scholarly journals Large deviations of a forced velocity-jump process with a Hamilton–Jacobi approach

2021 ◽  
pp. 1-23
Author(s):  
Nils Caillerie
Keyword(s):  
Large Deviations ◽  
Jump Process ◽  
Velocity Jump Process ◽  
Velocity Jump

scholarly journals Spreading in kinetic reaction–transport equations in higher velocity dimensions

2018 ◽  
Vol 30 (2) ◽  
pp. 219-247
Author(s):  
EMERIC BOUIN ◽  
NILS CAILLERIE
Keyword(s):  
Large Scale ◽  
Jump Process ◽  
Travelling Wave ◽  
Transport Equations ◽  
Kinetic Reaction ◽  
Velocity Jump Process ◽  
Minimal Speed ◽  
Crucial Difference ◽  
Velocity Jump ◽  
Reaction Transport

In this paper, we extend and complement previous works about propagation in kinetic reaction–transport equations. The model we study describes particles moving according to a velocity-jump process, and proliferating according to a reaction term of monostable type. We focus on the case of bounded velocities, having dimension higher than one. We extend previous results obtained by the first author with Calvez and Nadin in dimension one. We study the large time/large-scale hyperbolic limit via an Hamilton–Jacobi framework together with the half-relaxed limits method. We deduce spreading results and the existence of travelling wave solutions. A crucial difference with the mono-dimensional case is the resolution of the spectral problem at the edge of the front, that yields potential singular velocity distributions. As a consequence, the minimal speed of propagation may not be determined by a first-order condition.

scholarly journals Large Deviations of Jump Process Fluxes

2019 ◽  
Vol 22 (3) ◽  
Author(s):  
Robert I. A. Patterson ◽  
D. R. Michiel Renger
Keyword(s):  
Large Deviations ◽  
Jump Process

scholarly journals Heavy-tailed distributions in a stochastic gene autoregulation model

2021 ◽  
Author(s):  
Pavol Bokes
Keyword(s):  
Large Deviations ◽  
Stationary Distribution ◽  
Burst Size ◽  
Cumulative Effect ◽  
Jump Process ◽  
Gene Products ◽  
Tail Region ◽  
Expression Variability ◽  
Heavy Tailed Distributions ◽  
Heavy Tailed

Synthesis of gene products in bursts of multiple molecular copies is an important source of gene expression variability. This paper studies large deviations in a Markovian drift--jump process that combines exponentially distributed bursts with deterministic degradation. Large deviations occur as a cumulative effect of many bursts (as in diffusion) or, if the model includes negative feedback in burst size, in a single big jump. The latter possibility requires a modification in the WKB solution in the tail region. The main result of the paper is the construction, via a modified WKB scheme, of matched asymptotic approximations to the stationary distribution of the drift--jump process. The stationary distribution possesses a heavier tail than predicted by a routine application of the scheme.

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