A stochastic process driven by the quadratic Ornstein-Uhlenbeck noise: generator, propagators and all that

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Abstract
A non-Markovian stochastic process modelled by a linear first-order differential equation involving quadratic Ornstein-Uhlenbeck noise is investigated. The generator of an evolution operator of the process is constructed and linear propagators of a one-dimensional probability distribution are built. The initial correlation functions are presented and evolution equations for the moments of the process are derived. Some approximative methods are verified.