Systems of interacting diffusions with partial annihilation through membranes (Q516118)

The paper under review builds a new continuous state stochastic model for interactive particle systems, establishes the functional law of large numbers and obtains the hydrodynamic limit, which reflects diffusions in domains with mixed-type boundary conditions. Subsection 1.3 gives a brief literature review on interacting diffusion systems. Let \(N\) be the common initial number of particles in each underlying domain \(D_{\pm}\), and each particle in \(D_{\pm}\) performs a Brownian motion with drift in the interior of \(D_{\pm}\). When a particle hits the boundary, it is absorbed (harvested) on \(\Lambda_{pm}\) and instantaneously reflected on \(\partial D_{\pm} \setminus \Lambda_{\pm}\) along the inward normal direction of \(D_{\pm}\). When two particles of different types come within a small distance \(\delta_N\), they disappear with intensity \(\frac{\lambda}{N \delta_N^{d+1}}\), where the scaling \(\frac{\lambda}{N \delta_N^{d+1}}\) for the per-pair annihilation intensity guarantees that a nontrivial proportion of particles is killed during the time interval \([0, t]\) as the initial number becomes sufficiently large. The main result of the paper is Theorem 5.1 (hydrodynamic limit), stating that under certain assumptions ((2.4)--(2.6)) the following holds: if the normalized empirical measures at initial time \(t=0\) converge in law, then the normalized empirical measures converge in law over \([0, T]\) for any \(T>0\) to the probabilistic solution of an initial value problem for the coupled heat equation. The key idea is the interchange limit from Lemma 7.3 that characterizes the mean of any subsequential limit of the normalized empirical measures by comparing the integral equations satisfied by the hydrodynamic limit with its stochastic counterpart. Section 2 lays out the basic set-up for this new model, and basic properties for the transition density (\S 2.1), and lists the assumptions 2.4--2.7 and notations for the annihilating diffusion system (\S 2.2). Section 3 constructs the configuration space \(S_N\), the configuration process \(X^{(N)}\) and the normalized empirical measure process for the new system. Proposition 3.1 shows that the configuration process is a strong Markov process under the ambient probability space, as well as the normalized empirical measure process (Proposition 3.2). In Section 4, the authors consider the Banach space of continuous functions on \([0, T]\) taking values in \(C_{\infty}(\overline{D}_{\pm} \setminus \Lambda_{\pm})\), and prove the existence and uniqueness of solutions for the coupled integral equation by the Banach fixed point theorem. In Section 5, Theorem 5.1 (hydrodynamic limit) is stated rigorously with an example and a question on relaxing some assumptions. In Section 6, the authors start with the martingales for annihilating diffusion systems (Theorem 6.2) with the infinitesimal generator \(L + K\) of the normalized empirical measures, and then show that the normalized empirical measure process is also tight by Prohorov's theorem and the analysis of Feller generators of the reflected diffusion processes \(X^{\Lambda_{\pm}}\), thus completing the proof of the main result Theorem 5.1. In Section 7, the authors prove the mean-variance result of one the key propositions (Proposition 6.8) through the Minkowski content that identifies the first and second moments of subsequential limits of empirical distributions.