Research of the solutions of the partial differential equation of elliptic type or parabolic type in unbounded domains and its stochastic analysis consideration

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In this research project the boundary behavior of the solutions of elliptic or parabolic partial differential equations at infinity of unbounded domains was investigated from the view point of potential theory or the theory of probability.In particular, qualitative or quantitative properties of the exceptional sets on which the solutions behave irregularly e.g. rarefied sets and a-minimally thin sets at infinity of typical unbounded domains were treated.About qualitative characterization of these exceptional sets, there are "the judgment condition of Winner type" and "the sets of determination" of these thin sets. Although these were made to conical domains by the last research task, these were extended to cylindrical domains this time.When quantitative characterization of these exceptional sets are considered, there is a problem to find certain kinds of countably infinite of balls with which these thin sets are covered. The result obtained about these thin sets in the half space by Essen was extended to the results in conical domains.All results were published in American journals (Proc. Amer. Math. Soc. and Complex variables), a Czech journal (Czech. Math. J.) and domestic journals (Hiroshima Math. J. and Advanced Studies in Pure Math.), and will be published in a domestic journal (Hokkaido Math. J.).With respect to solutions (temperatures) of the equations of parabolic type, i.e. heat equations, although the results corresponding to ones obtained about harmonic functions are not obtained yet, it is ready to prepare to get them.