Grant amount：\4810000 （ Direct Cost: \3700000 、 Indirect Cost：\1110000 ）

We have obtained certain duality theorems on large deviation estimates for controlled semi-martingales. Then, we have extended the results to the cases which correspond to the theorems for the robust large deviation estimates in actual financial models when admitting model unceratainty. Here, key analysis was that of the H-J-B equations of ergodic type and its derivatives for the corresponding dual problems, and by regarding them as the H-J-B equations of certain stochstic differential games, the estimates were obtained. On the other hand, we obtained the existence and uniqueness theorems for the solutions to the H-J-B equations of optimal consumption-investment, and verification theorems.

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Grant amount：\13520000 （ Direct Cost: \10400000 、 Indirect Cost：\3120000 ）

The theory of Dirichlet forms has been developed as a useful tool for studying symmetric Markov processes. The theory of Dirichlet forms is an L-2-theory, and which is a reason why the theory is suitable for treating singular Markov processes. However, the theory of Markov processes is, in a sense, an L-1-theory. To bridge this gap, we prove the L-p-independence of growth bounds of Markov semi-groups under the conditions for the Markov processes to be strong Feller and to be tight. By applying the L-p-independence to time changed Markov processes, we show the exponential integrability of positive continuous additive functionals (PCAF's in short) and the large deviation principle of PCAF's. Moreover, we give a necessary and sufficient condition for heat kernel estimates being stable by perturbation of potential terms.

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Grant amount：\16770000 （ Direct Cost: \12900000 、 Indirect Cost：\3870000 ）

We analysed Markov chains on disordered media and their scaling limits in a unified manner by using probabilistic and analytic methods. From the viewpoint of constructing general theory, we established the following results; i) heat kernel estimates for non-symmetric Markov chains satisfying some cycle condition and analyzing their scaling limits, ii) equivalence of the sub-Gaussian heat kernel estimates and generalized Parabolic Harnack inequalities on symmetric diffusions for general metric measure spaces, iii) convergence of jump-type processes for general metric measure spaces. From the viewpoint of concrete examples, we established the following results; i) asymptotic behavior and scaling limits of biased random walks on critical percolation clusters (conditioned to survive forever) on trees, ii) convergence of scaled mixing times on Markov chains on random finite graphs.

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Grant amount：\16770000 （ Direct Cost: \12900000 、 Indirect Cost：\3870000 ）

We defined a class of stochastic ranking models, which mathematically models the rankings found on the web, e.g., as sales ranks of online bookstores, and proved existence of hydrodynamic limits and propagation of chaos. The result has implications on the analysis of so called long-tail structure of online retails. The results are reported in the book `Solving the mystery of Amazon ranking' (in Japanese), as well as in scientific papers for journals on mathematics.

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Grant amount：\32500000 （ Direct Cost: \25000000 、 Indirect Cost：\7500000 ）

To account of not only the members but also of the researchers on probability theory in Japan, we held several conferences every year, including two international ones, and made connections on study and joint research. Moreover we invited several foreign researchers and the members of this grant attended conferences outside Japan and visited foreign researchers. Through these activities we contributed some progress on theory of stochastic analysis and on applications to study on questions with origins from statistical physics, study of differential equations, spectra of manifolds, mathematical finance and so on.

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Grant amount：\4290000 （ Direct Cost: \3300000 、 Indirect Cost：\990000 ）

We gave maxima and maximizer of joint distribution function of maximally dependent random variables and its application. We defined a generalization of the Knothe-Rozenblatt Rearrangement (KRR)and its stochastic version Knothe-Rozenblatt process (KRP), proved the existence and the uniqueness and gave a characterization as the limit of the minimizer of a class of variational problem with a small parameter. We also gave a representation theorem for a random probability density function of a stochastic flow of KRP. We proved that the mean of a logarithm of the random probability density function with space variable replaced by KRP is convex.

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Grant amount：\10780000 （ Direct Cost: \8800000 、 Indirect Cost：\1980000 ）

The theory of Dirichlet forms is an L^2-theory, while the theory of Markov processes is, in a sense, an L^1-theory. To bridge this gap, we study the L^p-independence of growth bounds of Markov semigroups, more generally, of generalized Feynman-Kac (Schroedinger) semigroups. A key idea for the proof of the L^p-independence is to employ arguments in the Donsker-Varadhan large deviation theory. The L^p-independence enables us to control L^∞-properties of the symmetric Markov process ; in fact, we can state, in terms of the bottom of L^2-spectrum, a necessary and sufficient conditions for the integrability of Feynman-Kac functionals and for the stability of Gaussian both side estimates of Schroedinger heat kernels.

