The Diffusion Approximation for Linear Nonautonomous Reaction-Hyperbolic Equations

A number of biological processes can be modeled as nonautonomous jump-velocity processes of the form $dx/dt = V(x,t,S_i)$ ($1\le i\le N$), where $S_i$ are discrete states evolving by space-time jump Markov process. The probabilities $p_i(x,t)$ of being in state $S_i$ at $(x,t)$ then satisfy a reaction-hyperbolic system of the form $ \partial p_i/\partial t + \partial (v_ip_i)/\partial x = \frac1\varepsilon\sum_{j=1}^n k_{ij} p_j$, where $v_i=v_i(x,t)$ and $\varepsilon$ is positive and small; the $k_{ij}$ form an irreducible matrix which is annihilated by a unit vector $(\lambda_1,\ldots,\lambda_n)$ with positive components. We prove that, as $\varepsilon\to 0$, $p_i\to \lambda_i Q_0$, where $Q_0$ satisfies a diffusion equation.