{'fraction_non_alphanumeric': 0.01072961373390558, 'fraction_numerical': 0.0, 'mean_word_length': 6.184615384615385, '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}}
Probabilistic Representations and Hyperbound Estimates for Semigroups

In this paper, we study lower order perturbations of a symmetric second order diier-ential operator generating a hypercontractive semigroup. We give a probabilistic representation of the in general not sub-Markovian semigroup associated with the perturbed operator and prove that the perturbed semigroup is also hypercontractive under some exponential integrability conditions on the coeecients.