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Grant amount：\13570000 （ Direct Cost: \11500000 、 Indirect Cost：\2070000 ）

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Grant amount：\14860000 （ Direct Cost: \13000000 、 Indirect Cost：\1860000 ）

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Grant amount：\47710000 （ Direct Cost: \36700000 、 Indirect Cost：\11010000 ）

Our purpose is to contribute to Japanese society of probability theory by the total sum of research results of our investigators, who are experts of each field, in order to develop and make use of the traversality that probability theory originally has.
We sponsored or co-sponsored 4 international conferences (Probability and Number theory, Large Scale Interaction Systems, 2 conferences on Mathematical Finance), 26 domestic symposia, in which we financially supported 285 participants including 13 from overseas.
We got many results. Among them, we here present 3 results obtained by the head investigator. (1) By formulating the Monte Carlo method as stochastic game (gambling), and applying Kolmogorov's random number theory, we revealed the essential problem of sampling in the Monte Carlo method (2) The probability of two monic polynomials over any finite fields to be coprime is computed by an extended ergodic theorem in the adelic completion of the ring of polynomials. (3) We considered a randomized digamma function, and investigated the central limit theorem for them, whose results can be regarded as the fluctuated potential caused by randomly located electric charges on the half line.

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Grant amount：\2600000 （ Direct Cost: \2600000 ）

結晶格子上のランダムウォークの大偏差に関して,幾何学的考察を行った.結晶格子とは,自由アーベル群が自由に作用し,商空間が有限グラフとなる無限グラフである.この周期性から,結晶格子を無限遠から観察すると一様な図形に見える.より正確に述べる.結晶格子をグラフ距離によって距離空間と考え,距離をスケール変換した距離空間の1パラメーター族を得る.このスケールをゼロに近付けたときのグロモフ・ハウスドルフ位相による極限距離空間を,結晶格子の無限遠での接錐という.結晶格子のようなアーベル周期性をもつ距離空間の無限遠での接錐の存在は,グロモフによって知られているが,この極限距離空間を具体的に,また,グラフの幾何の言葉で特徴つけた.距離球はコンパクト凸多面体となり,その端点が,具体的に商空間の閉路の構造で決まることを調べた.更に,この極限空間の単位距離球が,ランダム・ウォークの大偏差源氏に現れるレート関数の本質的定義域と一致することを示し,Math. Z.に発表した.これを非アーベル被覆の場合に拡張することを目標に,ランダム・スネーク,R樹木,超準極限に関するグロモフ,シャピロ,ドルータの結果など,関連研究について情報収集し,検討を行った.その結果,超準解析による定式化を書き下すことができたが,論文として公表するには至らなかった.ランダム・ウォークの長時間挙動,及び磁場付き推移作用素のスペクトルと結晶格子の幾何に関する総説をまとめAmer .Math. Soc. Sugaku Expositoryに公表した.

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Grant amount：\3600000 （ Direct Cost: \3600000 ）

We proved that the integrability of Feynman-Kac functionals (gaugeability) is equivalent to that the principal eigenvalue of time-changed process is greater than 1, which is also equivalent to the subcriticality of Schroedinger operators. This fact says that the principal eigenvalue of time-changed process accurately measures the size of measures. Using this fact, we obtained three results : The first result is that the ultracontractiyity of Schroedinger semigroups holds if and only if the princilal eigenvalue of time-changed process is greater than 1. The second result is that the expectation of the number of branches hitting a closed set in a branching symmetric stable process is finite if and only if the princilal eigenvalue is greater than 1. The final result is as foolows : Suppose that the heat kernel on a complete Riemannian manifold satisfies the global Gaussian bounds, so called Li-Yau estimate. Then the heat kernel of the Schroedinger operator also possesses the global Gaussian bounds, if and anly if the princilal eigenvalue is greater than 1.

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Grant amount：\1700000 （ Direct Cost: \1700000 ）

Second order arithmetic was first introduced by D.Hilbert around 1920's to set a firm foundation of analysis. Among many different formalizations of second order arithmetic, RCA_0 and WKL_0 are particularly important with respect to Hilbert's program. Tanaka and Yamazaki, in cooperation with Simpson, proved that one can eliminate, the compactness argument from proofs for a certain sort of theorems in WKL_0, namely WKL_0 is conservative over RCA_0 for the formulas in a certain form. They also showed that the bounded dependent choice scheme, which is often used in the computation of real numbers, can not be eliminated likewise. Thus, Tanaka and Yamazaki constructed an appropriate truth definition for the real number system via the quantifier elimination method, by which one can manipulate the reals freely in RCA_0. By sophisticating this argument, Tanaka, jointly with Sakamoto, could prove Hilbert's Nullstellensatz within RCA_0. Yamazaki and Sakamoto also studied uniform versions of WKL_0 and other statements within higher order arithmetic. Yamazaki investigated the logical strength of completeness theorems for intuitionistic logic.

