Geodesic
Almost sure polynomial asymptotic stability of stochastic difference equations
Contemporary Mathematics. Fundamental Directions, Proceedings of the Fourth International Conference on Differential and Functional-Differential Equations (Moscow, August 14–21, 2005). Part 3, Tome 17 (2006), pp. 110-128

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In this paper, we establish the almost sure asymptotic stability and decay results for solutions of an autonomous scalar difference equation with a nonhyperbolic equilibrium at the origin, which is perturbed by a random term with a fading state–independent intensity. In particular, we show that if the unbounded noise has tails which fade more quickly than polynomially, then the state–independent perturbation dies away at a sufficiently fast polynomial rate in time, and if the autonomous difference equation has a polynomial nonlinearity at the origin, then the almost sure polynomial rate of decay of solutions can be determined exactly. 
@article{CMFD_2006_17_a7,
     author = {J. Appleby and D. Mackey and A. Rodkina},
     title = {Almost sure polynomial asymptotic stability of stochastic difference equations},
     journal = {Contemporary Mathematics. Fundamental Directions},
     pages = {110--128},
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     volume = {17},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2006_17_a7/}
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J. Appleby; D. Mackey; A. Rodkina. Almost sure polynomial asymptotic stability of stochastic difference equations. Contemporary Mathematics. Fundamental Directions, Proceedings of the Fourth International Conference on Differential and Functional-Differential Equations (Moscow, August 14–21, 2005). Part 3, Tome 17 (2006), pp. 110-128. http://geodesic.mathdoc.fr/item/CMFD_2006_17_a7/