確率論と幾何学(Geometry and Probability)
日程 2016 年 10 月 24 日 (月)〜28 日 (金)(24–28 October 2016)
会場 数理解析研究所 420 室(Room 420, Research Institute for Mathematical Science)

Program
10 月 24 日 (月) Oct. 24 (Mon.)
13:00–13:50

Nicola Gigli (La Scuola Internazionale Superiore di Studi Avanzati)
Nonsmooth diﬀerential geometry I

14:00–14:50

田中 勝 Masaru Tanaka (福岡大学 Fukuoka University)τ -Information geometry endowed with a contact structure

15:10–16:00

Young-Heon Kim (University of British Colombia)
Optimal martingale transport in general dimensions I

16:10–17:00

Short communications I10 月 25 日 (火) Oct. 25 (Tue.)
9:50–10:40 Nicola Gigli (La Scuola Internazionale Superiore di Studi Avanzati)
Nonsmooth diﬀerential geometry II
10:50–11:40

白井 朋之 Tomoyuki Shirai (九州大学 Kyushu University)
Persistent homology of random complexes built over point processes

13:00–13:50

Young-Heon Kim (University of British Colombia)
Optimal martingale transport in general dimensions II

14:00–14:50

稲濱 譲 Yuzuru Inahama (九州大学 Kyushu University)
Short time full asymptotic expansion of hypoelliptic heat kernel at thecut locus

15:10–16:00

Paul Woon Yin Lee (Chinese University of Hong Kong)
The Harnack inequality for the Jordan-Kinderlehrer-Otto/minimizing
movement scheme of the heat equation

16:10–17:00

Short communications II10 月 26 日 (水) Oct. 26 (Wed.)
9:50–10:40 Nicol Gigli (La Scuola Internazionale Superiore di Studi Avanzati)
Nonsmooth diﬀerential geometry III
10:50–11:40

Laurent Saloﬀ-Coste (Cornell University)
The isoperimetric profile for random walks on finitely generated groups I

10 月 27 日 (木) Oct. 27 (Thu.)
9:50–10:40 白石 大典 Daisuke Shiraishi (京都大学 Kyoto University)
Geometry of Brownian motion
10:50–11:40

Laurent Saloﬀ-Coste (Cornell University)
The isoperimetric profile for random walks on finitely generated groups II

13:00–13:50

利根川 吉廣 Yoshihiro Tonegawa (東京工業大学 Tokyo Institute of Technology)
Long-time existence of mean curvature flow of grain boundaries

14:00–14:50

Paul Woon Yin Lee (Chinese University of Hong Kong)
Ricci curvature type lower bounds on Sasakian manifolds

15:10–16:00

加須栄 篤 Atsushi Kasue (金沢大学 Kanazawa University)
Ideal boundaries of open Riemannian manifolds and convergence of
induced Dirichlet forms

16:10–17:00

Short communications III10 月 28 日 (金) Oct. 28 (Fri.)
9:50–10:40 正宗 淳 Jun Masamune (北海道大学 Hokkaido University)
On the L1 Liouville property of a manifold with ends
10:50–11:40

Laurent Saloﬀ-Coste (Cornell University)
The isoperimetric profile for random walks on finitely generated groups III

世話人:(Organizers)

石渡聡 Satoshi Ishiwata (山形大学 Yamagata University)桒田 和正 Kazumasa Kuwada (東京工業大学 Tokyo Institute of Technology)
桑江 一洋 Kazuhiro Kuwae (福岡大学 Fukuoka University)
太田 慎一 Shin-ichi Ohta (京都大学 Kyoto University)
塩谷 隆 Takashi Shioya (東北大学 Tohoku University)

