The Laplacian in a stochastic model for spatiotemporal reaction systems

We demonstrate how the Laplace operator can be combined with a stochastic jump process to describe the evolution of a polymer network. Growth processes in polymer networks can proceed through the transfer of rather large oligomeric subunits from a pool of soluble molecules into the laments forming the network. In such a situation stochastic jump processes are a natural tool for the description of network formation at the nanometer scale. However, modeling of the reaction system in high spatiotemporal resolution also needs to capture the evolution of the soluble lament precursor molecules, which controls local properties of the growth process. We illustrate how this may be achieved by combining a deterministic di usion equation with a stochastic jump process within the setting of a piecewise-deterministic Markov process. Since in many applications the di usion equation will be considered with periodic boundary conditions, we apply the theory of bilinear forms to prove existence and uniqueness of a solution, which is also shown to be positive, mass-preserving, and convergent to an equilibrium.