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Abstract We consider particle-based stochastic reaction-drift-diffusion models where particles move via diffusion and drift induced by one- and two-body potential interactions. The dynamics of the particles are formulated as measure-valued stochastic processes (MVSPs), which describe the evolution of the singular, stochastic concentration fields of each chemical species. The mean field large population limit of such models is derived and proven, giving coarse-grained deterministic partial integro-differential equations (PIDEs) for the limiting deterministic concentration fields’ dynamics. We generalize previous studies on the mean field limit of models involving only diffusive motion, with care to formulating the MVSP representation to ensure detailed balance of reversible reactions in the presence of potentials. Our work illustrates the more general set of PIDEs that arise in the mean field limit, demonstrating that the limiting macroscopic reactive interaction terms for reversible reactions obtain additional nonlinear concentration-dependent coefficients compared to the purely diffusive case. Numerical studies are presented which illustrate that two-body repulsive potential interactions can have a significant impact on the reaction dynamics, and also demonstrate the empirical numerical convergence of solutions to the PBSRDD model to the derived mean field PIDEs as the population size increases. 

Abstract We develop a broadly applicable class of coevolving latent space network with attractors (CLSNA) models, where nodes represent individual social actors assumed to lie in an unknown latent space, edges represent the presence of a specified interaction between actors, and attractors are added in the latent level to capture the notion of attractive and repulsive forces. We apply the CLSNA models to understand the dynamics of partisan polarization in US politics on social media, where we expect Republicans and Democrats to increasingly interact with their own party and disengage with the opposing party. Using longitudinal social networks from the social media platforms Twitter and Reddit, we quantify the relative contributions of positive (attractive) and negative (repulsive) forces among political elites and the public, respectively. 

Abstract We introduce a novel geometry-informed irreversible perturbation that accelerates convergence of the Langevin algorithm for Bayesian computation. It is well documented that there exist perturbations to the Langevin dynamics that preserve its invariant measure while accelerating its convergence. Irreversible perturbations and reversible perturbations (such as Riemannian manifold Langevin dynamics (RMLD)) have separately been shown to improve the performance of Langevin samplers. We consider these two perturbations simultaneously by presenting a novel form of irreversible perturbation for RMLD that is informed by the underlying geometry. Through numerical examples, we show that this new irreversible perturbation can improve estimation performance over irreversible perturbations that do not take the geometry into account. Moreover we demonstrate that irreversible perturbations generally can be implemented in conjunction with the stochastic gradient version of the Langevin algorithm. Lastly, while continuous-time irreversible perturbations cannot impair the performance of a Langevin estimator, the situation can sometimes be more complicated when discretization is considered. To this end, we describe a discrete-time example in which irreversibility increases both the bias and variance of the resulting estimator. 

We study the asymptotic behavior, uniform-in-time, of a nonlinear dynamical system under the combined effects of fast periodic sampling with period [Formula: see text] and small white noise of size [Formula: see text]. The dynamics depend on both the current and recent measurements of the state, and as such it is not Markovian. Our main results can be interpreted as Law of Large Numbers (LLN) and Central Limit Theorem (CLT) type results. LLN type result shows that the resulting stochastic process is close to an ordinary differential equation (ODE) uniformly in time as [Formula: see text] Further, in regards to CLT, we provide quantitative and uniform-in-time control of the fluctuations process. The interaction of the small parameters provides an additional drift term in the limiting fluctuations, which captures both the sampling and noise effects. As a consequence, we obtain a first-order perturbation expansion of the stochastic process along with time-independent estimates on the remainder. The zeroth- and first-order terms in the expansion are given by an ODE and SDE, respectively. Simulation studies that illustrate and supplement the theoretical results are also provided.