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Register a new userScaling Limit of Moderately Interacting Particle Systems with Singular Interaction and Environmental Noise
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We consider moderately interacting particle systems with singular interaction kernel and environmental noise. It is shown that the mollified empirical measures converge in strong norms to the unique (local) solutions of nonlinear Fokker-Planck equations. The approach works for the Biot-Savart and Poisson kernels.
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In the nonlinear diffusion framework, stochastic processes of McKean-Vlasov type play an important role. In some cases they correspond to processes attracted by their own probability distribution: the so-called self-stabilizing processes. Such diffusions can be obtained by taking the hydrodymamic limit in a huge system of linear diffusions in interaction. In both cases, for the linear and the nonlinear processes, small-noise asymptotics have been emphasized by specific large deviation phenomenons. The natural question, therefore, is: is it possible to interchange the mean-field limit with the small-noise limit? The aim here is to consider this question by proving that the rate function of the first particle in a mean-field system converges to the rate function of the hydrodynamic limit as the number of particles becomes large.
We consider the asymptotic behavior of the fluctuations for the empirical measures of interacting particle systems with singular kernels. We prove that the sequence of fluctuation processes converges in distribution to a generalized Ornstein-Uhlenbeck process. Our result considerably extends classical results to singular kernels, including the Biot-Savart law. The result applies to the point vortex model approximating the 2D incompressible Navier-Stokes equation and the 2D Euler equation. We also obtain Gaussianity and optimal regularity of the limiting Ornstein-Uhlenbeck process. The method relies on the martingale approach and the Donsker-Varadhan variational formula, which transfers the uniform estimate to some exponential integrals. Estimation of those exponential integrals follows by cancellations and combinatorics techniques and is of the type of large deviation principle.
In this paper we consider three classes of interacting particle systems on $mathbb Z$: independent random walks, the exclusion process, and the inclusion process. We allow particles to switch their jump rate (the rate identifies the type of particle) between $1$ (fast particles) and $epsilonin[0,1]$ (slow particles). The switch between the two jump rates happens at rate $gammain(0,infty)$. In the exclusion process, the interaction is such that each site can be occupied by at most one particle of each type. In the inclusion process, the interaction takes places between particles of the same type at different sites and between particles of different type at the same site. We derive the macroscopic limit equations for the three systems, obtained after scaling space by $N^{-1}$, time by $N^2$, the switching rate by $N^{-2}$, and letting $Ntoinfty$. The limit equations for the macroscopic densities associated to the fast and slow particles is the well-studied double diffusivity model. This system of reaction-diffusion equations was introduced to model polycrystal diffusion and dislocation pipe diffusion, with the goal to overcome the limitations imposed by Ficks law. In order to investigate the microscopic out-of-equilibrium properties, we analyse the system on $[N]={1,ldots,N}$, adding boundary reservoirs at sites $1$ and $N$ of fast and slow particles, respectively. Inside $[N]$ particles move as before, but now particles are injected and absorbed at sites $1$ and $N$ with prescribed rates that depend on the particle type. We compute the steady-state density profile and the steady-state current. It turns out that uphill diffusion is possible, i.e., the total flow can be in the direction of increasing total density. This phenomenon, which cannot occur in a single-type particle system, is a violation of Ficks law made possible by the switching between types.
We propose an interacting particle system to model the evolution of a system of banks with mutual exposures. In this model, a bank defaults when its normalized asset value hits a lower threshold, and its default causes instantaneous losses to other banks, possibly triggering a cascade of defaults. The strength of this interaction is determined by the level of the so-called non-core exposure. We show that, when the size of the system becomes large, the cumulative loss process of a bank resulting from the defaults of other banks exhibits discontinuities. These discontinuities are naturally interpreted as systemic events, and we characterize them explicitly in terms of the level of non-core exposure and the fraction of banks that are about to default. The main mathematical challenges of our work stem from the very singular nature of the interaction between the particles, which is inherited by the limiting system. A similar particle system is analyzed in [DIRT15a] and [DIRT15b], and we build on and extend their results. In particular, we characterize the large-population limit of the system and analyze the jump times, the regularity between jumps, and the local uniqueness of the limiting process.
We consider on the torus the scaling limit of stochastic 2D (inviscid) fluid dynamical equations with transport noise to deterministic viscous equations. Quantitative estimates on the convergence rates are provided by combining analytic and probabilistic arguments, especially heat kernel properties and maximal estimates for stochastic convolutions. Similar ideas are applied to the stochastic 2D Keller-Segel model, yielding explicit choice of noise to ensure that the blow-up probability is less than any given threshold. Our approach also gives rise to some mixing property for stochastic linear transport equations and dissipation enhancement in the viscous case.
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