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Stochastic functional Kolmogorov equations, I: Persistence

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Summary: This study introduces a new framework for stochastic functional Kolmogorov equations, with a focus on long-term extinction in biological and ecological problems. The main techniques used are the functional Itô formula and a dynamic system approach for analyzing the long-term behavior of systems. The characterization of extinction is based on investigating random occupation measures and examining the behavior of functional systems around boundaries.

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