Hörmander型亚椭圆算子的小时间对角渐近性
Small time asymptotic on the diagonal for Hörmander's type hypoelliptic operators
摘要：
本文计算了具有漂移的Hörmander型亚椭圆算子基本解在漂移场的一个驻点$x_0$的小时间渐近性质。我们证明了渐近阶依赖于相关控制问题及其近似系统的可控性。如果近似系统的控制问题在$x_0$是可控的，那么原始控制问题也是可控的，在这种情况下，我们证明了基本解爆破为$t^{-N/2}$，其中$N$是由定义亚椭圆算子的域的$x_0$处的李代数确定的数。
英文摘要：
We compute the small time asymptotic of the fundamental solution of Hörmander's type hypoelliptic operators with drift, at a stationary point, $x_0$, of the drift field. We show that the order of the asymptotic depends on the controllability of an associated control problem and of its approximating system. If the control problem of the approximating system is controllable at $x_0$, then so is also the original control problem, and in this case we show that the fundamental solution blows up as $t^{-N/2}$, where $N$ is a number determined by the Lie algebra at $x_0$ of the fields, that define the hypoelliptic operator.
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