Brownian motion with Poissonian resetting ${\cal X}(t)$ from an interval $(0,b)$ and the first-exit area (FEA) $A,$ namely the area swept out by ${\cal X}(t)$ till the time $τ$; this type of diffusion process ${\cal X}(t)$ 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 ${\cal X}(0)= η\in (0,b)$ is deterministic and fixed, the direct FET problem consists in investigating the statistical properties of the FET $τ,$ whilst the direct FEA problem studies the probability distribution of the FEA $A$. The inverse FET problem regards the case when $η$ 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 $η,$ if it exists, such that $P[τ\le t ] = F(t), \ t >0.$ %In addition to the case of random initial position $η,$ we also study the case when the initial position $η$ and the resetting rate $r$ are fixed, whereas the reset position $x_R$ is random. Several explicit examples of solutions to the inverse FET problem are provided.

Study of direct and inverse first-exit problems for drifted Brownian motion with Poissonian resetting

TL;DR

The paper studies direct and inverse first-exit time (FET) and first-exit area (FEA) problems for a drifted Brownian motion with Poissonian resetting in the interval . It develops a generator-based ODE framework to characterize exit probabilities, Laplace transforms, and moments of the FET and FEA, with explicit elementary solutions available in the drifted BM with resetting case, and non-elementary cases handled via special functions. For the inverse problem, a Laplace-transform relation between the FET density and the initial-position density is derived, with symmetry constraints yielding uniqueness and explicit existence results demonstrated through several examples including uniform, Beta, and exponential-type densities. The results provide a systematic toolkit for direct and inverse exit problems in resetting diffusion models, with potential applications in biology, finance, and queueing theory, and suggest natural extensions to more general diffusions and random reset mechanisms.

Abstract

\noindent We address some direct and inverse problems, for the first-exit time (FET) of a drifted Brownian motion with Poissonian resetting from an interval and the first-exit area (FEA) namely the area swept out by till the time ; this type of diffusion process is characterized by the fact that a reset to the position can occur according to a homogeneous Poisson process with rate When the initial position is deterministic and fixed, the direct FET problem consists in investigating the statistical properties of the FET whilst the direct FEA problem studies the probability distribution of the FEA . The inverse FET problem regards the case when is randomly distributed in (while and are fixed); if is a given distribution function on the time axis, the inverse FET problem consists in finding the density of if it exists, such that %In addition to the case of random initial position we also study the case when the initial position and the resetting rate are fixed, whereas the reset position is random. Several explicit examples of solutions to the inverse FET problem are provided.

Paper Structure

This paper contains 12 sections, 2 theorems, 145 equations, 10 figures.

Table of Contents

  1. Introduction
  2. The direct first-exit time problem for drifted Brownian motion with resetting
  3. The exit probability of $\mathcal{X} (t)$ from the left end of the interval $(0,b)$
  4. The Laplace transforms of the survival function and of the first-exit time
  5. The moments of the FET
  6. The Laplace transform of the first-exit area
  7. The moments of the FEA
  8. Joint moment of $\tau_ \mu (x)$ and $A_ \mu(x)$
  9. The distributions of the maximum and minimum displacement till the exit time
  10. The inverse first-exit time (IFET) problem for BM with resetting
  11. Some examples of solutions to the IFET problem
  12. Conclusions and Final Remarks

Key Result

Proposition 3.1

Let $\mathcal{X}(t)$ be (undrifted) BM with resetting, starting from the random initial position $\eta \in (0, b),$ which is supposed to be independent of $\mathcal{X}(t),$ and let $f(t), \ t >0,$ be a given FPT density. Then, if there exists a solution $g$ to the IFET problem for $\mathcal{X}(t),$ $(\alpha _ \lambda = \sqrt {2(\lambda +r)}).$ If one requires that $g$ is symmetric with respect to

Figures (10)

  • Figure 1: Five examples of the graphs of $\pi_0^\mu (x)$ given by \ref{['pi0driftBMreset']}, as function of $x \in (0,b),$ for $b=1, \ \mu =1, \ r = 1, \ x_R= 1/4$ (lowest curve); $b=1, \ \mu =1, r = 0$ (slightly higher curve); $b=1, \ \mu =0, \ r = 0$ (straight line); $b=1, \ \mu =1, \ r = 5, \ x_R= 1/4$ (curve with inflection point); $b=1, \ \mu =1, \ r = 50, \ x_R= 1/4$ (highest, concave curve).
  • Figure 2: Four examples of plots of $E[\tau _ \mu (x)],$ as a function of $x \in (0,b);$ the curves are ordered from the lower to the higher peak height. The sets of parameters are: $b=1, \ \mu =1, \ r = 0$ (curve 1); $b=1, \ \mu =1, \ r = 1, \ x_R=1/4$ (curve 2); $b=1, \ \mu =1, \ r = 5, \ x_R=1/4$ (curve 3); $b=1, \ \mu = 0, \ r = 0$ (curve 4).
  • Figure 3: Four examples of plots of $E[\tau _ \mu ^2(x)],$ as a function of $x \in (0,b);$ the curves are ordered from the lower to the higher peak height. The sets of parameters are the same ones as in the Figure \ref{['expectedFET']}.
  • Figure 4: Three examples of graphs of $E[A _ 0 (x)],$ as a function of $x \in (0,b);$ the curves are ordered from the higher to the lower peak height. The sets of parameters are: $\mu =0, \ b=1, r = 0$ (curve 1); $\mu =0, \ b=1, r = 1/2, \ x_R= 1/8$ (curve 2); $\mu =0, \ b=1, r = 5, \ x_R= 1/8$ (curve 3).
  • Figure 5: Three examples of graphs of of $E[A ^2_ 0 (x)],$ as a function of $x \in (0,b);$ the curves are from the higher to the lower peak height. The sets of parameters are: $\mu =0, \ b=1, r = 0$ (curve 1); $\mu =0, \ b=1, r = 1/2, \ x_R= 1/8$ (curve 2); $\mu =0, \ b=1, r = 5, \ x_R= 1/8$ (curve 3).
  • ...and 5 more figures

Theorems & Definitions (4)

  • Remark 2.1
  • Proposition 3.1
  • Remark 3.2
  • Proposition 3.3