A LAW OF THE ITERATED LOGARITHM FOR STOCHASTIC-PROCESSES DEFINED BY DIFFERENTIAL-EQUATIONS WITH A SMALL-PARAMETER

Consider the following random ordinary differential equation: X(epsilon)(tau) = F(X(epsilon)(tau), (tau/epsilon, omega) subject to X(epsilon)(0) = x0, where {F(x, t, omega), t > 0} are stochastic processes indexed by x in R(d), and the dependence on x is sufficiently regular to ensure that the equation has a unique solution X(epsilon)(tau, omega) over the interval 0 less-than-or-equal-to T less-than-or-equal-to 1 for each epsilon > 0. Under rather general conditions one can associate with the preceding equation a nonrandom averaged equation: x0(tau) = F(x0(tau))BAR subject to x0(0) = x0, such that lim(epsilon --> 0) sup0 less-than-or-equal-to tau less-than-or-equal-to 1 E\X(epsilon)(tau) - x0(tau)\ = 0. In this article we show that as epsilon --> 0 the random function (X(epsilon)(tau) - x0(.))/ square-root 2epsilon log log epsilon-1 almost surely converges to and clusters throughout a compact set K of C[0,1].