Reflecting Brownian motion in three dimensions: a new proof of sufficient conditions for positive recurrence

Abstract By means of the Banach fixed point principle, we establish some sufficient conditions ensuring the existence of the global attracting sets of neutral stochastic functional integrodifferential equations with finite delay driven by a fractional Brownian motion (fBm) with Hurst parameter H ∈ ( 1 2 , 1 ) {H\in(\frac{1}{2},1)} in a Hilbert space.

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Existence and exponential stability in the pth moment for impulsive neutral stochastic integro-differential equations driven by mixed fractional Brownian motion

Journal of Inequalities and Applications ◽

10.1186/s13660-019-2213-5 ◽

2019 ◽

Vol 2019 (1) ◽

Cited By ~ 1

Author(s):

Xia Zhou ◽

Dongpeng Zhou ◽

Shouming Zhong

Keyword(s):

Brownian Motion ◽

Differential Equations ◽

Exponential Stability ◽

Fractional Brownian Motion ◽

Mild Solutions ◽

Semigroup Theory ◽

Sufficient Conditions ◽

Fractional Power ◽

Banach Fixed Point Theorem ◽

Mixed Fractional Brownian Motion

Abstract This paper consider the existence, uniqueness and exponential stability in the pth moment of mild solution for impulsive neutral stochastic integro-differential equations driven simultaneously by fractional Brownian motion and by standard Brownian motion. Based on semigroup theory, the sufficient conditions to ensure the existence and uniqueness of mild solutions are obtained in terms of fractional power of operators and Banach fixed point theorem. Moreover, the pth moment exponential stability conditions of the equation are obtained by means of an impulsive integral inequality. Finally, an example is presented to illustrate the effectiveness of the obtained results.

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The M/M/1 queue with mass exodus and mass arrivals when empty

Journal of Applied Probability ◽

10.1017/s0021900200100816 ◽

1997 ◽

Vol 34 (01) ◽

pp. 192-207 ◽

Cited By ~ 12

Author(s):

Anyue Chen ◽

Eric Renshaw

Keyword(s):

Sufficient Conditions ◽

High Order ◽

Necessary And Sufficient Conditions ◽

Positive Recurrence ◽

Powerful Technique ◽

Resolvent Approach ◽

Necessary And Sufficient ◽

Direct Attack

An M/M/1 queue is subject to mass exodus at rate β and mass immigration at rate when idle. A general resolvent approach is used to derive occupation probabilities and high-order moments. This powerful technique is not only considerably easier to apply than a standard direct attack on the forward p.g.f. equation, but it also implicitly yields necessary and sufficient conditions for recurrence, positive recurrence and transience.

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