| Co-Investigator(Kenkyū-buntansha) |
NAMIKI Takao Hokkaido Univ., Grad. School of Sci., Assist., 大学院・理学研究科, 助手 (40271712)
MATSUMOTO Kenji Hokkaido Univ., Grad. School of Sci., Asso. Prof., 大学院・理学研究科, 助教授 (80183953)
INOUE Akihiko Hokkaido Univ., Grad. School of Sci., Asso. Prof., 大学院・理学研究科, 助教授 (50168431)
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| Research Abstract |
Let p(t,x) (0【less than or equal】t【less than or equal】1,x∈R^d) be a solution to the Liouville equation, and L(t,x;u):[0,1] × R^d × R^d → [0,∞) be measurable and strictly conves in u.
(1) We showed that there exists a minimizer of E[∫^1_0L(t,X^ε(t);u(t))dt] over all R^d-valued continuous semimartingales {X^ε(t)=X^ε(0)+∫^t_0u(s)ds+εW(t)}_0【less than or equal】t【less than or equal】1 which has distributions p(t,x)dx for all t∈[0,1], and that the minimizer is Markovian and converges, as ε→ 0, to a R^d-valued absolutely continuous stochastic process {φ^0(t)}_0【less than or equal】t【less than or equal】1.
(2) We showed that {φ^0(t)}_0【less than or equal】t【less than or equal】1 is a minimizer of E[∫^1_0L(t,φ(t);dφ(t)/dt)dt] over all R^d-values absolutely continous stochastic processes which has distributions p(t,x)dx for all t∈[0,1]. We also showed that the minimizers satisfies the same ordinary differential equation.
Let R be a probability density function on (d【greater than or equal】1). We obtained the following result on R-anisotropic Gauss curvature flow of glaphs on R^d : the construction of approximating discrete stochastic processes of Gauss curvature flow : the existence and the uniqueness of a weak solution to the partial differential equation which describes Gauss curvature flow : the proof that the unique weak solution to the partial differential equation is a viscosity solution.
Let R be a probability density function on S^<n-1> (n【greater than or equal】2). We obtained the similar result on R-anisotropic Gauss curvature flow of hypersurfaces in R^n. The construction of a descrete approximation of Gauss curvature flow of hypersurfaces in R^n was a famous open problem when n【greater than or equal】3. We solved it completely by studying it in the framework of the probability theory.
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