解析学・幾何学に関連する確率論

1996 Fiscal Year Annual Research Report

• Project/Area Number: 07304064
• Research Institution: Saga University
• Principal Investigator: 小倉 幸雄 佐賀大学, 理工学部, 教授 (00037847)
• Co-Investigator(Kenkyū-buntansha): 舟木 直久 東京大学, 数理科学研究科, 教授 (60112174) ; 長井 英生 大阪大学, 基礎工学部, 教授 (70110848) ; 國田 寛 九州大学, 数理学研究科, 教授 (30022552) ; 渡辺 信三 京都大学, 理学研究科, 教授 (90025297) ; 池田 信行 立命館大学, 理工学部, 教授 (00028078)
• Keywords: 漸近理論 / 遺伝モデル / 計算数学 / 確率制御 / 量子カオス / リーマン多様体 / ブラウン運動 / 非平衡統計力学

Research Abstract

以下のテーマで研究集会を開催して、講演と討論を行ない、新しい研究成果を得た。
1.「確率論と計算数学」:物理学、工学の研究者を交え、理論、応用の両面からのアプローチを行った。特に杉田は、力学系の理論を使う擬似乱数発生法の提案、福山はこの乱数にたいして、モンテカルロ法の収束性の議論をおこなった。高嶋は、擬似乱数を確率過程と見る検定法について報告した。
2.「Topics on stochastic control」:制御確率過程が安全な場合にはリスク鋭感的確率制御問題がブレークダウンしないための条件が従来知られたものより、大幅に緩和されることがこの研究によりわかった。
3.「量子カオスの確率論的諸問題2」:数論的量子カオス、確率解析的量子カオスといった、量子カオスの数学的側面についての講演と討論を行なった。
4.「確率解析と幾何学」:金井雅彦は、離散群の作用の剛性問題への確率解析からの接近法を紹介し、Elworthyは、確率微分方程式の視点から、homologyおよびhomotopy消滅定理のを論じ、浦川肇は、グラフのspectre幾何を紹介し、久村裕憲は、Riemann多様体の本質的spectre構造を、また加須栄篤は、Riemannian submersion の幾何解析を論じた。
5.「集団遺伝学における確率過程」:本研究集会の討論において、S.N.Ethierおよび舘田氏自身の参加のもとに太田一舘田のHouse-of-Cardsモデルについて そのモデルと研究課題の数学的定式化を行った。
6.「非平衡統計力学と確率論」:H.Spohn(Ludwig-Maximilians-Univ.),川崎恭治(中部大・工)の連続講義と確率論研究者の講演を交え,非平衡統計力学と確率論との相互関係についての現状把握・今後の展望を行なった。
7.「拡散過程とrandom walkに関するいくつかの話題」:主にBrown運動の局所時間に関する新しい結果や、拡散過程の概観を中心に講演や討論を行った。
8.「確率解析と漸近理論」:確率漸近理論について、既に確立された分野である漸近展開・大偏差原理(ラプラスの方法)のみならず、今後の展開の待たれる確率振動積分に関連する漸近理論についても多くの話題の提供と討論を行った。

Research Products

• [Publications] T.Funaki: "Gibbs-Cox random fields and Burger's turbulence" Ann.Appl.Prob.5. 461-492 (1995)
• [Publications] T.Funaki: "The scaling limit for a stochastic PDE and the separation of phases" Probab.Theory Relat. Fields. 102. 221-288 (1995)
• [Publications] T.Funaki: "Hydrodynamic limit for lattice gas reversible under Bernoulli measures" "Nonlinear Stochastic PDEs: Hydrodynamic Limit and Burgers' Turbulence(eds.N.Funaki and W.A.Woyczynski)",Springer. 77. 1-40 (1995)
• [Publications] T.Funaki: "Equilibrium fluctuations for lattice gas" "Ito's Stochastic Calculus and Probability Theory"(eds.N.Ikeda,S.Watanabe,M.Fukushima and H.Kunita),Springer. 63-72 (1996)
• [Publications] T.Funaki: "Singular limit for reaction-diffusion equation with self-similar Gaussian noise" Proceedings of Taniguchi symposium "New Trends in Stochastic Analysis"(eds.K.D.Elworthy,S.Kusuoka and I.Shigekawa),World. (to appear). (1997)
• [Publications] T.Funaki: "Motion by mean curvature from the Ginzburg-Landau ▽φ interface model" Commun.Math.Phys.(to appear). (1997)
• [Publications] 舟木直久: "相分離の確率モデルと界面の運動方程式" 数学. (掲載予定). (1997)
• [Publications] Y.Hamana: "The fluctuation results for the single point range of random walk in low dimensions" Japanese Jour.Math.21. 287-333 (1995)
• [Publications] Y.Hamana: "The limit theorems for the single point range of strongly transient random walks" Osaka Jour.Math.32. 869-886 (1995)
• [Publications] Y.Hamana: "On the multiple point range of three dimensinalrandom walks" Kobe Jour.Math.12. 95-122 (1995)
• [Publications] Y.Hamana: "The fluctuation result for the multiple point range of two dimensional recurrent random walks" The Annals of Probability. (to appear).
