STOCHASTIC DIFFERENTIAL EQUATIONS AND LIE ALGEBRAS,LIE GROUPS

研究了由随机微分方程决定的流形上的随机过程，特别是李群上的随机过程，并通过对李群的作用证明了它们的几何性质。1)对于李群上的Levy过程，通过其特征（扩散系数、漂移、Levy测度）明确了使其稳定的条件。作为特例，我们得到了李群上布朗运动稳定的相关向量场所满足的一个充分必要条件。通过这一结果，证明了李群的非交换性引起的扭转会破坏稳定性。2)研究了由SDE生成的随机微分同态流的分布的不变性，即自相似。我们发现由控制SDE的向量场生成的李代数应该是幂零的，并且服从特定的交换关系。作为相关问题，我们研究了随机环境下的扩散过程、无限维空间上的随机分析、随机数值分析、随机游走多点的渐近分析、p进数上的加性过程等，并取得了许多成果。

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.