STOCHASTIC DIFFERENTIAL EQUATIONS AND LIE ALGEBRAS,LIE GROUPS

本文研究了由随机微分方程确定的流形上的随机过程，特别是李群上的随机过程，并通过李群上的作用，揭示了它们的几何行为。1）对于李群上的Levy过程，通过其特征（偏差系数，漂移，Levy测度），明确了其稳定的条件。作为一种特殊情况，我们得到了李群上布朗运动稳定的一个充分必要条件，并由此证明了李群的非交换性所引起的挠率会破坏稳定性。2）研究了李群生成的随机布朗流的挠率的不变性，这就是所谓的自相似性。作为相关问题，我们研究了随机微分方程中的扩散过程，无穷维空间中的随机分析，随机数值分析，随机游动多点的渐近分析，p-adic数上的可加过程，并得到了许多结果。

We studied stochastic processes on a manifold determined by stochastic differential equations, in particular, stochjastic processes on Lie groups and showed thier geometric behavior through the action on the Lie group.1) For a Levy process on a Lie group, we clearified the conditon that make it stable, through its characteristics (deffusion coefficients, drifts, Levy measure). As a special case, we obtained a necessary and sufficient condition for the associated vector fields to satisfy, in order that a Brownian motion on a Lie group is stable.Through this result, it turned out that the torsion caused by the non-commutativety of the Lie group should break the stability.2) We studied the invariance property of the destribution of a stochastic flow of diffeomorphisms generated by a SDE,which is called the self-similarity. We found that the Lie algebra generated by the vector fields governing the SDE should be nilpotent and obey a specific commutation relation.As the related problems, WE studied diffusion processes in random environmenrs, stochastic analysis on infinite dimensional spaces, stochastic numerical analysis, the asymptotic analysis of the multiple points of random walks, additive processes on the p-adic numbers and obtained many results.