• [Publications] H.C.Chae: "Invariant nuclear space of a second quantization operator" Hirosima Math.Journ.25. 541-559 (1995)
• [Publications] K.Handa: "On a stochastic PDE related to Burgers' equation with noise" "Nonlinear Stochastic PDEs: Hydrodynamic Limit and Burgers' Turbulence (eds.N.Funaki and W.A.Woyczynski)",IMA,Springer. 77. 147-156 (1995)
• [Publications] K.Handa: "Entropy production per site in (nonreversible)spin-flip processes" Journ.Stat.Phys.83. 555-571 (1996)
• [Publications] N.Ikeda: "Levy's stochastic area formula and related problems" Proc.Symposia Pure Math.,ed.by M.C.Cranston and M.Pinsky,A.M.S.57. 281-305 (1995)
• [Publications] N.Ikeda: "Van Vleck-Paili formula for Wiener integrals and Jacobi fields" "Ito's Stochastic Calculus and Probability Theory,ed.by N.Ikeda,S.Watanabe,M.Fukushima,H.Kunita,Springer. 141-156 (1996)
• [Publications] N.Ikeda: "Short time asymptotics for fundamental solutions for heat equations with boundary conditions" New Trends in Stochastic Analysis,Proceedings of Taniguchi International Workshop,ed.by K.D.Elworthy,S.Kusuoka and I.Shigekawa,World Scientific. (appear). (1997)
• [Publications] A.Kasue: "Spectral convergence of Riemannian manifolds II" Tohoku Math.J.48. 71-120 (1996)
• [Publications] A.Kasue: "Riemannian submersions and isoperimetric inequalities" Geometriae Dedicata. (to appear).
• [Publications] T.Kumagai: "Rotation invariance and characterization of a class of self-similar diffusion processes on the Sierpinski gasket" Algorithms,fractals,and dynamics(ed.Y.Takahashi),Plenum. 131-142 (1995)
• [Publications] T.Kumagai: "Homogenization on nested fractals" Probab.Theory Relat.Fields. 104. 375-398 (1996)
• [Publications] T.Kumagai: "Percolation on pre-Sierpinski carpets" New Trends in Stochastic Analysis-Proceeding of a Taniguchi international workshop(eds.K.D.Elworthy,S.Kusuoka and I.Shigekawa),World Scientific. (to appear).
• [Publications] B.M.Hambly: "Heat kernel estimates and homogenization for asymptotically lower dimensional processes on some nested fractals" Potentian Anal.(to appear).
• [Publications] T.Kumagai: "Short time asymptotic behavior and large deviations for Brownian motion on some affine nested fractals" Publ.Res.Inst.Math.Sci.(to appear).
