一般社団法人 日本数学会 Application Server

# アブストラクト事後公開 — 2018年度年会(於:東京大学)

## 統計数学分科会

 2017年度(第16回)日本数学会解析学賞受賞特別講演 緊密性をもつ対称マルコフ過程の性質 竹田雅好 (東北大理) We consider irreducible, strong Feller, symmetric Markov processes with tightness property, and call the class of symmetric Markov processes having such properties Class (T). If a symmetric Markov process in Class (T) is conservative, then it has a very strong recurrence property, and if not, then it explodes very fast. Using this fact, we can derive some spectral properties, for example, $p$-independence of the growth bound of its semi-group, compactness of semi-group, bounded continuity of eigenfunctions. As an application of these properties, we can prove the existence and uniqueness of quasi-stationary distribution. By checking time-changed processes to be in Class (T), we give an analytic characterization of the criticality (or subcriticality) for Schroedinger forms. Moreover, we show that Green function for Schroedinger-type operator satisfies some principles in potential theory (eg. Ugaheri’s maximum principle, the continuity principle) if the principal eigenvalue of the time-changed process is greater than one. PDF 特別講演 数理ファイナンスの第1基本定理 ―確率積分とマルチンゲール測度― 高岡浩一郎 (一橋大商) The first fundamental theorem of asset pricing roughly states that a stochastic process satisfies a property concerning the Itô integrals with respect to it, called the no-arbitrage property, if and only if the process has an equivalent martingale measure. The theorem for discrete-time finite-dimensional processes on finite probability spaces, which goes back to Harrison and Kreps (1979), is a restatement of Stiemke’s lemma of finite-dimensional linear algebra. The theorem in the general setting can thus be viewed as an infinite-dimensional extension of Stiemke’s lemma. In the continuous-time setting, there is more than one way to formulate the “no-arbitrage property.” In this presentation, an emphasis will be put on the $L^0$ space of random variables and the technique of numeraire change. PDF 特別講演 BIBデザインの組加法性と関連する組合せ構造 松原和樹 (中央学院大商) A set of $\ell$ balanced incomplete block (BIB) designs with common parameters having pairwise additivity is called an $\ell$-pairwise additive BIB design. The $\ell$-pairwise additivity of BIB designs can be regarded as a decomposition of a BIB design and/or a composition of several BIB designs, and further yields BIB designs with $\ell$ distinct block sizes. In this talk, we discuss comprehensively the existence of $\ell$-pairwise additive BIB designs through the relationship with other combinatorial structures. Especially, as related combinatorial structures, we focus on nested BIB designs, orthogonal arrays (mutually orthogonal latin squares), perpendicular arrays, difference matrices and finite geometries. Furthermore, we present some results on splitting-balanced block designs named newly. The pairwise additivity also relates to the splitting-balanced property. The splitting-balanced block design (also called a splitting BIB design in literature) was introduced for applications to authentication codes by Ogata et al. in 2004. Finally, the bound of the number of blocks and some existence of splitting-balanced block designs are provided. PDF 特別講演 ヒルベルト空間における弱収束理論とその応用 佃　康司 (東大総合文化) In this presentation, weak convergences of random processes taking values in a separable Hilbert space are discussed. Especially, by using a common approach based on the limit theory in $L^2$ spaces, we study partial sum processes of dependent random variables which appear in the following two topics: (i) statistical change point testing; (ii) functional central limit theorem for logarithmic combinatorial assemblies. (i) Statistical change point testing. When there exist chronologically obtained data, it is in interest to determine whether there are structural changes or not. In the domain of statistics, tests for such hypothesis is considered in the so-called change point testing. For parametric change point problems, several works have proposed test procedures based on score functions. Following these works, we propose a procedure based on the functional of a weighted random process whose weak convergence is discussed in $L^2(0,1)$, which is a modification of test statistics in previous works. (ii) Functional central limit theorem for logarithmic combinatorial assemblies. Assemblies are a class of random combinatorial structures which includes random permutations, random mappings, random forests of labeled trees, and so on. The law of component counts of assemblies is provided from independent Poisson variables combined with the conditioning relation. In previous works, the weak convergence of the partial sum process of component counts of logarithmic assemblies, which are assemblies satisfying the logarithmic condition, has been considered in the Skorokhod space. On the other hand, we consider the partial sum process involving a weight function, which makes the limit different, and derive its weak convergence in $L^2(0,1)$. PDF 1. Fluctuation theory for level-dependent Lévy processes 山崎和俊 (関西大システム理工)・​I. Czarna (Univ. of Wroc{\l}aw)・​J.