Certain random motion of a ball colliding with infinite particles of jump type (Q1826238)

We consider a system of a hard ball with radius r and of infinitely many point particles in \({\mathbb{R}}^ d\) according to the following rules: (i) There are no particles in the r-neighborhood of the center x(t) of the hard ball at time t. (ii) The ball or a particle at x waits an exponential holding time with mean one. It jumps to the position y where y is distributed according to p(\(| x-y|)dy\), except that the jump is suppressed, if the particle comes to lie within the region occupied by the hard ball. We prove that \(\epsilon x(t/\epsilon^ 2)\to \sigma B(t)\) as \(\epsilon\) \(\to 0\), in the sense of distribution in \({\mathbb{D}}[0,\infty)\), where B(t) is a d-dimensional Brownian motion and \(\sigma\) is a positive constant.