## The Annals of Probability

- Ann. Probab.
- Volume 22, Number 2 (1994), 659-679.

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

M. A. Kouritzin and A. J. Heunis

#### 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 $C\lbrack 0, 1\rbrack$.

#### Article information

**Source**

Ann. Probab., Volume 22, Number 2 (1994), 659-679.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176988724

**Digital Object Identifier**

doi:10.1214/aop/1176988724

**Mathematical Reviews number (MathSciNet)**

MR1288126

**Zentralblatt MATH identifier**

0806.60017

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F15: Strong theorems

Secondary: 60F17: Functional limit theorems; invariance principles 93E03: Stochastic systems, general

**Keywords**

Ordinary differential equation mixing processes central limit theorem laws of the iterated logarithm

#### Citation

Kouritzin, M. A.; Heunis, A. J. A Law of the Iterated Logarithm for Stochastic Processes Defined by Differential Equations with a Small Parameter. Ann. Probab. 22 (1994), no. 2, 659--679. doi:10.1214/aop/1176988724. https://projecteuclid.org/euclid.aop/1176988724