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Small-time fluctuations for the bridge in a model class of hypoelliptic diffusions of weak Hörmander type. (English) Zbl 1419.35009

The diffusion process whose generator is an operator of the form \(L=\sum_{i=1}^{d}(Ax)_{x_{i}}\frac{\partial }{\partial x_{i}}+\sum_{i,j=1}^{d}(BB^{*})_{ij}\frac{\partial^{2}}{\partial x_{i} \partial x_{j}} \) is studied. Here \(A\) and \(B\) are \(d\times d\) and \(d\times m\) (\(m\geq d\)) constant matrices such that the operator \(L\) satisfies the weak Hörmander condition and thus hypoelliptic. Fix \(x\in {\mathbb R}^{d}\) and let \(\varepsilon>0\).There exists a diffusion process \(x_{t}^{\varepsilon}\) (\(t\in [0;1]\)) starting from x and having the generator \(\varepsilon L\). For \(y\in {\mathbb R}^{d}\), let \(z^{\varepsilon}_{t}\) be the process \(x_{t}^{\varepsilon}\) such that \(x_{1}^{\varepsilon} = y\). The main results of the article are the description of the diffusion bridges \(z^{\varepsilon}_{t}\) in the limit as \(\varepsilon \to 0\). The leading order behaviour of the diffusion bridge \(z^{\varepsilon}_{t}\) as \(\varepsilon \to 0\) is described. The case of the iterated Kolmogorov diffusion is considered as an example.

MSC:

35H10 Hypoelliptic equations
60F05 Central limit and other weak theorems
60J60 Diffusion processes
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