A Law of the Iterated Logarithm for Stochastic Processes Defined by Differential Equations with a Small Parameter

Abstract

Consider the following random ordinary differential equation: $\dot{X}^\epsilon(\tau) = F(X^\epsilon(\tau), \tau/\epsilon, \omega) \text{subject to} X^\epsilon(0) = x_0$, where $\{F(x, t, \omega), t \geq 0\}$ are stochastic processes indexed by $x$ in $\mathfrak{R}^d$, and the dependence on $x$ is sufficiently regular to ensure that the equation has a unique solution $X^\epsilon(\tau, \omega)$ over the interval $0 \leq \tau \leq 1$ for each $\epsilon > 0$. Under rather general conditions one can associate with the preceding equation a nonrandom averaged equation: $\dot{x}^0(\tau) = \overline{F}(x^0(\tau)) \text{subject to} x^0(0) = x_0,$ such that $\lim_{\epsilon\rightarrow 0} \sup_{0\leq\tau\leq 1}E|X^\epsilon(\tau) - x^0(\tau)| = 0$. In this article we show that as $\epsilon \rightarrow 0$ the random function $(X^\epsilon(\cdot) - x^0(\cdot))/\sqrt{2\epsilon\log\log\epsilon^{-1}}$ almost surely converges to and clusters throughout a compact set $K$ of $C\lbrack 0, 1\rbrack$.