Pathwise McKean-Vlasov Theory with Additive Noise

Abstract

We take a pathwise approach to classical McKean-Vlasov stochastic differential equations with additive noise, as e.g. exposed in Sznitmann [38]. Our study was prompted by some concrete problems in battery modelling [23], and also by recent progrss on rough-pathwise McKean-Vlasov theory, notably Cass-Lyons [10], and then Bailleul, Catellier and Delarue [4]. Such a "pathwise McKean-Vlasov theory" can be traced back to Tanaka [40]. This paper can be seen as an attempt to advertize the ideas, power and simplicity of the pathwise appproach, not so easily extracted from [4,10,40], together with a number of novel applications. These include mean field convergence without a priori independence and exchangeability assumption; common noise, càdlàg noise, and reflecting boundaries. Last not least, we generalize Dawson-Gärtner large deviations and the central limit theorem to a non-Brownian noise setting. * ⇀ λ ∈ P p (I).

Solving mean field rough differential equations

“…Remark 4.4. Recently, the authors in [13] obtained a quantified propagation of chaos result for mean field stochastic equations with additive noise…”

A Law of Large Numbers for interacting diffusions via a mild formulation

“…The theory of weakly interacting particle systems has received great attention in the last fifty years. On the one hand, its mathematical tractability has allowed to obtain a deep understanding of the behavior of the empirical measure for such systems: law of large numbers [28,5], fluctuations and central limit theorems [31,12], large deviations [11,19] and propagation of chaos properties [30] are by now established. On the other hand, the theory of weakly interacting particles enters in several areas of applied mathematics such as mean-field games or finance models [4], making it an area of active research.…”

“…This richness in models is reflected in a variety of different techniques implemented in their study (see e.g. [5,28,30] for three very different approaches).…”

“…[19]) or to apply a fixed-point argument in a suitable topological space (e.g. [5]). The only exceptions are given by [30,31], although they require independent and identically distributed (IID) initial conditions.…”

“…Whenever (x i 0 ) i∈N are taken either to be IID random variables sampled from ν 0 , or such that ν n 0 weakly converges to ν 0 with some p 1 finite moment, it is well known (e.g. [30,Theorem 1.4] and [5,Theorem 3.1]) that ν n converges (in a precise sense depending on the setting) to the solution of the following PDE, known as non-linear Fokker-Planck (or McKean-Vlasov) equation…”