-L. Perez (Centro de Investigaci\'on en Matem\'aticas)・​T. Rolski (Univ. of Wroc{\l}aw) We consider a level-dependent Lévy process that changes its drift depending on the position of the process. We first generalize the refracted Lévy process of Kyprianou and Loeffen (2010), which changes its drift above a given threshold, to the multi-refracted case. We then extend the results for more general level-dependent Lévy processes. We show how fluctuation identities of these processes can be expressed via scale functions. PDF 2. 負スペクトルLévy過程における Poisson的配当の最適化問題 野場　啓 (京大理)・​J.-L. Pérez (CIMAT)・​山崎和俊 (関西大システム理工)・​矢野孝次 (京大理) We study the optimal periodic dividend problem where dividend payments can only be made at the jump times of an independent Poisson process. PDF 3. Upper rate functions of Brownian motion type for symmetric jump processes 塩沢裕一 (阪大理)・​Jian Wang (Fujian Normal Univ.) We are concerned with upper rate functions for a symmetric jump process on the Euclidean space generated by a regular Dirichlet form. We give a condition on the jumping function for the process to enjoy upper rate functions of Brownian motion type. Our condition implies that the second moment of the jumping function is finite. PDF 4. 正規分布の自由自己分解可能性 長谷部高広 (北大理)・​佐久間紀佳 (愛知教育大教育)・​S. Thorbjørnsen (Aarhus Univ.) We prove that the (classical) normal distributions are freely selfdecomposable. More generally it is established that the Askey–Wimp–Kerov distributions are freely self-decomposable. The main ingredient in the proof is a general characterization of the freely selfdecomposable distributions in terms of the derivative of their free cumulant transform. PDF 5. The maxima for the generalized St. Petersburg game 中田寿夫 (福岡教育大) In this talk, we consider the maxima of payoffs for the generalized St. Petersburg game. The maxima for the original St. Petersburg game cannot be normalized to converge to a nondegenerate limit distribution. However, tuning the parameters appearing in the generalization, we show the normalized maxima converge to the Fréchet distribution. PDF 6. 平坦な臨界点を持つ単峰写像力学系の大偏差原理について 鄭　容武 (広島大工)・​高橋博樹 (慶大理工) We study a smooth unimodal map whose critical point is non-recurrent and flat. Assuming the critical order is either polynomial or logarithmic, we establish the large deviation principle and give a partial description of the zeroes of the corresponding rate function. PDF 7. Relation between mixing properties and chaos in the sense of Devaney 波止元仁 (東京工高専) In this talk, we consider continuous maps of a metric space and introduce relations between mixing properties such as decay of correlations and topologically mixing and Devaney’s chaos. PDF 8. 無限保測変換に対する厳格エルゴード的相対モデル定理 湯浅久利 (大阪教育大教育) Every factor map between given ergodic, measure-preserving transformations on infinite Lebesgue spaces can be realized as a proper, factor map between strictly ergodic, locally compact Cantor systems. A locally compact Cantor system is a topological dynamical system of a homeomorphism on a locally compact (non-compact) metric space, whose one-point compactification is a Cantor set. Such a system, or a homeomorphism, is said to be strictly ergodic if the homeomorphism has a unique, up to scaling, invariant Radon measure and every orbit of it is dense in the metric space. PDF 9. On convergence of the Gibbs measures of perturbed graph iterated function systems with degeneration 田中晴喜 (和歌山県立医大) We study a perturbation of graph iterated function systems (graph IFS). In this study, we consider a family of graph IFSs with small parameter $\epsilon>0$ such that some functions which compose this IFS converge to constant values as $\epsilon\to 0$. Therefore, the limit graph IFS (unperturbed system) may have several Gibbs measures associated with the dimension of the limit set even if the parametrized graph IFS (perturbed system) possess a unique Gibbs measure $\mu(\epsilon,\cdot)$ for each $\epsilon>0$. In this situation, we give a necessary and sufficient condition for convergence of $\mu(\epsilon,\cdot)$ in the case when the limit graph IFS has $2$ or $3$ Gibbs measures. In particular, this condition is composed of Perron eigenvalues of Ruelle operators. PDF 10. On $p$-frame potential of random point configurations on the sphere 平尾将剛 (愛知県立大情報) In this talk we deal with two types of random point configurations, spherical ensemble and the jittered sampling on the sphere. The former is a well-studied determinantal point process on the sphere, and the latter is one of the famous random sampling method. We compare these random point configurations with spherical designs, which are one of the non-random ^^ ^^ good” point configurations on the sphere in the viewpoint of $p$-frame potential. We also discuss other random point configurations on the sphere if possible. PDF 11. 