Towards a unified dynamical theory of the Brownian particle in an ideal gas
We consider the trajectory \(Q^ M(t)\) of a Brownian particle of mass M in an ideal gas of identical particles of mass 1 and of density 1 in equilibrium at inverse temperature 1 (the dynamics is uniform motion plus elastic collisions with the Brownian particle). Our theory, in dimension one, describes a variety of limiting processes - containing the Wiener process and the Ornstein-Uhlenbeck process - for \(A^{-1/2} Q^{M(A)} (At)\) depending on the asymptotic behaviour of M(A). Part of the theory is hypothetical while another part relies upon known results. We also prove that, if \(A^{+\epsilon}\ll M(A)\ll A\), then \(A^{-1/2} Q^{M(A)} (At)\) converges to a Wiener process whose variance is known from papers of Sinai-Soloveichik and of the present authors.
scientific article; zbMATH DE number 4043020