First-passage properties of the jump process with a drift. The general case

TL;DR

This work analyzes the first-passage properties of a jump process with constant drift by mapping the continuous-time dynamics to an effective discrete-time random walk with light-tailed inter-arrival and jump distributions. Using a triple Laplace transform and Pollaczek-Spitzer-type factorization, it derives explicit exponential decay rates in the survival and absorption regimes and algebraic decay at the critical point, complemented by Brownian-scaling limits. It also provides systematic, Mellin-transform-based expansions for the means and variances of the first-passage time and jump count, valid for general distributions and accessible to numerical evaluation via Fourier moments. The results identify a drift threshold $\alpha_c=\langle M\rangle/\langle t\rangle$ separating regimes, with detailed asymptotics for $X_0\to0$ and $X_0\to\infty$ and robust scaling forms at criticality, offering a powerful framework beyond exactly solvable cases. The methodology and explicit integral expressions facilitate applications to queuing, risk, and related stochastic-resetting problems where first-passage phenomena are central.

Abstract

We study the first-passage properties of a jump process with constant drift where jump amplitudes and inter-arrival times follow arbitrary light-tailed distributions with smooth densities. Using a mapping to an effective discrete-time random walk, we identify three regimes determined by the drift strength: survival (weak drift), absorption (strong drift), and critical. We derive explicit expressions for exponential decay rates in the survival and absorption regimes, and characterize algebraic decay at the critical point. We also obtain asymptotic behavior of the mean first-passage time, number of jumps, and their variances for processes starting either close to the origin or far from it.

Figures (11)

• Figure 1: An example of the trajectory. Starting at $X_0$ the process instantaneously undergoes a jump $M_1$, then moves toward the origin with constant velocity $\alpha$ for the time $t_1$ when the next jump $M_2$ occurs. This pattern continues until the process crosses the origin at time $\tau$ after $n$ jumps (here $n=6$). The inter-jump intervals $\{t_1,\ldots,t_6\}$ are i.i.d. random variables drawn from $p(t)$, and the jump amplitudes $\{M_1,\ldots,M_6\}$ are also i.i.d. random variables following $q(M)$.
• Figure 2: Schematic representation of the survival probability $S_\infty(X_0)$ as a function of the drift strength $\alpha$ for fixed initial position $X_0$. Below the critical value $\alpha_c$, the system is in the survival regime with $S_\infty(X_0) > 0$. The critical point occurs at $\alpha = \alpha_c$. Above $\alpha_c$, the system is in the absorption regime with $S_\infty(X_0) = 0$.
• Figure 3: The structure of the Fourier transform $F(k;\rho)$ in the complex $k$-plane. The function is analytic in the horizontal strip $\, \mathrm{Im}(k) \in (\zeta_-, \zeta_+)$, where $\zeta_\pm$ are constants determined by the decay rates of the probability densities $p(t)$ and $q(M)$, as given in \ref{['eq:zeat_pm=def']}.
• Figure 4: Schematic plot of $F(k; \rho)$ along the imaginary axis. The function is convex with the minimum at $\zeta_*\equiv\zeta_*(\rho)$ which is negative if $\mu_1(\rho)<0$ and positive if $\mu_1(\rho)>0$. If $s \, c(\rho)<1$, then there are two real solutions $\zeta_{1,2}\equiv\zeta_{1,2}(\rho,s)$ to \ref{['eq:zeta12=def']} located on opposite sides of the origin.
• Figure 5: The structure of $\log[1-s c(\rho)F(k;\rho)]$ in the complex $k$-plane. The function is analytic in the horizontal strip $\, \mathrm{Im}(k) \in (\zeta_1, \zeta_2)$ with branching points at $k=\mathrm{i} \zeta_{1,2}$. The edges of the strip $\zeta_{1,2}$ are two real solutions of \ref{['eq:zeta12=def']} as shown in Fig. \ref{['fig:F_imaginary_line']}. Note that \ref{['eq:zeta12=def']} may have other non-real solutions, leading to more singularities in the logarithmic term, but due to \ref{['eq:|scF|<1 logStrip']} these singularities lie in the shaded region.