On large deviations of coupled diffusions with time scale separation (Q317501)

Consider the coupled diffusions given by NEWLINE\[NEWLINE\begin{aligned} dX^\varepsilon_ t &= A(X^\varepsilon_t, x^\varepsilon_t)\,dt+ \sqrt{\varepsilon} B(X^\varepsilon_t, x^\varepsilon_t)\,dW^\varepsilon_t,\\ dx^\varepsilon_t &= {1\over \varepsilon} a(X^\varepsilon_t,x^\varepsilon_t)\,dt+ {1\over\sqrt{\varepsilon}} b(X^\varepsilon_t, x^\varepsilon_t)\,dW^\varepsilon_t,\end{aligned}NEWLINE\]NEWLINE where \(\varepsilon> 0\) is a small parameter, \(A(u,x)\) an \(n\)-vector with \(u\in\mathbb{R}^n\) and \(x\in \mathbb{R}^1\), \(B(u,x)\) an \(n\times k\)-matrix, \(a(u,x)\) an \(\ell\)-vector, \(b(u,x)\) an \(\ell\times k\)-matrix, and \(W^\varepsilon\) is an \(\mathbb{R}^k\)-valued standard Wiener process. The processes \(X^\varepsilon\) and \(x^\varepsilon\) are evolving on different time scales, the time for \(x^\varepsilon\) being accelerated by a factor \(1/\varepsilon\). Denote by \(\mu^\varepsilon\) the empirical process associated with \(x^\varepsilon\).NEWLINENEWLINE The author obtains a large deviation principle (LDP) for the distribution of \((X^\varepsilon,\mu^\varepsilon)\) and the large deviation rate function. Projections on the first and the second component lead to LDPs for \(X^\varepsilon\) and \(\mu^\varepsilon\), respectively. This result generalizes that of \textit{R. Liptser} [Probab. Theory Relat. Fields 106, No. 1, 71--104 (1996; Zbl 0855.60030)], who treated the case of one-dimensional \(X^\varepsilon\) and \(x^\varepsilon\) with \(a\) and \(b\) not depending on \(u\) and the Wiener processes being independent.