Large deviations and functional iterated logarithm law for diffusion processes (Q1072251)

Let y be the diffusion process on \(U\subset R^ m\) satisfying the equation \(dy_ t=b(y_ t)+\sigma (y_ t)dB_ t,\) \(y_ 0=x\), where \(B_ t\) is the Brownian motion in \(R^ k\), \(\sigma\) is an \(m\times k\) matrix field and b is a vector field on U. Let for every \(u>0\) \(z_ u(t)=\Gamma_{\sqrt{u \log \log u}}(y_{ut})\), \(0\leq t\leq 1\), where \(\Gamma_{\sqrt{u \log \log u}}(\cdot)\) is a certain family of contractions from U into U \((\Gamma_{\alpha}y=\alpha^{-1}y\) is a special case). Using a large deviation principle, it is shown that under some assumptions \(\{z_ u\}\) is relatively compact a.s. as \(u\to \infty\) in the set of continuous functions \(\{f: [0,1]\to U| \quad f(0)=x\}\) with the uniform convergence, and the limit set of \(\{z_ u\}\) is explicitly given. The above is a generalization of Strassen's law of the iterated logarithm for Brownian motion. Applications for iterated integrals and diffusions on Lie groups are discussed.