Large deviation estimates and functional law for diffusions in modulus spaces (Q2722269)

The \(m\)-dimensional Itô stochastic differential equation NEWLINE\[NEWLINE X_{t}^{\varepsilon,i}=x^{\varepsilon,i}(t)+ \varepsilon\sum_{j=1}^{d}\int_{0}^{t}\sigma^{i}_{\varepsilon,i} (s,X^{\varepsilon}_{s})dW^{j}_{s}+ \int_{0}^{t}b^{j}_{\varepsilon}(s,X^{\varepsilon}_{s}) ds, NEWLINE\]NEWLINE is considered with a family \(\sigma_{\varepsilon}\) of \(m\times d\) matrix fields and vector fields \(b_{\varepsilon}\) on \(R_{+}\times R^{m}\) being uniformly continuous and with \(W\) as a standard \(d\)-dimensional Wiener process; \(x^{\varepsilon}(\cdot)\) is a sequence of functions in the modulus space which converges in this space when \(\varepsilon\to 0\) to a function \(x(\cdot).\) It is proved that the Freidlin-Wentzel large deviation principle (LDP) for small random perturbations of the above-defined dynamical system and the Strassen's functional laws can be extended to the topology of modulus spaces under weaker hypotheses than the Lipschitz one. The main tool of this work is to prove that Schilder's theorem, which gives the LDP for the Wiener measures, and Strassen's theorem, which gives the relative compactness of the family of functions \(\{(2u\log\log u)^{-1/2}W(ut): u>3\}\) of \(t\in [0,1],\) usually announced in the topology of uniform convergence, remain true for diffusions and w.r.t. the topology of modulus spaces. As an application, the LDP estimates are used to extend to the topology of modulus spaces the functional law of the iterated logarithm for diffusion processes and for some functionals of the Brownian motion.