Small deviations for Gaussian Markov processes under the sup-norm
It is well known that if \(\{ B_t\), \(t\geq 0\}\) is a linear Brownian motion starting from 0, then \(B\) verifies the following ``small deviations principle'': \[ \lim_{\varepsilon\rightarrow 0} \varepsilon^2 \log {\mathbf P}\Bigl(\sup_{0\leq t\leq 1} \|B_t \|<\varepsilon\Bigr)=-\pi^2/8. \] The purpose of the paper is to extend this property to the Gaussian Markov processes on the line. Such processes may be represented as follows: for every \(t \in (0,1)\), \(X_t= h_t B_{g_t}\), where \(g\) is a positive nondecreasing function, \(h\) a positive function, and \(B\) a linear Brownian motion starting from 0. Then, if \(H = h\) and \(G = gh\), the main theorem says that \[ \lim_{\varepsilon\rightarrow 0} \varepsilon^2 \log {\mathbf P}\Bigl(\sup_{0\leq t\leq 1} \|X_t \|<\varepsilon\Bigr)=-\pi^2/8\int_0^1 (G'H - GH'). \] Except in the trivial deterministic case the integral is always positive, and some finer study is also carried out when it is infinite. The method relies on a link with the small ball estimates for Brownian motion under a weighted sup norm, which were first studied by \textit{A. A. Mogul'skij} [Theory Probab. Appl. 19, 726-736 (1974); translation from Teor. Veroyatn. Primen. 19, 755-765 (1974; Zbl 0326.60061)].
scientific article; zbMATH DE number 1418047