減衰因子のあるホワイトノイズをポテンシャルとする1次元シュレーディンガー作用素について 南　就将 (慶大医) We consider the Schrödinger operator whose potential consists of the white noise multiplied by a decaying factor. This model can be defined as a symmetric operator for any realization of white noise, and it is almost surely self-adjoint under some mild conditions. The nature of the positive part of this operator is similar to the model considered by Kotani and Ushiroya (CMP vol. 115 (1988)). PDF 12. Diffusion processes with random potentials consisting of specially contracted self-similar processes 鈴木由紀 (慶大医) Limiting behaviors of one-dimensional diffusion processes with random potentials are studied. The potentials consist of specially contracted self-similar processes. The minimum processes and the maximum processes of the processes are also investigated. PDF 13. Wiener汎関数のSFCからの広義の再構成式について 星野浄生 (阪府大理) We discuss whether a random function (or a stochastic differential as an extension) is identified and how it is reconstructed from the stochastic Fourier coefficients (SFCs). In this talk, we focus on Skorokhod type SFC and give simple reconstruction formulas using Wiener chaos decomposition from SFCs of Wiener functionals on a space with Haar measure. PDF 14. 逆準凸制約を持つ準凸計画問題について 鈴木　聡 (島根大総合理工) In this talk, we study quasiconvex programming with a reverse quasiconvex constraint. We introduce affine and quasiaffine characterizations of a reverse quasiconvex constraint. By using these characterizations, we show necessary optimality conditions for the problem in terms of Greenberg–Pierskalla subdifferential. Additionally, we investigate surrogate duality for quasiconvex programming with a reverse quasiconvex constraint. PDF 15. Adaptive approach in a multivariate Bayesian control chart 堀口正之 (神奈川大理) In this talk, we consider an adaptive control approach in Markov decision process in order to solve a problem of multivariate Bayesian control chart. We show that there exist an average optimal adaptive and asymptotically discounted optimal policies. PDF 16. 合流型推移をもつ決定過程について —3つの再帰的アプローチ— 藤田敏治 (九工大工) In this study, we consider a decision process model with a converging branch system which is one of the nonserial transition systems. The model is treated by three approaches. Thus we introduce three types of recursive equations by using dynamic programming technique. PDF 17. A超幾何分布からの直接抽出とランダムYoung図形への応用 間野修平 (統計数理研) A distribution whose normalization constant is an A-hypergeometric polynomial is called an A-hypergeometric distribution. Such a distribution is in turn a generalization of the generalized hypergeometric distribution on the contingency tables with fixed marginal sums. For sampling from an A-hypergeometric distribution, the first choice may be use of Markov chain Monte Carlo (MCMC) with moves generated by a Markov basis. In this talk, as an alternative to MCMC methods, a direct sampling algorithm for general A-hypergeometric distribution will be presented. As an application of the exact sampler, sampling from random Young tableaux will be discussed. The Ferguson’s Dirichlet process is an example of such random Young tableaux. A popular direct sampler, such as the Blackwell–MacQueen’s urn scheme, does not work for random Young tableaux without infinite exchangeability. In contrast to the urn schemes, our direct sampler still works without exchangeability. PDF 18. Probabilistic loop path integral for spins 山下秀康 (愛知学院大教養) Let ${\rm Spin}(2\ell+1)$ denote the spin group, represented as a subgroup of ${\rm SU}(2^{\ell})$ (spin representation). Let $G$ be ${\rm SU}(\ell)$ or ${\rm Spin}(2\ell+1)$, and $V$ be $\mathbb{C}^{\ell}$ or $\mathbb{C}^{2^{\ell}}$, respectively. Fix $T>0$ and% $S^{1}$ denote the loop, viewed as the interval $[0,2T]\subset\mathbb{R}$ where the endpoints identified. Let $H$ be a self-adjoint operator on $V$. For any operator $A$ on $V$, let $A(t):=e^{-itH}Ae^{itH},\ t\in\mathbb{R}.$ We show some formulas which give the value of \[ {\rm Tr} A_{n}(t_{n})\cdots A_{0}(t_{0})B_{1}(t_{1}')\cdots B_{m}(t_{m}'),\ 0=t_{0}<\cdots