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Grant amount：\11900000 （ Direct Cost: \11900000 ）

1. We considered risk-sensitive portfolio optimization problems on infinite time horizon for linear Gaussian models and general factormodels. Proving existence of solutions of ergodic type Bellman equations we got the results constructing explicitly the optimal strategies from the solutions. As for linear Gaussian models we got the same results in the case of partial information as well by only using the informations of security prices.
2. In the case of partial information, using the information of only security prices, we obtained maximum principle as necessary conditions for optimality for the problems on a finite time horizon
3. In the above case we showed that optimal strategies could be expressed explicitly by using the solution of Bellman equation with degenerate coefficients for conditionally Gaussian models
4. We showed semi-classical behavior of the minimum eigenvalues of Schrodinger operators on Wiener space can be captured in a similar way to the case of finite dimensions. By using similar idea we proved rough lower estimates holds for the minimum eigenvalues of the operators on path spaces (not pinned) on Riemannian manifolds. We also proved, by considering semi-classical limits on the pinned pathe space on Lie groups, that it implies that harmonic forms vanishes
5. We studied estimates of log derivatives of the heat kernels on Riemannian manifolds in which curvatures rapidly decrease enough and proved log Sobolev inequalities on path spaces. We also studied relationships between Brownian rough path and weak type poincare inequalities.
6. We studied optimization problems concerning exponential hedging in mathematical finance. In particular we calculated asymptotic expansion of the backward stochastic differential equations with respect to small parameter and obtained asymptotics of the optimal controls
7. We constructed optimal portfolio by getting higher order differentiability of the solutions of nonlinear partial differential equations arising from mathematical finance
8. We got interested in solving optimization problem by the methods of convex duality in mathematical finance and extended known. results in applying the methods to the case of partial information, or super hedging under constraints with respect to delta
9. We got the results on exsistence and uniqueness of viscosity solutions by deriving Euler equations as singular limits of minimum elements of minimization problems of functionals topologically equivalent. We got the Holder estimates of Lp viscosity solutions of fully nonlinear elliptic partial differential equation with super-linear growth with respect to first order derivatives.
10. We discussed hydrodynamic limits of critical surface models on walls and derived variational inequalities of evolution type. We also derived Alt-Caffarelli variational problems by proving large deviation principles for equilibrium systems of the critical surfaces with pinning.

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Grant amount：\1700000 （ Direct Cost: \1700000 ）

確率的外力項(stochastic forcing term)を持つKorteweg-de Vries方程式は,雑音のかかったプラズマ中を通過するイオン音波ソリトンを記述するなど,様々な分野で数理物理的モデル方程式として用いられている重要な方程式である.昨年度は周期境界条件の下で,Besov型のBourgain空間を用いて,時間変数に対してはホワイトノイズであり,空間変数に関してはホワイトノイズにかなり近い確率項を持つ場合の初期値問題を考え,解の一意存在を証明した.しかし,扱える確率摂動項とホワイトノイズの間には,必ずしも小さいとは言えないギャップがあった.
今年度は,Besov型のBourgain空間の定義において,2進分解の取り方を変更することにより,新しい不等式を証明することに成功した.この不等式は,通常よく使われている2進分解から定義されるBesov型Bourgain空間では成立しないと思われ,調和解析的にも興味深いものである.今回この不等式を用い,時間変数についてはホワイトであり,空間変数についてはホワイトにいくらでも近い確率摂動項に対し,初期値問題の強い解の一意存在定理を証明した.
時間変数と空間変数の両方についてホワイトである確率項を扱うことは,依然未解決問題であり今後の課題であろう.しかし,コンピュータによる数値シミュレーションは普通強い解を計算することから,ホワイトノイズにいくらでも近いノイズに対し,強い解を扱うことが可能な数学的定理を証明できたことは,数値シミュレーションを理論的に補強する際に応用できるのではないかと期待される.

