Mathematics – Probability
Scientific paper
2011-09-22
Mathematics
Probability
Scientific paper
The time evolution of an infinite system of interacting point particles on $\mathbb{R}^d$ is described on both micro- and mesoscopic levels as the evolution $\mu_0 \mapsto \mu_t$ of probability measures on the configuration space $\Gamma$. The particles are supposed to hop over $\mathbb{R}^d$ and repel each other, similarly to the Kawasaki dynamics on the lattice $\mathbb{Z}^d$. The microscopic description is based on solving linear equations for the correlation functions by means of a combination of methods including an Ovcyannikov-type technique, which yields the evolution in a scale of Banach spaces. Then the evolution of the corresponding measures is obtained therefrom by a special procedure based on local approximations. The mesoscopic description is performed within the Vlasov scaling method, which yields a linear infinite chain of equations obtained from those for the correlation function of the model. Its main peculiarity is that for the initial $r_0$ being the correlation function of the inhomogeneous Poisson measure with density $\varrho_0$, the solution $r_t$ is the correlation function of such a measure with density $\varrho_t$ which solves a nonlinear differential equation of convolution type.
Berns Christoph
Kondratiev Yuri
Kozitsky Yuri
Kutoviy Oleksandr
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