Logarithmic estimates for the density of hypoelliptic two-parameter diffusions
The authors study a hypoelliptic two-parameter diffusion. In the first paragraph two propositions provide tools for the analysis of the upper and lower bounds, respectively, for the behaviour of \(p^\varepsilon\) as \(\varepsilon \downarrow 0\). For example Proposition 1.1 (see D. Nualart (1998)). Assume that (i) \(\sup_{x\in[0,1]} \|F^\varepsilon \|_{k,p}< \infty\) for each integer \(k\geq 1\), \(p\in [1,\infty)\), (ii) for any \(p\in [1,\infty)\), there exist \(\varepsilon>0\) and \(N(p)\in [1,\infty]\) such that \(\|\Gamma_{p \varepsilon}\|\leq\varepsilon^{-N(p)}\), for any \(\varepsilon\leq \varepsilon_p \), (iii) \(\{F^\varepsilon,\;\varepsilon\in (0,1)\}\) obeys a large deviation principle on \(\mathbb{R}^m\) with rate function \(I(y)\), \(y\in\mathbb{R}^m\). Then \(\sup_{\varepsilon \downarrow 0}\varepsilon^2 \log p^\varepsilon(y) \leq-I(y)\). Using a similar approach to that in R. Leandre (1987), the authors prove that the upper and lower bounds provided in Propositions 1.1 and 1.2, respectively, coincide. In Section 2 the authors use the hypothesis (P\(1_x)\): Set \({\mathcal D}_0 =\{A_h,1\leq h\leq d\}\) and \({\mathcal D}_j= \{A_h^\nabla V,1\leq h\leq d,\;V\in {\mathcal D}_{j-1}\}\) for any \(j\geq 1\). The property (P\(1_x)\) yields the existence of a positive integer \(j\), such that the linear span of \(\bigcup^{j_0}_{j=0} {\mathcal D}_j\) at the point \(x\) has dimension \(m\). \(x\in R^m\) being fixed and assuming (P\(1_x)\) the authors prove a sharp upper bound in small time \(z\) for the \(L^p\) moments of the inverse of the Malliavin of \(X_z(x)\). In Section 3 with (P\(1_x)\) and using Propositions 1.1 and 1.2 the authors establish upper and lower bounds for the lim sup and the lim inf of \(\varepsilon^2 \log^\varepsilon_{1,1} (x,y)\), respectively. In Section 4 the authors assume that (P\(2_x)\) holds for any \(x\in \mathbb{R}^m\). For the essentials of Malliavin calculus needed for reading this note, see \textit{D. Nualart} [``The Malliavin calculus and related topics'' (1995; Zbl 0837.60050)].
scientific article; zbMATH DE number 1764704
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