• [Publications] 熊谷隆: "フラクタル上の確率過程とその周辺" 数学. 49. 131-145 (1997)
• [Publications] H.Kunita: "Stable limit distribution over a nilpotent Lie groups" Proc.Japan Acad.Ser.A. 71. 1-5 (1995)
• [Publications] H.Kunita: "Some problems concerning Levy processes on Lie groups" Proc.Symposia Pure Math.,ed.by M.C.Cranston and M.Pinsky,A.M.S.57. 377-408 (1995)
• [Publications] H.Kunita: "Stochastic flows with selfsimilar properties" Stochastic Analysis and Applications,(ed.by Davies,Truman and Elworthy),World Scientific. 286-300 (1996)
• [Publications] Y.Kifer: "Random positive semigroups and their infinitesimal generators." Stochastic Analysis and Applications,(ed.by Davies,Truman and Elworthy),World Scientific. 270-285 (1996)
• [Publications] H.Kunita: "Stochastic differential equations with jumps and stochastic flows of diffeomorphisms" Ito's Stochastic Calculus and Probability Theory,(ed.by Ikeda,Watanabe,Fukushima,Kunita),Springer. 197-212 (1996)
• [Publications] Y.Hashimoto: "Short time asymptotics and an approximation for the heat kernel of a singular diffusion" Ito's Stochastic Calculus and Probability Theory,ed.by N.Ikeda,S.Watanabe,M.Fukushima and H.Kunita,Springer. 129-139 (1996)
• [Publications] H.Nagai: "Bellman equations of risk sensitive control" Proceedings of the 34-th IEEE Conference on Decision and Control. 1048-1051 (1995)
• [Publications] A.Bensoussan: "Some results on risk-sensitive control with full observation" Proceedings of the 34-th IEEE Conference on Decision and Control. 1657-1661 (1995)
• [Publications] H.Nagai: "Bellman equations of risk sensitive control" SIAM Journal control and Optimization. 34. 74-101 (1996)
• [Publications] H.Ishii: "A singular limit on risk sensitive control and semi-classical analysis" Proceedings of the Japan-Russian Symposium on Probability Theory and Mathematical Statistics,World Scientific. 164-173 (1996)
• [Publications] A.Bensousan: "Min-max characterization of a small noise limit on risk sensitive control" SIAM J.Control and Optimization. 35 (to appear). (1997)
• [Publications] A.Bensousan: "Some results on risk-sensitive with full observation" Apll.Math.Optimization.(to appear).
• [Publications] Y.Ogura: "A probabilistic scheme for collapse of metrics" J.Math.Kyoto Univ.36. 73-92 (1996)
• [Publications] S.Li: "A convergence theorem of fuzzy martingales" Proceedings of FUZZ-IEEE '96 Internatinal Conference on Fuzzy Systems. 290-294 (1996)
• [Publications] S.Li: "The space of fuzzy random variables" Methodologies for the Conception,Design,and Application of Intelligent Systems (eds.T.Yamakawa and G.Matsumoto),World Scientific. 596-599 (1996)
• [Publications] S.Li: "Fuzzy random variables,conditional expectations and fuzzy valued martingales" Jour.Fuzzy Mathematics. 4. 905-927 (1996)
• [Publications] S.Li: "Convergence of set valued and fuzzy valued martingales" Fuzzy Sets and Systems. (to appear).
• [Publications] N.Saitoh: "An analysis of a family of rational maps containing integrable and nonintegrable difference analogue of the Logistic equation" Journal Physics A. (to appear).
