Brownian motion with a discrete drag coefficient and its analysis in terms of characteristic functionals

A functional formalism is presented to determine multi-time statistics of the stochastic process of a Brownian particle whose drag coefficient varies directly depending on whether its velocity is positive or not. The nonlinear feature of the Langevin equation describing the process makes it difficult to employ the conventional Fokker-Planck equation method. A new approach is developed here based on functional differential equations for a characteristic functional (CHFL), which is a generator of multi-time moment functions, and for an incomplete characteristic functional (ICHFL) which bears information about statistics of particle velocities of positive values alone. A pseudo-projection operator p is devised to project CHFL onto ICHFL. Novikov’s formula allows one to derive from the Langevin equation a set of functional differential equations governing CHFL and ICHFL. Putting CHFL to be a linear combination of two auxiliary functionals, one perpendicular to p, and the other invariant under p, one is able to find an approximate solution to the set of equations. The resulting velocity correlation function agrees with simulated data closely, in contrast with the poor agreement given by statistical linearization techniques. [In Japanese.]