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July  2006, 6(4): 667-696. doi: 10.3934/dcdsb.2006.6.667

Pathwise non-exponential decay rates of solutions of scalar nonlinear stochastic differential equations

John A. D. Appleby 1, , Alexandra Rodkina 2, and  Henri Schurz 3,

1. 

School of Mathematical Sciences, Dublin City University, Dublin, Ireland
2. 

Department of Mathematics and Computer Science, The University of the West Indies, Mona, Kingston 7
3. 

Southern Illinois University, Department of Mathematics, MC 4408, 1245 Lincoln Drive, Carbondale, IL 62901-7316

Received  January 2005 Revised  September 2005 Published  April 2006

This paper studies the pathwise asymptotic stability of the zero solution of scalar stochastic differential equation of Itô type. Specifically, we provide conditions for solutions to converge to zero at given non-exponential rates. The results completely classify the rates of decay of many parameterised families of stochastic differential equations.
Keywords: Almost sure asymptotic stability, asymptotic stability with probability p, stochastic differential equations., wedge function, weak exponential asymptotic stability, exponential asymptotic stability, polynomial asymptotic stability.
Mathematics Subject Classification: Primary: 93E15; Secondary: 34F05, 37H10, 60H1.
Citation: John A. D. Appleby, Alexandra Rodkina, Henri Schurz. Pathwise non-exponential decay rates of solutions of scalar nonlinear stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 667-696. doi: 10.3934/dcdsb.2006.6.667
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