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We consider the fully-coupled McKean-Vlasov equation with multi-time-scale potentials, and all the coefficients depend on the distributions of both the slow component and the fast motion. By studying the smoothness of the solution of the non-linear Poisson equation on Wasserstein space, we derive the asymptotic limit as well as the quantitative error estimate of the convergence for the slow process. Extra homogenized drift term containing derivative in the measure argument of the solution of the Poisson equation appears in the limit, which seems to be new and is unique for systems involving the fast distribution.

This work focuses on a class of stochastic damping Hamiltonian systems with state-dependent switching, where the switching process has a countably infinite state space. After establishing the existence and uniqueness of a global weak solution via the martingale approach under very mild conditions, the paper next proves the strong Feller property for regime-switching stochastic damping Hamiltonian systems by the killing technique together with some resolvent and transition probability identities. The commonly used continuity assumption for the switching rates $q_{kl}(\cdot)$ in the literature is relaxed to measurability in this paper. Finally the paper provides sufficient conditions for exponential ergodicity and large deviations principle for regime-switching stochastic damping Hamiltonian systems. Several examples on regime-switching van der Pol and (overdamped) Langevin systems are studied in detail for illustration.

In this work, we study a class of mean-field linear quadratic Gaussian (LQG) problems. Under suitable conditions, explicit solutions of the distribution-dependent optimal control problems are obtained. Riccati systems are derived by directly solving the associated master equations. Some extensions on controls with partial observations are also considered., Comment: 13 pages

This paper examines convergence and stability of the two classes of theta-Milstein schemes for stochastic differential equations (SDEs) with non-global Lipschitz continuous coefficients: the split-step theta-Milstein (SSTM) scheme and the stochastic theta-Milstein (STM) scheme. For \theta\in[1/2,1], this paper concludes that the two classes of theta-Milstein schemes converge strongly to the exact solution with the order 1. For \theta \in [0,1/2], under the additional linear growth condition for the drift coefficient, these two classes of the theta-Milstein schemes are also strongly convergent with the standard order. This paper also investigates exponential mean-square stability of these two classes of the theta-Milstein schemes. For \theta\in(1/2, 1], these two theta-Milstein schemes can share the exponential mean-square stability of the exact solution. For \theta\in[0, 1/2], similar to the convergence, under the additional linear growth condition, these two theta-Milstein schemes can also reproduce the exponential mean-square stability of the exact solution., Comment: 28 pages

In this note, the distributed consensus corrupted by relative-state-dependent measurement noises is considered. Each agent can measure or receive its neighbors' state information with random noises, whose intensity is a vector function of agents' relative states. By investigating the structure of this interaction and the tools of stochastic differential equations, we develop several small consensus gain theorems to give sufficient conditions in terms of the control gain, the number of agents and the noise intensity function to ensure mean square (m. s.) and almost sure (a. s.) consensus and quantify the convergence rate and the steady-state error. Especially, for the case with homogeneous communication and control channels, a necessary and sufficient condition to ensure m. s. consensus on the control gain is given and it is shown that the control gain is independent of the specific network topology, but only depends on the number of nodes and the noise coefficient constant. For symmetric measurement models, the almost sure convergence rate is estimated by the Iterated Logarithm Law of Brownian motions.