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Geometric Optics Approach to First‐Passage Distributions: Caustic Boundaries and Exponentially Small Eigenvalues
Author(s) -
Knessl Charles
Publication year - 2000
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/1467-9590.00153
Subject(s) - caustic (mathematics) , eigenvalues and eigenvectors , geometrical optics , exponential growth , mathematics , singular perturbation , mathematical analysis , diffusion , perturbation (astronomy) , distribution (mathematics) , diffusion process , physics , optics , quantum mechanics , knowledge management , innovation diffusion , computer science
We develop singular perturbation methods for computing the first passage time distribution for one‐dimensional diffusion processes. Detailed results are given for an Ornstein–Uhlenbeck process, and the method is sketched for more general problems. For some parameter values, we find the presence of caustic boundaries; whereas, for other parameter values, there are exponentially small eigenvalues. We use the ray method of geometric optics and asymptotic matching.