Jump locations of jump-diffusion processes with state-dependent rates

Abstract

We propose a general framework for studying statistics of jump-diffusion systems driven by both Brownian noise (diffusion) and a jump process with statedependent intensity. Of particular natural interest in many physical systems are the jump locations: the system evaluated at the jump times. As an example, this could be the voltage at which a neuron fires, or the so-called "threshold voltage." However, the state-dependence of the jump rate provides direct coupling between the diffusion and jump components, making it difficult to disentangle the two to study individually. In this work, we provide an iterative map formulation of the sequence of distributions of jump locations. The distributions computed by this map can be used to elucidate other interesting quantities about the process, including statistics of the interjump times. Ultimately, the limit of the map reveals that knowledge of the stationary distribution of the full process is sufficient to recover (but not necessarily equal to) the distribution of jump locations. We propose two biophysical examples to illustrate the use of this framework to provide insight about a system. We find that a sharp threshold voltage emerges robustly in a simple stochastic integrate-and-fire neuronal model. The interplay between the two sources of noise is also investigated in a stepping model of molecular motor in intracellular transport pulling a diffusive cargo. arXiv:1701.01920v2 [cond-mat.stat-mech] 26 Sep 2018 Jump Locations of Jump-Diffusion Theorem Appendix B.1 (Easterling 1998, [19]). Suppose thatG ∈ L 2 and is nonnegative. If there exists an α > 0, β > 0, u 0 such thatfor all x, ξ, then the integral operator A has a dominant eigenvalue with associated eigenfunction.This establishes conditions for the existence of a dominant eigenvalue and eigenvector, which establish the long-term behavior.

Order out of Randomness: Self-Organization Processes in Astrophysics

“…An alternative yet analogous scenario involving elastic stresses (star quakes) has also been proposed (Middleditch et al 2006). A promising theoretical framework that is applicable to a wide variety of self-orgnaizing systems of this kind is the mean-field model of a state-dependent Poisson process, indroduced originally in the context of forest fires (Daly and Porporato 2006), and solar flares (Wheatland 2008), and generalized recently to neutron stars (Fulgenzi et al 2017) and biological applications (Miles and Keener 2017). The model makes quantitative predictions of size and waiting time distributions and size-waiting time correlations as a function of the driving rate, independent of the detailed microphysics.…”

Large deviations for Markov processes with stochastic resetting: analysis via the empirical density and flows or via excursions between resets

“…The recent interest into stochastic resetting (see the review [1] and references therein) can be explained via its many applications : within the field of intermittent search strategies (see the review [2] and references therein), the resetting procedure is clearly the simplest one; jump-diffusions processes have been also much studied in mathematical finance [3,4]), in biology for integrate-and-fire neuronal models [5,6] and in ecology to describe fires [7] or soil moisture [8,9].…”

“…The recent interest into stochastic resetting (see the review [1] and references therein) can be explained via its many applications: within the field of intermittent search strategies (see the review [2] and references therein), the resetting procedure is clearly the simplest one; jump-diffusions processes have been also much studied in mathematical finance [3,4]), in biology for integrate-and-fire neuronal models [5,6] and in ecology to describe fires [7] or soil moisture [8,9]; finally, for open quantum systems, the unraveling of the Lindblad dynamics in terms of quantum trajectories involve quantum jumps that are analogous to resetting procedures [10], and it is thus interesting to better understand the similarities and differences with resetting in classical stochastic models [10][11][12].…”

“…As a consequence, besides the analysis of large deviations at level 2.5 for the empirical density and of the empirical flows mentioned above, we will also consider the large deviations for the empirical excursions between two consecutive resets. These two points of view will be then used to analyze the large deviations of the most general time-additive observables involving both the position and the increments of the dynamical trajectory, when the reset probabilities or reset rates are not uniform anymore but space-dependent, since this is relevant in many applications [6,7,9,[77][78][79][80].…”

Jump-Drift and Jump-Diffusion Processes : Large Deviations for the density, the current and the jump-flow and for the excursions between jumps

“…Jump-drift and jump-diffusion processes play a major role in many applications, in particular in mathematical finance [1,2]), in biology for integrate-and-fire neuronal models [3,4] and in ecology to describe fires [5] or soil moisture [6][7][8][9]. They have also attracted a lot of interest in the field of intermittent search strategies (see the review [10] and references therein) and in the context of stochastic resetting (see the review [11] and references therein).…”