Splitting Probabilities of Jump Processes

Abstract

We derive a universal, exact asymptotic form of the splitting probability for symmetric continuous jump processes, which quantifies the probability ${\pi }_{0,\underline{x}}(x_0)$ that the process crosses $x$ before 0 starting from a given position $x_0\in [0,x]$ in the regime $x_0\ll x$. This analysis provides in particular a fully explicit determination of the transmission probability ($x_0=0$), in striking contrast with the trivial prediction ${\pi }_{0,\underline{x}}(0)=0$ obtained by taking the continuous limit of the process, which reveals the importance of the microscopic properties of the dynamics. These results are illustrated with paradigmatic models of jump processes with applications to light scattering in heterogeneous media in realistic 3$d$ slab geometries. In this context, our explicit predictions of the transmission probability, which can be directly measured experimentally, provide a quantitative characterization of the effective random process describing light scattering in the medium.

arXiv comment5 pages + 7 pages for supplementary information

References in corpus (20)

• Active Brownian Particles. From Individual to Collective Stochastic Dynamics
• Lévy walks
• Light fields in complex media: mesoscopic scattering meets wave control
• Probing microscopic origins of confined subdiffusion by first-passage observables
• Universal First-passage Properties of Discrete-time Random Walks and Levy Flights on a Line: Statistics of the Global Maximum and Records
• Universal survival probability for a $d$-dimensional run-and-tumble particle
• Optimizing persistent random searches
• Record statistics and persistence for a random walk with a drift
• Hitting probability for anomalous diffusion processes
• Lévy walks and scaling in quenched disordered media
• Unified Solution of the Expected Maximum of a Random Walk and the Discrete Flux to a Spherical Trap
• Capture of particles undergoing discrete random walks
• Helix or Coil? Fate of a Melting Heteropolymer
• Optimal non-Markovian search strategies with n-step memory
• Rare events in generalized Lévy Walks and the Big Jump principle
• Survival Probability of Random Walks and Lévy Flights on a Semi-Infinite Line
• Universal first-passage statistics of aging processes
• Signatures of Lévy flights with annealed disorder
• Superdiffusion and Transport in 2d-systems with Lévy Like Quenched Disorder
• Lévy flights of photons with infinite mean free path

Cited by in corpus (13)

• An encounter-based approach to the escape problem
• Fast rare events in exit times distributions of jump processes
• Effects of mortality on stochastic search processes with resetting
• Optimizing Leapover Lengths of Lévy Flights with Resetting
• Optimal conditions for first passage of jump processes with resetting
• Optimizing the random search of a finite-lived target by a Lévy flight
• Leftward, Rightward and Complete Exit Time Distributions of Jump Processes
• Martingale approach for first-passage problems of time-additive observables in Markov processes
• Splitting probabilities for dynamics in corrugated channels: passive VS active Brownian motion
• First-passage and extreme value statistics for overdamped Brownian motion in a linear potential
• Run-and-tumble particles in slit geometry as a splitting probability problem
• Transition in Splitting Probabilities of Quantum Walks
• Stochastic kinetics under combined action of two noise sources