{'fraction_non_alphanumeric': 0.014453477868112014, 'fraction_numerical': 0.0, 'mean_word_length': 5.087912087912088, '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}}
White-Noise Paraxial Approximation for a General Random Hyperbolic System

In this paper we consider a general hyperbolic system subject to random perturbations which models wave propagation in random media. We consider the paraxial white-noise regime in which the propagation distance is much larger than the diameter of the input beam or source, which is itself much larger than the typical wavelength, and in which the correlation length of the medium is of the same order as the diameter of the input beam or source. We introduce a general framework that allows us to consider the general random hyperbolic system. Using invariant imbedding and asymptotic analysis, we show how to derive the system of Itoź--Schroźdinger equations driven by Brownian fields that govern the wave propagation. The forms of the diffraction operator and the covariance matrix of the Brownian fields are computed from the eigenvalues and eigenvectors of the unperturbed system and from the two-point statistics of the random fluctuations of the medium. Applications are given for acoustic, elastic, and electromagnetic waves.