Quenched large deviations for one dimensional nonlinear filtering

Consider the standard, one dimensional, nonlinear filtering problem for a diffusion processe ${\mathrm{\Xi }}_t$ observed in small additive white noise. Denote by $q_1^{ϵ}(\cdot )$ the density of the law of ${\mathrm{\Xi }}_1$ conditioned on $\sigma (Y_t^{ϵ}:0\le t\le 1)$. We provide “quenched” large deviation estimates for the random family of measures $q_1^{ϵ}(x)dx$: there exists a continuous, explicit mapping $\bar{J}:R^2\to R$ such that for almost all $B_{\cdot },V_{\cdot }$, $\bar{J}(\cdot ,X_1)$ is a good rate function and for any measurable $G\subset R$, $$-\underset{x\in G^o}{\inf }\bar{J}(x,X_1)\le \mathrm{lim\hspace{0.17em}inf}ϵ\log {\int }_G{q}_1^{ϵ}(x)dx\le \mathrm{lim\hspace{0.17em}sup}ϵ\log {\int }_G{q}_1^{ϵ}(x)dx\le -\underset{x\in \bar{G}}{\inf }\bar{J}(x,X_1)\mathrm{.}$$