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Grant amount：\3200000 （ Direct Cost: \3200000 ）

The main purpose of this research is to construct and give the properties of the diffusion processes on the domains which move depending on time. We assume that not only the domain but also the generator allow depending on time. The typical processes are absorbing or reflecting diffusion processes. In our case, difficulty arises because the boundary changes depending on time. In this research, after reorganizing the basic theory of time dependent Dirichlet forms, we constructed the processes on moving domains and get some properties of them. The fundamental settings are as follows. The domains are related by a time dependent maps of a fixed domain. The generator may depend upon the time. We first construct the diffusion processes on a fixed domain. For the construction, we use the general theory of time dependent Dirichlet forms. After that, we map the process to get the desired processes. In this argument, since the surface elements depend on time, to show that the mapped processes satisfy the desired property, we need the stochastic calculus including the Girsanov type transformations. Such stochastic calculus is given separately but not systematically in the time dependent case. Since we use such calculus in many places, the fundamental calculus will be published. As an important case, we consider the reflecting diffusion processes. Since the boundary changes depending on time, it is not easy to specify the local time and the normal derivative of the boundary. In this research, by using the map among the domains, we give the notions and define the local time on the boundary. Using the local time, we give the Skorohod representation of the diffusion processes. It is possible to apply our processes. One of the applications is an optimal stopping problem of time inhomogeneous diffusion processes. In the time inhomogeneous case, since there exist semipolar sets, it is not easy to characterize the optimal stopping time. For this problem, we can give the characterization of the time. We are considering that further related applications are possible.

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Grant amount：\2900000 （ Direct Cost: \2900000 ）

The uniformization theorem foe Riemann surfaces claims that complex structures on non-compact simply connected surfaces are equivalent to either the standard complex structure on the complex plain or the one on the unit disk. In terms of Brownian motions this classification is completely described by the recurrence of the process on the domain. On the complex plain the L^∞-norm of Beltrami-coefficients must be 1 as for those complex structures not equivalent to the standard one, in other words, if the corresponding Brownian motion is transient then the L^∞-norm of Beltrami-coefficient is 1. However this does not mean that every complex structure on the complex plain whose Beltrami-coefficient has unit L^∞-norm corresponds to transient Brownian motions. The aim of the present research project is to study such complex structures. One can construct a one-parameter family of complex structures with critical point and at the critical point the recurrence and the transience switch to the other. The model of the complex structures discussed in the present research is parameterized by the closed unit disk and the modulus of the parameter describes the L^∞-norm of the corresponding Beltrami-coefficient. When the parameter lies in the open unit disk the prescribed complex structure is equivalent to the standard one. On the boundary of the unit disk except for one point the complex structure is equivalent to the standard one on the unit disk and thus the Brownian motion is transient. At the critical point the complex structure is recurrent but the freedom of deformation that preserves recurrence is relatively low. A natural direction for further direction is as follows : Study the action of the modular group on the space of Beltrami-coefficients and the action of the modular group on the boundary.

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Grant amount：\2700000 （ Direct Cost: \2700000 ）

The integrability of Feynman-Kac functional is called gaugeability or conditional gaugeability. The conditional gaugeability is closely related to the subcriticality of Schrodinger operators, that is, the existence of the Green function. In this study, we obtained a necessary and sufficient condition for conditional gaugeability; moreover we showed the equivalence between conditional gaugeability and subcriticality of generalized Schrodinger operators. We applied this condition to show the subcriticality of concrete Schrodinger operators; for example, we considered the Schrodinger operator whose potential is the surface measure of sphare. We obtained a necessary and sufficient condition the Schrodinger operator being subcritical in terms of the radius of the sphere. To prove the necessary and sufficient condition for conditional gaugeability, we extended the full large deviation principle of Donsker-Varadhan type to symmetric Markov processes with finite lifetime and applied it to a time changed process of the Brownian motion.

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Grant amount：\29200000 （ Direct Cost: \26200000 、 Indirect Cost：\3000000 ）

This research was accomplished by 25 members under strong helps from many researchers in probability theory and related fields. During three years of the research period, 27 meetings were organized and 21 researchers were invited from abroad. A lot of research products were obtained in broad area:
1. Related to the basic theory in probability theory, extension of the inverse arcsine law to one dimensional diffusion processes, properties of Brownian motion/heat kernel/Green function on several spaces, Markov processes and Dirichlet form, infinite dimensional stochastic analysis, stationary phase method and asymptotic theory and others were discussed.
2. As applications of probability theory, mathematical physics such as analysis of phase separating surface arising under phase transitions, derivation of free boundary problem, motion of interacting Brownian hard balls, scaling limit of percolation cluster, random matrix, stochastic processes on fractals, problem of risk sensitive stochastic control, ergodid theory especially law of large numbers for ψ-mixing random variables, numerical calculation for stochastic differential equations and nonlinear diffusion equations, asymptotic expansion and applications of Malliavin calculus in mathematical statistics, problems related to differential geometry like collapse of manifolds.
Concrete explanations on each research result can be found in the booklet of the research report in details. As is stated above, under the project of this research, many foreign researchers were invited and it was very fruitful and extremely important for the development of the probability theory in Japan in future.

