Stochastic differential equations

Gaussian stochastic processes in physics

A cumulant expansion for stochastic linear differential equations. I

The relation between the time-convolutionless projection operator formalism and the methods developed by Kubo and van Kampen in the context of linear stochastic differential equations is discussed in detail. It is shown how this formalism may be used to develop a systematic procedure for the elimination of “fast variables”. This procedure is applied to the case of a free Brownian particle and to that of a heavily damped Brownian particle in the presence of an external force to obtain a reduced description in terms of the position variable: the results are found to be in complete agreement with the previous works on this problem. The rotational Brownian motion of an asymmetrical top in the presence of an external torque is also considered and an equation for the reduced probability distribution is derived which contains all the previous works on this problem as special cases.

Time-convolutionless projection operator formalism for elimination of fast variables. Applications to Brownian motion

A detailed discussion of the path-integral formalism for stochastic processes described by a stochastic differential equation driven by a nonwhite noise is given. The path-integral representation in the configuration space of the transition probability for a process driven by Ornstein-Uhlenbeck noise is derived. We show how to treat in this approach any kind of initial conditions, including the question of the coupling with the noise at initial time. Known approximations are reobtained in this context. Markovian approximations based on the Lagrangian are also discussed. The stationary distribution of the process in the weak-noise limit is obtained from the Lagrangian without relying on the use of Fokker-Planck or Markovian approximations.

R.H. Terwiel

Papers

Projection operator method applied to stochastic linear differential equations