Outflow Probability for Drift–Diffusion Dynamics

The proposed explanations are provided for the one–dimensional diffusion process with constant drift by using forward Fokker–Planck technique. We present the exact calculations and numerical evaluation to get the outflow probability in a finite interval, i.e. first passage time probability density distribution taking into account reflecting boundary on left hand side and absorbing border on right hand side. This quantity is calculated from balance equation which follows from conservation of probability. At first, the initial-boundary-value problem is solved analytically in terms of eigenfunction expansion which relates to Sturm–Liouville analysis. The results are obtained for all possible values of drift (positive, zero, negative). As application we get the cumulative breakdown probability which is used in theory of traffic flow.

[1]  H. Risken.  The Fokker-Planck equation : methods of solution and applications , 1985 .

[2]  Anton Zettl,et al.  Sturm-Liouville theory , 2005 .

[3]  V. Linetsky.  The spectral representation of Bessel processes with constant drift: applications in queueing and finance , 2004, Journal of Applied Probability.

[4]  Evolution of escape processes with a time-varying load. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  R. Fox,et al.  Rectified Brownian motion and kinesin motion along microtubules. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  S. Redner.  A guide to first-passage processes , 2001 .

[7]  M. Marletta,et al.  Floquet theory for left-definite Sturm-Liouville problems , 2005 .

[8]  Ihor Lubashevsky,et al.  Probabilistic Description of Traffic Flow , 2001 .

[9]  C. Gardiner.  Handbook of Stochastic Methods , 1983 .

[10]  Sidney Redner,et al.  A guide to first-passage processes , 2001 .

[11]  C. W. Gardiner,et al.  Handbook of stochastic methods - for physics, chemistry and the natural sciences, Second Edition , 1986, Springer series in synergetics.

[12]  A.P.S. Selvadurai,et al.  Partial differential equations in mechanics , 2000 .

[13]  Vincenzo Capasso,et al.  An Introduction to Continuous-Time Stochastic Processes , 2004, Modeling and Simulation in Science, Engineering and Technology.

[14]  V. Linetsky.  On the transition densities for reflected diffusions , 2005, Advances in Applied Probability.

[15]  K. Vahala.  Handbook of stochastic methods for physics, chemistry and the natural sciences , 1986, IEEE Journal of Quantum Electronics.