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We develop a new generalized coupling approach to the study of stochastic delay equations with Holder continuous coefficients, for which analytical PDE-based methods are not available. We prove that such equations possess unique weak solutions, and establish weak ergodic rates for the corresponding segment processes. We also prove, under additional smoothness assumptions on the coefficients, stabilization rates for the sensitivities in the initial value of the corresponding semigroups

Well-posedness, stability and sensitivities for stochastic delay equations: A generalized coupling approach