Averaged run-and-tumble walks

A random walk consisting of a run phase at constant speed interrupted by tumble events is analyzed and analytically solved for arbitrary time distributions. A general expression is given for the Laplace-Fourier transform of the probability density function and for the mean square displacement averaging over initial conditions. Run-and-tumble bacteria and Levy walks are considered as particular cases. The effects of an underlying Brownian noise are also discussed. Derived expressions can be used for a direct comparison with experimentally measured quantities.