A Hilbertian approach for fluctuations on the McKean-Vlasov model
Let us consider a mean field interacting particle system in \(\mathbb R^{d}\) given by \[ X^{i,n}_{t} = X^{i,n}_{0} + \int^{t}_{0} b[X^{i,n}_{s},\mu^{n}_{s}] ds + \int^{t}_{0} \sigma [X^{i,n}_{s},\mu^{n}_{s}] dB^{i}_{s}, \quad 1\leq i\leq n, \;0\leq t\leq T, \] where \(\mu^{n}_{t}\) denotes the empirical measure \(\mu^{n}_{t} = n^{-1}\sum^{n}_{i=1}\delta _{X^{i,n} _{t}}\), \(B^{1},\ldots ,B^{n}\) are independent \(k\)-dimensional Wiener processes and \(b[y,m] = \int _{\mathbb R^{d}} b(y,z) dm(z)\), \(\sigma [y,m] = \int _{\mathbb R^{d}} \sigma (y,z) dm(z)\) for any probability measure \(m\). It is known that under some assumptions on the smoothness of the functions \(b\), \(\sigma \), and on the initial data, the measures \(\mu^{n}\) converge weakly as \(n\to \infty \) to the law \(P\) of the solution to the equation \[ X_{t} = X_{0} + \int^{t}_{0} b[X_{s},P_{s}] ds + \int^{t}_{0}\sigma [X_{s},P_{s}] dB_{s}, \quad 0\leq t\leq T, \] \(P_{s}\) denoting the law of \(X_{s}\). The authors investigate the fluctuation process \(\eta^{n} = \sqrt {n}(\mu^{n} -P)\). The tightness of the sequence \(\{\eta^{n}\}\) in the space \({\mathcal C}([0,T];W^{-(2+2D),D}_{0})\) of continuous \(W^{-(2+2D),D}_{0}\)-valued functions on \([0,T]\) is established, \(D = 1 + [\frac {d}2]\), and under slightly more restrictive assumptions it is shown that \(\eta^{n}\) converge to an Ornstein-Uhlenbeck process solving a Langevin-type equation in \(W^{-(4+2D),D} _{0}\). Here, \(W^{k,\beta }_{0}\) denotes the weighted Sobolev space of functions on \(\mathbb R^{d}\) whose distributional derivatives up to the order \(k\) belong to the weighted \(L^2\) space with the weight \((1+|x|^{2\beta })^{-1}\), and \(W^{-k,\beta } _{0}\) is its dual space.
infinite-dimensional Ornstein-Uhlenbeck process
scientific article; zbMATH DE number 1409133
scientific article