Flux and First-Passage Time Distributions in One-Dimensional Integrated Stochastic Processes with Arbitrary Temporal Correlation and Drift

The arrival of tracers at boundaries with defined distances from the origin of their motion in stochastically fluctuating advection processes is investigated. The advection model is a stationary one-dimensional integrated stochastic process with an arbitrary a priori known correlation and with possible mean drift. The current (direction-sensitive), the total flux (direction-insensitive) of tracers through a non-absorbing boundary, and the first-passage times of the tracers at an absorbing boundary are derived depending on the correlation function of the carrying flow velocity. While the general derivations are universal with respect to the distribution function of the advection’s increments, the current and the total flux are explicitly derived for a Gaussian distribution. The first-passage time is derived implicitly through an integral that is solved numerically in the present study. No approximations or restrictions to special cases of the advection process are used. One application is one-dimensional Gaussian turbulence, where the one-dimensional random velocity carries tracer particles through space. Finally, subdiffusive or superdiffusive behavior can temporarily be reached by such a stochastic process with an adequately designed correlation function.