A stochastic perturbation of Wright's delay equation with bounded Lipschitz noise admits a trivial invariant measure at -1 and a nontrivial invariant measure supported on (-1, infinity).
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Sufficient conditions are given for invariant probability measures in stochastic delay differential equations driven by integrable Lévy noise, with a key reduction from bounded solutions to bounded segments under a one-sided bound on the deterministic coefficient.
Non-trivial invariant measures exist for stochastic Mackey-Glass and Nicholson's blowflies equations if and only if solutions remain bounded away from zero in probability for at least one initial condition.
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Stochastic Wright's Equation: Existence of Invariant Measures
A stochastic perturbation of Wright's delay equation with bounded Lipschitz noise admits a trivial invariant measure at -1 and a nontrivial invariant measure supported on (-1, infinity).