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On the Euler equation for minimal geodesics on Riemannian manifoldshaving discontinuous metrics
On stochastic stabilization of difference equations
1.  School of Mathematical Sciences, Dublin City University, Dublin, Ireland 
2.  Department of Statistics and Modelling Science, University of Strathclyde, Glasgow, G1 1XH, Scotland, United Kingdom 
3.  Department of Mathematics and Computer Science, The University of the West Indies, Mona, Kingston 7, Jamaica 
$x_{n+1}=x_n(1+a_nf(x_n))$, $n\ge 1$, $x_0=a$.
We show how this equation can be stabilized by adding the random noise term $\sigma_ng(x_n)\xi_{n+1}$ where $\xi_n$ takes the values +1 or 1 each with probability $1/2$. We also prove a theorem on the almost sure asymptotic stability of the solution of a scalar nonlinear stochastic difference equation with bounded coefficients, and show the connection between the noise stabilization of a stochastic differential equation, and a discretization of this equation.
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