From Cox processes to the chemical Lèvy-Langevin equation

We develop a new statistical model for biochemical reaction with non-stationary conditions. The new model is an extension of the well known Poisson model to Cox processes. The Cox process can be considered as a scale (and mean) mixture of Poisson processes. It is shown that the property of the convergence of the Poisson distribution to Gauss distribution for large rate parameter is paralleled in Cox processes to a convergence into scale (and mean) mixture of Gaussian distribution. We identify a special case namely alpha-stable distribution that can model skewed distributions as well as impulsive distributions and satisfy a generalized version of the central limit theorem. Based on this observation, we extend the classical Chemical Langevin Equation to Chemical Levy-Langevin equation, a stochastic process that is modelling Levy-walks as opposed to the Brownian motion modelled by classical Chemical Langevin Equation.