By Jianhai Bao, George Yin, Chenggui Yuan

This short treats dynamical platforms that contain delays and random disturbances. The learn is prompted by means of a large choice of structures in genuine existence within which random noise needs to be considered and the impression of delays can't be missed. targeting such platforms which are defined by way of useful stochastic differential equations, this paintings specializes in the examine of huge time habit, specifically, ergodicity.This short is written for probabilists, utilized mathematicians, engineers, and scientists who have to use hold up platforms and sensible stochastic differential equations of their paintings. chosen subject matters from the short is usually utilized in a graduate point issues path in likelihood and stochastic processes.

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**Example text**

51) for some λ > 0. 51) hold, respectively. 52) s eνu (ν − λ1 ) (ε + Λ(u, s) 2 H) + b(Λu (s) + Υu (s)) H du. 53) s t + Leντ eνu Υ (u, s) H du s + Leντ s eνu ξ(u − s) H du. 12)] and (C2) that ∞ E Υ (t, s) H βkα k=1 1 − e−αλk (t−s) αλk 1/α < ∞. 55) by choosing ν ∈ (0, λ1 ) sufficiently small such that λ1 − ν − Leντ > 0 due to λ1 > L. 52), we deduce that E sup (eνr Λ(r, s1 ) − Λ(r, s2 ) t−τ ≤r ≤t ≤ eνs2 E Λ(s2 , s1 ) − ξ(0) H H) + eντ L t s2 + eντ L s2 s2 −τ eνu E Γ (u) H du s2 e−(λ1 −ν)u+λ1 s2 du E e(s2 −r )A dZ (r ) s1 + 2eντ L s2 s2 −τ eνu E Υ (u, s1 ) H H du.

6 Under dissipative conditions, by the Arzelà–Ascoli tightness characterization, Es-Sarhir et al. [54] and Kinnally-Williams [79] exploited existence of invariant measures for FSDEs with super-linear drift terms and positivity constraints, respectively. Applying the Itô formula, they gave the uniform boundedness for higher moments of the segment processes, which plays a key role in analyzing the diffusion terms by the Kolmogrov tightness criterion. 4) need not satisfy any dissipative conditions, and therefore the techniques adopted in [54, 79] no longer work.

5], and in the second step utilized the elementary inequality: (a + b)2 ≤ (1 + θ )a 2 + (1 + θ1 )b2 with θ = ε. 11). Thus, by Gronwall’s inequality, we have t e2λt (E|X (t)|2 p )1/ p ≤ cε e2λs ds + ξ 2 ∞ 0 t + 2cε βκ,ε ( p) s 0 e2λu du + ξ 0 2 ∞ e2βκ,ε ( p)(t−s) ds. 11). 14) t−2τ 1+ ξ 2p ∞, t ≥ τ. Similarly, E 1+ ξ sup |X (t)|2 p 2p ∞. 15) immediately. 10). Let Θ(t) = X (t, ξ ) − X (t, η) for notational simplicity. 21), followed by utilizing Hölder’s inequality for the time integrals and Itô’s isometry for the term involving martingale, we obtain that E|Θ(t)|2 ≤ 1+α Γ (t)(ξ(0) − η(0)) + α t + (1 + α) [−τ,0] 0 μ(dθ ) θ Γ (t + θ − s)(ξ(s) − η(s))ds E Γ (t − s)(σ (X s (ξ )) − σ (X s (η))) 2 H S ds 0 1 + α −2λt e ξ −η α t + (1 + α)cλ2 λ3 + ρ([−τ, 0]) 2 ∞ 2 e−2λ(t−s) |Θ(s)| + |Θ(s + θ )|ρ(dθ ) ds 0 1 + α −2λt e ξ −η α [−τ,0] cα e−2λt ξ − η 2 ∞ 2 2 ∞ + 2(1 + α)cλ2 λ3 [−τ,0] t −2λ(t−s) |Θ(s)|2 e 0 |Θ(s + θ )|2 ρ(dθ ) ds t + 2β(α) e−2λ(t−s) E|Θ(s)|2 ds, 0 where β(α) := (1 + α)cλ2 λ3 (1 + e2λτ ρ([−τ, 0])).