{'fraction_non_alphanumeric': 0.022452504317789293, 'fraction_numerical': 0.0, 'mean_word_length': 6.073170731707317, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Geometric Optics Approach to First‐Passage Distributions: Caustic Boundaries and Exponentially Small Eigenvalues

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.