Image-dependent conditional McKean–Vlasov SDEs for measure-valued diffusion processes

Abstract

We consider a special class of mean field SDEs with common noise which depend on the image of the solution (i.e. the conditional distribution given noise). The strong well-posedness is derived under a monotone condition which is weaker than those used in the literature of mean field games, the Feynman-Kac formula is established to solve Schrördinegr type PDEs on P 2 , and the ergodicity is proved for a class of measurevalued diffusion processes.

Derivative formulas in measure on Riemannian manifolds

“…Let us compare (2.10) with the corresponding formula presented in [3] for and . In this case, formula (2.10) is established for the probability space being Polish and with bounded , see also [6, Proposition A.2] and [14, Lemma 2.3] for this formula with more general functions on . Theorem 2.2 establishes (2.10) to on Riemannian manifolds and .…”

“…To develop analysis on the space of measures, some derivatives in measure have been introduced by different means, where the intrinsic and extrinsic derivatives defined in [1, 8] have been used to investigate measure‐valued diffusion processes over Riemannian manifolds (see [7, 10, 11, 13, 14] and references therein), and the ‐ and linear functional derivatives were investigated in [3, 4] on the Wasserstein space (the set of all probability measures on with finite second‐order moments). See [2] and references therein for calculus and optimal transport on the space of probability measures, and see [9, 12] for the Bismut formula and estimates on the ‐derivative of distribution dependent SDEs.…”

A new particle approximation to the diffusive Dean-Kawasaki equation with colored noise

“…Generally, for a spatially correlated noise with such kernel, we denote it as ( Qβ ) 1 2 −Wiener process. Q−Wiener process can be naturally seen as a infinite dimensional counterpart of Bownian motion in K. On the other hand, it is known (see [vRS09], [AvR10], [Wan21]) that the solution of (1.1) or its regularised form can be seen as a Wasserstein diffusion. To introduce the motivation of the particle model in section 3 , we start from the viewpoint of Q−Wiener process on Wasserstein space.…”

Geometry on the Wasserstein Space Over a Compact Riemannian Manifold

“…Remark 2.5 Comparing this to [14], as well to [15], we do not suppose the Lipschitz continuity with respect to µ; accordingly, we have no uniqueness of solutions of (2.6).…”