Strong approximation and central limit theorems for multiscale stochastic gene networks
Baptiste Nicolas Huguet
TL;DR
This paper studies a multiscale stochastic gene-network model as a hybrid jump process with a continuous (abundant) and a discrete (scarce) scale, Z^N=(X^N,Y^N). By employing a Poisson-random-measure coupling, it proves strong convergence in the uniform topology of Z^N to a piecewise deterministic Markov process Z=(X,Y) with generator $\mathcal{L}_\infty$, and, under additional regularity, a central limit theorem for fluctuations of the continuous scale toward an SDE $dV_t = \sigma_t dB_t + \langle \nabla_x F(Z(t)), V_t\rangle dt$. An $L^1$ convergence result is obtained under bounded rates, and a truncation argument extends the strong convergence to general rates. The central limit theorem constitutes a novel result for hybrid jump processes, providing a rigorous description of fluctuations in the continuous (concentration) component and informing efficient reduced-model analyses for gene networks. Together, the results offer a principled framework for both accurate approximation and uncertainty quantification in multiscale stochastic gene-regulatory dynamics.
Abstract
We study a mutliscale jump process introduced in a work by Crudu, Debussche, Muller and Radulescu. Using an adequate coupling, we are able to prove the strong convergence, for the uniform topology, to a piecewise deterministic Markov process. Under some additional regularity, we also obtain a central limit theorem and prove that the fluctuations of the continuous scale converge, in a weaker sense, to the solution of a stochastic differential equation.