First passage time distributions for finite one-dimensional random walks

We present closed expressions for the characteristic function of the first passage time distribution for biased and unbiased random walks on finite chains and continuous segments with reflecting boundary conditions. Earlier results on mean first passage times for one-dimensional random walks emerge as special cases. The divergences that result as the boundary is moved out to infinity are exhibited explicitly. For a symmetric random walk on a line, the distribution is an elliptic theta function that goes over into the known Lévy distribution with exponent 1/2 as the boundary tends to ∞.

[1]  E. Hansen.  A Table of Series and Products , 1977 .

[2]  F. Oberhettinger,et al.  Tables of Laplace Transforms , 1973 .

[3]  Daniel T. Gillespie.  On the calculation of mean first passage times for simple random walks , 1981 .

[4]  Bruce J. West,et al.  Analytic theory of extrema. III. Results for master equations with application to unimolecular decomposition , 1980 .

[5]  J. Bernasconi,et al.  Excitation Dynamics in Random One-Dimensional Systems , 1981 .

[6]  Bruce J. West,et al.  Fractal dimensionality of Lévy processes. , 1982, Proceedings of the National Academy of Sciences of the United States of America.

[7]  D. Darling,et al.  THE FIRST PASSAGE PROBLEM FOR A CONTINUOUS MARKOFF PROCESS , 1953 .

[8]  G. Weiss.  First passage time problems for one-dimensional random walks , 1981 .

[9]  J. Bernasconi,et al.  Spectral diffusion in a one-dimensional percolation model , 1978 .

[10]  G. Pólya.  Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straßennetz , 1921 .

[11]  A. Siegert.  On the First Passage Time Probability Problem , 1951 .

[12]  R. G. Medhurst,et al.  Topics in the Theory of Random Noise , 1969 .

[13]  Emanuel Parzen,et al.  Stochastic Processes , 1962 .

[14]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[15]  H. Heinrich,et al.  F. Oberhettinger / L. Badii, Tables of Laplace Transforms. VII. + 428 S. Berlin/Heidelberg/New York 1973. Springer‐Verlag. Preis brosch, DM 39,— , 1975 .

[16]  R. L. Stratonovich,et al.  Topics in the theory of random noise , 1967 .