A note on the LaSalle-type theorems for stochastic differential delay equations

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Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation (2009, background): “The following hypothesis is imposed in [22]: (*) For any initial value x(0) ∈ R, equation (28) has a unique solution denoted by x(t, x(0)) on t ≥ 0.”
Khasminskii-Type Theorems for Stochastic Differential Delay Equations (2005, background): “Mao, X. 2002.”
Stochastic differential delay equations of population dynamics (2005, methodology): “Since the coefficients of the SDDE (1.4) are locally Lipschitz continuous, for given initial data{x(t): −τ t 0} ∈ C([−τ,0];Rn+) there is a unique maximal local w ly, ne ns solutionx(t) on t ∈ [−τ, τe), whereτe is the explosion time (cf. Mao [15, p. 95]).”
Almost surely asymptotic stability of neutral stochastic differential delay equations with Markovian switching (2008, background): “For more detailed account on hybrid systems please see [1, 4, 5, 9, 10, 11, 19, 20, 22, 23, 24, 26, 27, 29].”
Neutral Stochastic Differential Delay Equations with Markovian Switching (2003, background): “For more detailed account on hybrid systems please see [1, 4, 5, 9, 10, 11, 19, 20, 22, 23, 24, 26, 27, 29].”
Stochastic Population Dynamics Driven by Levy Noise (2011, background): “In [6], under the condition (H2) σii > 0 if 1 ≤ i ≤ n while σij ≥ 0 if i 6= j. Mao, Marion and Renshaw reveal the important fact that Brownian motion noise can suppress a potential population explosion.”