Mean first passage times of higher-dimensional velocity jump processes

Abstract

First passage phenomena arise across physics, biology, and finance when stochastic processes first reach a threshold, triggering downstream events. Examples include the irreversible exit from a domain, a biochemical reaction, a financial selloff. While typical formulations involve diffusive motion, many stochastic processes are better described as velocity jump processes, characterized by persistent motion interrupted by stochastic velocity changes. Despite their ubiquity, first-passage properties of velocity jump processes remain underdeveloped in higher dimensions, especially under directional bias. We present a general framework to estimate the mean first passage time (MFPT) and higher moments of the survival probability for fixed-speed velocity jump processes where possible reorientations range from strong alignment to full angular anisotropy. For low Knudsen numbers, when the mean free path is small compared to the distance to the target, we derive a universal form for the MFPT in which two bias functions encode broad classes of angular distributions, including von Mises-Fisher, wrapped Cauchy, and elliptical families. In the narrow capture limit of a vanishingly small target, directional persistence induces anomalous scaling, including regimes where the MFPT remains finite whereas standard diffusion would predict divergence. Finally, we obtain a Langevin representation that accurately reproduces first-passage statistics. Analytical predictions are confirmed by numerical simulations.

Figures (3)

• Figure 1: The von Mises angular distribution in Eq. \ref{['VM']} for $d=2$ and three values of the concentration $\kappa(\bf x)$ that quantifies the bias towards $\hat{\gamma}( {\bf x})$. Larger $\kappa (\bf x)$ yields sharper alignment.
• Figure 2: Analytical estimates from Eq. \ref{['MFPTsol']} (curves) and particle simulations (squares) of the MFPT to the boundary of a ball of radius $R_0 = 6$ in $d=2$ (left) and $d=3$ (right). Reorientations follow the von Mises ($d=2$) or Fisher ($d=3$) distributions given in Table \ref{['table1']}. The speed $\sigma = 0.06$ and turning rate $\mu=1$ yield the diffusion constant $D = \sigma^2 / (d \mu)$, mean free path $\ell = \sigma/\mu = 0.06$, and Knudsen number $\varepsilon = \ell/R_0 = 0.01$. Deviations may arise when the initial distance to the exit boundary is comparable to $\ell$, i.e. for $r \gtrsim R_0 -\ell = 5.94$, denoted by the red vertical line. For unbiased motion (black dotted curves), when $\kappa(r) = \alpha(r) = \beta(r) =0$, the diffusive form $T(r) = (R_0^2 - r^2)/ (2 d D)$ is recovered from Eq. \ref{['MFPTsol']}. In all other curves motion is biased along the positive or negative radial direction $\hat{\gamma}(r) = \pm \hat{r}$ via $\kappa_1(r) = r/\lambda_1$ (increasing as the boundary is approached) or $\kappa_2(r) = A e^{-r/\lambda_2}$ (decreasing as the boundary is approached). The limit $\kappa(r) \to \infty$, $\gamma(r) = \hat{r}$, $\alpha(r), \beta(r) \to 1$ yields the ballistic form $T(r) = (R_0 - r)/ \sigma$. Compared to purely diffusive trajectories, positive biases reduce the MFPT, negative biases induce detours that delay exit. Even relatively modest biases can alter the MFPT by several orders of magnitude. We set $A = 0.1, \lambda_1 =50, \lambda_2 = 2$. Simulations are averaged over $10^4$ runs. Units are arbitrary. Other parameter choices are in the SM.
• Figure 3: Two-dimensional particles searching for the locus ${\bf {x}}_T$ of maximal concentration of the Gaussian plume $\phi{(\bf x)} = Q /(2 \pi K x) e^{-u (y^2 + H^2) /(4 K x)}$ with $Q=2$, $K =0.5$, $u=5$, $H=1$ and ${\bf {x}}_T = (u H^2 / 4K, 0) = (2.5, 0)$. Particles start at ${\bf x}_0=(3,0.5)$ and follow the von Mises distribution in Eq. \ref{['VM']} with $\hat{\gamma}(\bf x) = \nabla \phi(\bf x) / | \nabla \phi(\bf x)|$ and uniform $\kappa$. Speed and turning rate are $\sigma = \sqrt{\mu}= 10^2$. The trajectories in panels A,B, derived from Eq. \ref{['forward']} and the Langevin process in Eq. \ref{['Langevin']} respectively, are qualitatively similar. Panel C shows strong agreement between the MFPT $T({\bf x}_0)$ from Eq. \ref{['superfinal']} and simulations of Eq. \ref{['forward']} averaged over $10^4$ runs. Panel D shows that Eq. \ref{['Langevin']} accurately captures the survival probability $S({\bf x}_0, t)$. Units are arbitrary.