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Grant amount：\5200000 （ Direct Cost: \5200000 ）

(a) It is important to know the conditions for no breakdown in risk-sensitive stochastic control problems since the value function has not always a finite value. In this research we have obtained the condition as that of the size of risk-sensitive parameter in the case of a finite time horizon and also shown the solvability of the corresponding Bellman equation under the condition. It is applicable for the case that the risk-sensitive parameter is large and has great meaning in application. In the case of infinite time horizon we have shown existence and uniqueness of the corresponding ergodic type Bellman equation under a similar condition by noticing the relationships between the problems and the eigenvalue problems for Schrodinger operator. Besides, we have derived a first order partial differential equation relating to game theoretical approach to nonlinear H_∞ control as its singular limit.
(b) We have considered portfolio optimization problem for a factor model as application to mathematical finance of risk-sensitive stochastic control and got the results giving the explicit representation to optimal portfolio for the problem in the case of partial information. In the case of infinite time horizon we obtained the condition under which the solution of corresponding ergodic type Bellman equation defines the optimal portfolio and constructed it by the solution. We have also found that the solution does not always give the optimal portfolio without any condition.
(c) We have shown existence of the spectral gap of Schrodinger opeartor by using log Sobolev inequality and gave the estimate.
(d) We have shown that the large deviation principle holds for additive functional of Brownian motion corresponding to measures in Kato class. As its application we obtained necessary and sufficient condition under which additive functionals converges exponentially fast.
(e) We have shown Trotter product formula with respect to L_p and trace norm and their error estimates for the Schrodinger operator with a potential bounded below.
(f) We have shown that the sequence of symmetric statistics defined by Weyl transformation on infinite dimensional torus converges to the limit represented by multiple Wiener integral under the probability measures.

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Grant amount：\2700000 （ Direct Cost: \2700000 ）

The objective of this study is to investigate symmetric Markov processes by using Dirichlet form theory. Symmetric Markov processes are a special class in Donsker-Varadhan type large deviation theory in the sense that the rate functions of large deviation principle are given by the associated Dirichlet forms. In 1984, Fukushima and I showed that symmetric Markov processes can be transformed to ergodic processes by some supermartingale multiplicative functionals even if a symmetric Markov process is explosive or has the killing inside. As a result, Donsker-Varadhan type large deviation principle could be extended to symmetric Markov processes with finite lifetime. In this study, I found a new sufficient condition for the upper estimate holding for not only compact sets but also for closed sets. In fact, I showed that the full large deviation principle holds if the Markov process explodes so fast that the 1-resolvent of the identity function belongs to the space of continuous functions vanishing at infinity. As a corollary of this result, I showed LィイD1pィエD1-independence of the spectral radius of symmetric Markov semigroups. And I applied it to obtain a necessary and sufficient condition for the integrability of Feynman-Kac functionals. This result also gives us an criterion whether a Schrodinger operators is subcritical or not.
We further extended the large deviation principle to Markov processes with Feynman-Kac functional, and consider asymptotic properties of Feynman-Kac semigroups.

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Grant amount：\3000000 （ Direct Cost: \3000000 ）

We had studied the following two subjects for the period of July, 1997-March, 2000.
We first studied the well-posedness of the Cauchy problem for the system of nonlinear wave equations with different propagation speeds. One of the most important problems in the field of partial differential equations is to look for the largest possible function space in which the wave equations with quadratic nonlinearity is well-posed. This problem is closely related to the Lorentz invariant for the wave equation. When we consider the system of nonlinear wave equations with different propagation speeds, the discrepancy of propagation speeds breaks the Lorentz symmetry. We classified the quadratic nonlinear terms from a point of view of the time local well-posedness.
Second, we studied the unique solvability of the Cauchy problem for the Korteweg-de Vries equation with stochastic forcing term. The stochastic forcing term is regarded as a nonsmooth perturbation from a mathematical point of view. Especially, in general, the inverse scattering method is inapplicable to the Korteweg-de Vries equation with forcing term. We investigated the time local existence of solution for a natural class of stochastic forces.

