Journal of the
Korean Mathematical Society JKMS

We consider a system of particles with locations $$\{X^n_i(t):t\ge 0, i=1,\dots,n\}$$ in $R^d$, time-varying weights $\{A^n_i(t):t\ge 0,i=1,\dots,n\}$ and weighted empirical measure processes $V^n(t){\hskip-0.1cm}={\hskip-0.1cm}\frac 1n \sum_{i=1}^n A^n_i(t)\delta_{X^n_i}(t),$ where $\delta_x$ is the Dirac measure. It is known that there exists the limit of $\{V_n\}$ in the $\text{week}^*$ topology on $M(R^d)$ under suitable conditions. If $\{X^n_i, A^n_i, V^n\}$ satisfies some diffusion equations, applying Ito formula, we prove a central limit type theorem for the empirical process $\{V^n\}$, i.e., we consider the convergence of the processes $\eta^n_t\equiv \sqrt n(V^n-V)$. Besides, we study a characterization of its limit.

Keywords: central limit theorem, Ito formula, SDE, weighted Sobol-ev space

MSC numbers: 60G35, 60H10, 60F25