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Law of large numbers for stochastic multiscale spatial gene networks

Arnaud Debussche, Baptiste Huguet

Abstract

We study a stochastic multiscale spatial gene network. These naturally arise in molecular biology. In our model, the reactants are subject to on-site reactions on both scales and diffusion on the continuous scale only, although diffusion on both scales could easily be handled. We obtain, under a light condition on the scales between the total population size and the mesh discretisation, the convergence of the stochastic system to a deterministic system consisting of a PDE coupled to a ODE. This is in contrast with the well-stirred case where jumps remain at the limit. In order to prove this convergence result, we develop some moments control for martingales in discrete Sobolev topologies and use products rule in discrete Sobolev spaces.

Law of large numbers for stochastic multiscale spatial gene networks

Abstract

We study a stochastic multiscale spatial gene network. These naturally arise in molecular biology. In our model, the reactants are subject to on-site reactions on both scales and diffusion on the continuous scale only, although diffusion on both scales could easily be handled. We obtain, under a light condition on the scales between the total population size and the mesh discretisation, the convergence of the stochastic system to a deterministic system consisting of a PDE coupled to a ODE. This is in contrast with the well-stirred case where jumps remain at the limit. In order to prove this convergence result, we develop some moments control for martingales in discrete Sobolev topologies and use products rule in discrete Sobolev spaces.

Paper Structure

This paper contains 14 sections, 32 theorems, 225 equations.

Table of Contents

  1. Introduction
  2. The spatial stochastic gene network
  3. The stochastic model
  4. Classical and discrete Sobolev spaces
  5. The main result
  6. A truncation argument
  7. Tightness of the discrete scale
  8. Regularised jumps of the continuous scale
  9. Convergence
  10. The discrete PDE
  11. Convergence of the discrete PDE
  12. Uniqueness of the limit point and conclusion
  13. Product rules for discrete Sobolev space
  14. Existence and uniqueness of the solution

Key Result

Lemma 2.3

Theorems & Definitions (61)

  • Lemma 2.3: Blount91
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • Corollary 2.7
  • Lemma 2.9
  • Theorem 2.10
  • Proposition 2.11
  • ...and 51 more