STOCHASTIC DIFFERENTIAL EQUATIONS AND LIE ALGEBRAS,LIE GROUPS

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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.