Moderate deviations for fully coupled multiscale weakly interacting particle systems

Abstract

We consider a collection of fully coupled weakly interacting diffusion processes moving in a two-scale environment. We study the moderate deviations principle of the empirical distribution of the particles' positions in the combined limit as the number of particles grow to infinity and the time-scale separation parameter goes to zero simultaneously. We make use of weak convergence methods, which provide a convenient representation for the moderate deviations rate function in terms of an effective mean field control problem. We rigorously obtain equivalent representations for the moderate deviations rate function in an appropriate "negative Sobolev" form, which is reminiscent of the large deviations rate function for the empirical measure of weakly interacting diffusions obtained in the 1987 seminal paper by Dawson-Gärtner. In the course of the proof we obtain related ergodic theorems and we rigorously study the regularity of Poisson type of equations associated to McKean-Vlasov problems, both of which are topics of independent interest.

Importance sampling for stochastic reaction-diffusion equations in the moderate deviation regime

“…In this section we demonstrate the theoretical results of this paper by a series of simulation studies for (7). As explained in Section 3, we start the process X x at a stable equilibrium x = x * and develop a scheme that computes exit probabilities of the form…”

“…Remark 8. Theorem 3.1 is essentially equivalent to an MDP for the family {X } of solutions of (7), in the space C([0, T ]; E). The latter is an asympotic statement for exponential functionals of g(X ), where g : C([0, T ]; E) → R is continuous and bounded (see Definition 3.1), while the former covers exit probabilities and corresponds to the choice g = g with…”

“…Moderate Deviation Principles (MDPs) have been studied in many different contexts such as multiscale and interacting particle systems, Markov processes with jumps, small-noise stochastic dynamics, statistical estimation, option pricing and stochastic recursive algorithms see e.g. [35,61] for SRDEs as well as [7,13,26,34,39,40,42,44,51].…”

Poisson equation on Wasserstein space and diffusion approximations for McKean-Vlasov equation

“…The key difficulties caused by these features are that, compared with the previous results (e.g. [3,4,5,8,15,18,26]), we need to control the fluctuations estimate of the central limit type, and handle a totally non-linear frozen equation as well as the corresponding non-linear PDEs to serve as a corrector.…”

“…In particular, multiple scales can leads to hysteresis loops in the bifurcation diagram and induce phase transitions of certain McKean-Vlasov equation as studied in [8,14,15]. Existing averaging results for slow-fast McKean-Vlasov SDEs can be found in [3,4,5,18,26]. However, the coefficients in all the previous works are not allowed to depend on the distribution of the fast motion.…”