Abstract
We discuss first passage time problems for a class of one dimensional master equations with separable kernels. For this class of master equations, the integral equations for first passage time moments (FPTM) have been transformed exactly into ordinary differential equations by Weiss and Szabo. There the boundary conditions for the first passage time moments have been introduced separately. In our study we use the imbedding method to derive the first passage time densities(FPTD) by imbedding the boundary conditions for crossing the barriers into the master equations. We also derived the results of Weiss and Szabo model from our results for first passage time densities and FPTM for crossing of the barriers. In our study we also derived the results the FPTD and FPTM by taking one of the barriers as absorbing and the other barrier as reflecting. When the separable kernel has only single term the equation for FPTD and FPTM obtained are exactly the same as that of simple diffusion. Generalisation of the model is studied to obtain the differential equations for FPTD and FPTM. The generalisation in the separable kernel leads to higher order differential equation for FPTD and FPTM. The differential equation for FPTM is identified with that of time homogeneous Fokker-Plank equation.
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This work was completed by one of the authors (P.R. Vittal) during his stay at the Indian Statistical Institute, Bangalore as a Visiting Scientist.
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Vittal, P.R., Jagadesan, T. & Muralidhar, V. Stochastic dynamics and passage times for diffusion approximations. Differ Equ Dyn Syst 16, 145–161 (2008). https://doi.org/10.1007/s12591-008-0009-z
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DOI: https://doi.org/10.1007/s12591-008-0009-z