• [Publications] 清水邦夫: "修正ベータ・2項分布とその雑誌購買度数データへの応用" 応用統計学. 25. 49-60 (1996)
• [Publications] M.Minami: "Estimation for a common intraclass correlation in bivariate normal distributions with missing observations" Amer.Jour.Math.Management Sciences. 16 (to appear). (1996)
• [Publications] K.Shimizu: "The multivariate inverse trinomial distribution as a Lagrangian probability model" Comm.Statistics: Theory Methods. 26 (to appear). (1997)
• [Publications] S.Taniguchi: "Holomorphic functions on balls in an almost complex abstract Wiener space" J.Math.Soc.Japan. 47. 656-670 (1995)
• [Publications] P.Malliavin: "Extension holomorphe des fonctionnelles analytiques definies sur un espace de Wiener reel,formule de Cauchy,phase stationarie" C.R.Acad.Sci.Paris. 322. 261-265 (1996)
• [Publications] S.Taniguchi: "On Ricci curvatures of hypersurfaces in abstract Wiener spaces" J.Funct.Anal.136. 226-244 (1996)
• [Publications] S.Taniguchi: "On almost complex structure on abstract Wiener spaces" Osaka J.Math.33. 189-206 (1996)
• [Publications] S.Taniguchi: "Almost complex structures on path groups" Stochastic Analysis and Applications,Proc.Fifth Gregynog Symp.(ed.by I.M.Davis,A.Truman,and K.D.Elworthy). 419-428 (1996)
• [Publications] M.Fukushima: "On reflecting diffusions on Lipschitz domains with cusps-analytic construction and Skorohod representation" Potential Analysis. 4. 377-408 (1995)
• [Publications] M.Tomisaki: "Construction of reflecting diffusion processes on singular domains" Probability Theory and Mathematical Statistics,Proceedings of the Seventh Japan-Russian Symposium (eds.S.Watanabe,M.Fukushima,Yu.V.Prohorov,A.N.Shiryaev),World Scientific. 496-505 (1996)
• [Publications] M.Fukushima: "On decomposition of additive functionals of reflecting Brownian motions" Ito's Stochastic Calculus and Probability Theorey(eds.N.Ikeda,S.Watanabe,M.Fukushima H.Kunita),Springer. 51-61 (1996)
• [Publications] M.Fukushima: "Construction and decomposition of reflecting diffusions on Lipschitz domains with Holder cusps" Probab.Theory Relat.Fields. 106. 521-557 (1996)
• [Publications] S.Watanabe: "Generalized arc-sine laws for one-dimensional diffusion processes and random walks" Proc.Symp.Pure Math.(eds.M.C.Cranston and M.Pinsky),AMS. 57. 157-172 (1995)
• [Publications] J.Ren: "A convergence theorem for probability densities and conditional expectations of Wiener functionals" Dirichlet Forms and Stochastic Processes,Proc.Intern.Conf.Beijing,Walter de Gruyter. 335-344 (1995)
• [Publications] Y-Z.Hu: "Donsker's delta functions and approximation of heat kernels by time discretization method" J.Math.Kyoto Univ.36. 499-518 (1996)
• [Publications] S.Watanabe: "Levy's stochastic area formula and Brownian motion on compact Lie groups" Ito's Stochastic Calculus and Probability Theory,Springer-Verlag,Tokyo. 401-411 (1996)
• [Publications] T.Funaki et al.(eds.): "Nonlinear Stochastic PDEs: Hydrodynamic Limit and Burgers' Turbulence,IMA volumes in Mathematics and its Applications,77" Univ.Minnesota,Springer, 330 (1995)
• [Publications] 舟木直久: "確率微分方程式(岩波講座「現代数学の基礎」)" 岩波書店, 203 (1997)
• [Publications] N.Ikeda et al.(eds.): "Ito's Stochastic Calculus and Probability Theory" Springer, 422 (1996)
• [Publications] 大矢雅則他編: "数理情報科学事典" 朝倉書店, 1164 (1995)
• [Publications] Y.Takahashi(ed.): "Algorithms,fractals、and dynamics" Plenum, 227 (1995)
• [Publications] 高橋陽一郎: "微分と積分2" 岩波書店, 191 (1995)
• [Publications] 高橋陽一郎: "力学と微分方程式" 岩波書店, 208 (1996)
• [Publications] 高橋陽一郎: "実関数とFourier解析" 岩波書店, 191 (1996)
• [Publications] 青本和彦: "現代数学の流れ2" 岩波書店, 184 (1996)
• [Publications] 高橋陽一郎(監訳): "フーリエ解析大全(上)" 朝倉書店, 320 (1996)
• [Publications] 高橋陽一郎(監訳): "フーリエ解析大全(下)" 朝倉書店, 350 (1996)
• [Publications] S.Watanabe et al.(eds.): "World Scientific" Proceedings of the Seventh Japan-Russian Symposium, 515 (1996)