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Grant amount：\2400000 （ Direct Cost: \2400000 ）

本研究代表者による従来の研究で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リスク鋭感的パラメーターの大きさに関する条件はかなり強い制約があり、応用上からみても条件を緩和すべき問題が残っていた。今回の本研究により強い勾配ベクトル場をずれの係数に持つ確率微分方程式で定義される制御確率過程のように、よい安定性を制御確率過程が持っている場合には、その問題がブレークダウンしないためのリスク鋭感的パラメータの大きさに関する条件が従来知られたものより大幅に緩和されることが示された。また、その条件の下で、対応するベルマン方程式の解の存在が変分法を用いて示された。一方、一般にエルゴード型ベルマン方程式は解を多数持つことが知られているが、一意性をもっている減価型方程式の解の極限として得られるものが応用上意味のあるものである。この観点から行った次のような研究を行なった。ある特別な場合に限定したとき、リスク鋭感的確率制御問題のエルゴード型ベルマン方程式の解で、減価型ベルマン方程式の解の極限として得られるものがシュレ-デインガー作用素の第一固有関数と第一固有値を使って表現されることを示した。さらにエルゴード型ベルマン方程式の特異極限を考察し、上で得られた解が極限方程式の粘性解に収束すること、並びに極限方程式の粘性解の一意性が、解の増大度(減少度)を規定する条件の下で得られることが示された。またエルゴード型ベルマン方程式の導出と大偏差原理との関連に関する新しい知見を得た。

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Grant amount：\2400000 （ Direct Cost: \2400000 ）

マルコフ過程の汎関数の分解定理の一般化や精密化とそれらの応用を図るという目的に沿った申請書記述の研究計画・方法に従って研究を進めた結果以下の研究実績を得ることが出来た。
代表者福島は対称なマルコフ過程の加法的汎関数がマルチンゲ-ル部分とエネルギー零の部分の和に分解できるという定理(福島分解定理)が、推移確率が基礎の測度に絶対連続な場合に、容量零の除外集合を許さない形に精密化出来るための必要十分条件を与え、更にその際にエネルギー零の部分が有界変動になるための使いやすい判定条件を与えた。福島はこの一般論を応用し、滑らかでない境界を持つ領域上の反射壁拡散過程のスカラホ-ド分解が除外集合を許さない形で導いた。そしてカスプ境界点におけるヘルダー指数が1/2より大であればエネルギー零の部分が境界の局所時間により積分として表され有界変動となることを示し、またそれが1/2より大でなければ有界変動とならない例があることを最近のDeblassie-Tobyの研究と関連づけることによって明らかにした。これによってまた、反射壁拡散過程へのヂリクレ形式による接近とスカラホ-ド型の確率微分方程式による接近とは初めて同定することが出来た。
福島分解の一般論は従来対称マルコフ過程に対応するヂリクレ形式の正則性の仮定の下で定式化されていたが、その仮定を外してもマルコフ過程が右連続ならば成立することが最近のMa-Rockner等の研究で明らかになっている。分担者の竹田は優マルチンゲ-ル乗法的汎関数によって変換された必ずしも正則ヂリクレ形式が対応しないマルコフ過程の再帰性をこの一般化された分解定理を適用して示すことに成功した。

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Grant amount：\2000000 （ Direct Cost: \2000000 ）

対称マルコフ過程の加法的汎関数の分解定理の精密化とその応用を図るという研究目的に沿った申請書の研究計画・方法に従って研究を進めた結果以下の研究業績を得ることが出来た。
代表者福島は分担者竹田および熊本大学の大島教授と共に、共著の著書(1994年発刊)において上記分解定理のみならず、その局所的な場合への拡張及び推移確率が絶対連続な場合における精密化についての基礎理論を展開している。特に精密化に関してはいくつかの十分条件が与えられているが、福島はこの一般論を徹底して進め、加法的汎関数u(X_t)-u(X_o)のマルチンゲ-ル部分M_tとエネルギー零の部分N_tとの和としてのstrictな分解が成立するための関数uに対する必要十分条件やN_tが有界変動になるためのuに対する必要十分条件を求めることに成功し、更にN_tのstrictな意味での台とuのスペクトルの関係を明らかにした。福島はこの一般論を山口大学の富崎教授と共同で応用して境界がヘルダー連続性しか持たないR^d上の領域に対し、ヘルダー指数がd/(d-1)より大きい時にはその領域上の反射壁ブラウン運動はスカラホ-ド分解を許容することを証明した。実はdの如何にかかわらずヘルダー指数が1/2より大きいときにこれは正しいという予想を持ち目下その証明に取り組んでいる。
分担者竹田は上記共著の著書で展開された対称マルコフ過程の乗法的汎関数による変換とデイリクレ形式の変換の関係に関する基礎理論の発展に取り組んでいる。特に上述の加法的汎関数の分解定理の精密化を用いることによって優マルチンゲ-ル的な乗法的汎関数による変換論の精密化に成功している。
また尾角は数理物理的な可解格子模型にたいして、稲垣、谷口、安芸は統計的モデルに対してそれぞれ汎関数の分解定理の有効性を示す研究を着実に進めた。

