A Brownian ball interacting with infinitely many Brownian particles in \(\mathbb{R}^ d\) (Q1321752)

We construct a system of a hard ball with radius \(r(\in(0,\infty))\) interacting with infinitely many point particles in \(R^ d\) \((d\leqq 2)\). All particles and the ball are undergoing Brownian motions and when the distance between a particle and the center of the ball attains a given constant \(r\), they repel each other instantly. We prove the existence and the uniqueness of a stochastic differential equation describing the model in the case where the environment process seen from the ball is the stationary process whose stationary measure is a Poisson distribution. \textit{Y. Saisho} and \textit{H. Tanaka} [Osaka J. Math. 23, 725-740 (1986; Zbl 0613.60057)] constructed a system of mutually reflecting finitely many hard balls by solving certain stochastic differential equation of Skorokhod type. Following the idea of Saisho and Tanaka, \textit{Y. Saisho} [Probability theory and mathematical statistics, Proc. 5th Jap.-USSR Symp., Kyoto/Jap. 1986, Lect. Notes Math. 1299, 444- 453 (1988; Zbl 0635.60068)] constructed a system of mutually repelling finitely many particles of \(m\) types: the number of particles of type \(k\) is \(n_ k\) \((\sum^ m_{k=1}n_ k=n<\infty)\) and when the distance between two particles of different type attains a constant \(r\), they repel each other instantly. In case each type consists of only one particle, the model of Saisho is reduced to that of Saisho and Tanaka. Our present model is formally regarded as the case of \(m=2\), \(n_ 1=1\) and \(n_ 2=\infty\) in the model of Saisho and Tanaka.