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Grant amount：\18500000 （ Direct Cost: \18500000 ）

As two years project from 1993 we carried out co-operative research on "Probability Theory and Related Topics" in co-operation with the joint researchers of this project and other researchers of the probability section of the Mathematical Society of Japan. First, aiming at futher development of theory of stochastic analysis which is most fundamental in modern probability theory, we organized two symposiums "Analysis on Wiener space" and "Stochastic Analysis". The former is originated by K.Ito and in the latter one a main theme is Dirichlet space theory due to M.Fukushima. In this field Japan is the leading country, and many pioneering works were produced motivated by these symposiums. We had also symposiums "Analysis related to Markov processes" and "Stochastic Analysis on Manifolds and Large Deviation Theory", which are related to analysis and geometry, jointly with invited analysts and geometricians. Based upon the discussions there new researchs were created. Furthermore, aiming at application of stochastic analysis to physics and biology we organized three symposiums "Stochastic Analysis in Physics and Biology", "Fractals and Related Fields" and "Infinite Particles Systems and Hydrodynamical LImit" in cooperation with phycists and biologists, and we could find out several fascinating mathematical problems. It has been proved that stochastic analysis is very powerful in the engineering involving control theory, mathematical economics such as finance theory and mathematical approach in sociology, for which we organized two symposiums "Stochastis Models in Applied Fields" and "Application of Stochastic Analysis". Moreover to discuss new results on miscellaneous problems of probability theory we organized "Gaussian and Stable Processes and their Applications", "Stochastic Processes and Related Fields", "Ergodic Theory and Related Fields", and "Summer School on Probability Theory". Stimulated by these symposiums many interesting results were produced, which are published as the report of research results by this grant.

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Grant amount：\1900000 （ Direct Cost: \1900000 ）

確率諸過程の関数解析的研究という目的に沿った申請書記述の研究計画・方法に従って研究を進めた結果以下の研究実績を得ることが出来た。
代表者福島は正則Dirichlet形式にHunt過程を容量0の集合を除いて一意的に構成するという20年前の方法を拡張して、(r,p)-容量0というもっと精細な集合を除いての構成をある擬微分作用素の族に対して実行した。福島と分担者竹田は熊本大学の大島洋一氏との平成5年出版予定の共著の著書に於いてDirichlet形式とマルコフ過程の関係についての基礎理論を更に充実させ、特にextended Dirichlet spaceの理論、マルコフ過程のtime changeとの対応理論、対称作用素のマルコフ拡大の理論をほぼ完成させて載せることができた。
福島と分担者島はSierpiski gasketという代表的なフラクタル集合上で最近導入されたラプラス作用素に対してその固有値、スペクトルを精細に調べいくつかの通常と異なるwildな性質を発見した。更により一般なフラクタル集合であるnested fractal上のラプラス作用素に対してもそのスペクトル分布関数の原点の近くでの増大度にフラクタルのスペクトル次元と呼ばれる量が関係していることを福島が見いだし、島はランダムなポテンシャルを持つ対応するシュレージンガー作用素についてもそのスペクトル分布関数のLifchitz tailとしてやはりフラクタル次元が現れることを発見した。これらの研究に於いてDirichlet形式論が極めて有効な働きをすることが確認され今やそれはフラクタル集合上の解析学や拡散過程の研究に欠かせない枠組みとなっている。
また伊達は数理物理学的な可解格子模型に対して、谷口、安芸は統計学的モデルに対して、それぞれ2次形式やエネルギー概念の有効性を示す研究を着実に押し進めた。

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Grant amount：\900000 （ Direct Cost: \900000 ）

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Grant amount：\700000 （ Direct Cost: \700000 ）

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Grant amount：\800000 （ Direct Cost: \800000 ）

放物型方程式の初期値問題の基本解が助変数の付いた擬微分作用素として構成される事はすでに明らかになっている。二階放物型混合問題の基本解の構成について同様の事を目ざした。この場合には、擬微分作用素として構成するのは無理であって、ある種の積分作用素を使って構成する事をめざした。その積分作用素は擬微分作用素の表象のcalculusと類似の性質を持つものであり、その結果形式的な表象の計算によって、基本解の漸近形が求められる。
本年度計画されていたディレクレ問題、ノイマン問題、第3種、斜交微分及び特異境界値問題について、上記の基本階の構成ができたので、論文としてまとめる予定である。基本解の漸近形を固有値の漸近挙動に応用できて、第3種、斜交微分及び特異境界値問題については、固有値分布について新しい結果が得られた。以後、漸近挙動の計算方法が明らかになったので、幾何学的意味付けを考察する事が残っている。
万一ノイマン問題、より一般の退化した放物型混合問題の基本解の構成については、以前研究していた退化した放物型初期値問題の基本解の構成が応用できると考えられる。これについては、より精密な積分作用素のクラスを考える必要がある事が明らかになった。
退化して放物型混合問題の基本解の構成とその漸近挙動への応用が残った問題である。さらに、関数解析的手法による発展作用素の構成との関係、さらに確率論的手法による基本解の構成との関係を調べる。万一ノイマン問題については関数論への応用についても探りたい。

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Grant amount：\800000 （ Direct Cost: \800000 ）

制御系の状態が簡単な確率過程にしたがって変化する適応型のゲ-ムとして、互に対立する相手プレ-ヤの行動に関する情報様式がある確率法則にしたがって、観測不可能な状態から可能な状態へ移行したり、逆に可能な状態から不可能な状態へ移行する。2人非0和タイミング・ゲ-ムを研究することから出発した。このモデルは、経営における新製品開発の最適時節の決定や飛行機における座席の各クラスへの最適配分、生物学における2種生物のナワバリ占有競争への最適努力持続時間の決定等、から抽出されたものである。このゲ-ムに対して、積分方程式論で開発された方法や定常確率過程で得られた結果を利用することにより、かなり広範囲なクラスの精度関数を持つ場合のナッシュの平衡戦略を厳密解の形で導くことに成功し、いくつかの数値例も与えた。
次いで、生物進化過程の理論的説明づけから抽出されたモデルとして、1つのナワバリを複占する2種の生物間の競争を適応型のゲ-ムとして提案し、時間の推移を考えない静的有限ゲ-ムと時間の推移を考えに入れた動的無限ゲ-ムを合わせた形で定式化し、条件やパラメ-タをより一般的な形に残したままで解析した。非0和双行列ゲ-ムの理論からの結果および微分方程式論からの結果を応用することでナッシュ平衡戦略を導くことに成功し、この結果生物進化学における進化的に安定な戦略の数学的意味を考察することにも成功した。
その他、確率系の研究として、対称拡散過程へのマルチンゲ-ルの方法に関する研究、ドンスカ-・バラダハンのエントロピ-の応用に関する研究、微分方程式に関連して、係数が無限回微分可能なセミグル-プの生成作用素となっている放物型発展方程式の研究も行ない、各専門誌や研究集会で発表することもできた。

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Grant amount：\700000 （ Direct Cost: \700000 ）

Banach空間Xに作用する一つの閉線形作用素Aについて、Aのスペクトル集合σ(A)の近傍で有界かつ正則なすべての関数fに対して、自然な方法により、fのAでの値f(A)がXの有界作用素として定義されるとき、この対応f→f(A)をAのH^∞関数計算という。Banach空間XがL^p(1<p<∞)空間で、Aのスペクトル集合σ(A)がある角領域Sに含まれている場合について次のような成果が得られた。AについてのH^∞関数計算が存在することと、Aに対する2乗積分の形をしたgー関数g(ψ)=S_Γ1A^<1/2>(λ-A)^<-1>|^2ψ|^2dλ、ΓはSの外を走る積分路、ψ〓L^p、に対して、評価式||g(ψ)||L^p【less than or equal】C||ψ||L^pが成立することとは同値であることが示された。これは、Hilbert空間の場合の同様な結果の、L^p空間への拡張である。次に、AがH^∞関数計算を持てば、Aの純虚数巾A^<2^^〜y>(-∞<y<∞)はXの有界作用素であることが分かる。Hilbert空間では、この逆、すなわちA^<2^^〜y>が有界作用素になれば、AはH^∞関数計算をもつことが示された。しかし、このことは、L^p空間ではもはや成立しないことが明らかになった。また、AがH^∞関数計算をもてば、Aの分数巾A^θ(0<θ<1)と、Aの定義域D(A)とXの補間空間に関して、D(A^θ)=[X,D(A)]_<1-θ>がすべての0<θ<1について成立することが示された。Hilbert空間ではこの逆も成立したが、L^p空間の場合では、成立するかどうか分からなかった。偏微分方程式への応用に関しては、係数が十分滑らかな楕円型微分作用素から定まる線形作用素は、L^p空間で、H^∞何数計算をもつことが分かった。イタリアの数学者達により、H^∞関数計算をもつような解析的半群の生成作用素Aについて、発展方程式du/dt+Au=ψ(t)はL^p(O,T;X)でMaximal Regularityをもつことが報告された。

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