この研究会は，数理解析研究所の援助を受けています．

AbstractsNicola Gigli (La Scuola Internazionale Superiore di Studi Avanzati)
Nonsmooth diﬀerential geometry I–III
Abstract: Aim of the lectures is to give an overview about diﬀerential calculus on metricmeasure spaces: we shall see in which sense general metric measure spaces posses a weak
first order diﬀerential structure and how on spaces with Ricci curvature bounded from below
a second order calculus is possible.
The construction is based on the well-known notion of Sobolev function on a metric measurespace and on the one of L2 -normed L∞ -module, for which I’ve got inspiration from a workof Weaver (the more general case of Lp -modules can also be discussed, but we shall focus
on p = 2 for these lectures). Such modules provide an abstraction of the notion of space of
L2 -sections of a normed vector bundle on a smooth manifold and we shall see that in a very
natural way every mm space admits a cotangent module and a diﬀerential operator acting
on Sobolev functions. Starting from this, by duality we can introduce the notion of tangent
module. We shall see that: - vector fields (intended as elements of the tangent module) are
in 1-1 correspondence with L2 derivations od Sobolev functions; - there is a tight relation
between the metric speed of curves and the norm of their velocities interpreted in the sense ofmodules. Thus, in an appropriate weak sense, the norm of vector fields induces the distanceon the base space.
Still concerning the first order structure, we shall discuss in which sense maps f between metric measure spaces which are both Lipschitz and of bounded compression (i.e. f∗ m1 ≤ Cm2
for some C > 0, where m1 , m2 are the reference measures on the given spaces) canonicallyinduce a pullback of 1-form and, by duality, have a diﬀerential acting on vector fields.
Then we introduce spaces with Ricci curvature bounded from below (RCD spaces in short):
for what concerns the topic of the lectures, the key feature of these spaces which allows tomove beyond first order calculus is the presence of the Bochner inequality. In particular,
exploiting the self-improving properties of such inequality we shall see that it is possible tointroduce the Sobolev space W 2,2 of functions having Hessian in L2 and prove that it is dense
in W 1,2 . Here the Hessian is intended as element of the - naturally defined - tensor productof cotangent bundle with itself.
Once this is done, it will be easy to speak about Sobolev vector fields, i.e. with covariant
derivative in L2 : the basic example of such vector field is given by ∇f with f ∈ W 2,2 .
Similarly, we can introduce exterior diﬀerentiation of k-forms and we shall verify that mimicking the constructions in the smooth setting the de Rham cohomology can be introduced
and a very natural version of Hodge theory can also be developed. As it turns out, these
constructions are functorial w.r.t. the Lipschitz maps of bounded compression previously
discussed.
We shall conclude the lectures by discussing in which sense RCD spaces possess a measurevalued Ricci tensor and how its study can give informations about the geometry of the space.
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Short time full asymptotic expansion of hypoelliptic
heat kernel at the cut locus
Yuzuru Inahama ∗This is a jointwork with Setsuo Taniguchi (Kyushu University) and can be found at arXiv
Preprint Server (arXiv:1603.01386).
We discuss a short time asymptotic expansion of a hypoelliptic heat kernel on a Euclidean space and on a compact manifold. We study the ”cut locus” case, namely, the casewhere energy-minimizing paths which join the two points under consideration form nota finite set, but a compact manifold. Under mild assumptions we obtain an asymptotic
expansion of the heat kernel up to any order. Our approach is probabilistic and the heat
kernel is regarded as the density of the law of a hypoelliptic diﬀusion process, which isrealized as a unique solution of the corresponding stochastic diﬀerential equation (SDE).
Our main tools are S. Watanabe’s distributional Malliavin calculus and T. Lyons’ roughpath theory.
Our work has the following three features. To our knowledge, there are no works which
satisfy all of these conditions simultaneously:
1. The manifold and the hypoelliptic diﬀusion process on it are rather general. In
other words, this is not a study of special examples.
2. The ”cut locus” case is studied. More precisely, we mean by this that the set ofenergy-minimizing paths (or controls) which connect the two points under consideration becomes a compact manifold of finite dimension.
3. The asymptotic expansion is full, that is, the polynomial part of the asymptotics isup to any order.
On a Euclidean space, however, there are two famous results which satisfy (2), (3)and the latter half of (1). Both of them are probabilistic and use generalized versions ofMalliavin calculus. One is Takanobu and Watanabe [3]. They use Watanabe’s distributional Malliavin calculus. The other is Kusuoka and Stroock [2]. They use their versionof generalized Malliavin calculus. We use the former.
Though we basically follow Takanobu-Watanabe’s argument in [3], the main diﬀerenceis that we use T. Lyons’ rough path theory together, which is something like a deterministic version of the SDE theory. The main advantage of using rough path theory is thatwhile the usual Itô map i.e., the solution map of an SDE is discontinuous, the Lyons-Itô
map i.e., the solution map of a rough diﬀerential equation (RDE) is continuous.
This fact enables us to do ”local analysis” of the Lyons-Itô map (for instance, restricting the map on a neighborhood of its critical point and doing a Taylor-like expansion)in a somewhat similar way we do in the Fréchet calculus. Recall that in the standard
SDE theory, this type of local operation is very hard and sometimes impossible, due to∗

Kyushu University, Japan.

Email:inahama@math.kyushu-u.ac.jp

the discontinuity of the Itô map. For this reason, the localization procedure in [3] looksso complicated that it might be diﬃcult to generalize their method if rough path theory
did not exist. Of course, there is a possibility that our main result can be proved withoutrough path theory, but we believe that the theory is quite suitable for this problem andgives us a very clear view (in particular, in the manifold case).
A detailed proof can be found in our preprint [1]. We first reprove and generalize
the main result in [3] in the Euclidean setting by using rough path theory. Then, westudy the manifold case. Recall that Malliavin calculus for a manifold-valued SDE wasstudied by Taniguchi [4]. Even in this Euclidean setting, many parts of the proof aretechnically improved, thanks to rough path theory. We believe that the following are worth
mentioning: (i) Large deviation upper bound. (ii) Asymptotic partition of unity. (iii) A
Taylor-like expansion of the Lyons-Itô map and the uniform exponential integrability
lemma for the ordinary and the remainder terms of the expansion. (iv) Quasi-sure analysis
for the solution of the SDE.

References[1] Y. Inahama, S. Taniguchi, Short time full asymptotic expansion of hypoelliptic heat
kernel at the cut locus, preprint, arXiv:1603.01386.
[2] S. Kusuoka, D. W. Stroock, Precise asymptotics of certain Wiener functionals, J.
Funct. Anal. 99 (1991), no. 1, 1–74.
[3] S. Takanobu, S. Watanabe, Asymptotic expansion formulas of the Schilder typefor a class of conditional Wiener functional integrations, Asymptotic problems inprobability theory: Wiener functionals and asymptotics (Sanda/Kyoto, 1990), 194–241, Pitman Res. Notes Math. Ser., 284, Longman Sci. Tech., Harlow, 1993.
[4] S. Taniguchi, Malliavin’s stochastic calculus of variations for manifold-valued Wiener
functionals and its applications, Z. Wahrsch. Verw. Gebiete 65 (1983), no. 2, 269–290.

Atsushi Kasue (Kanazawa University)
Ideal boundaries of open Riemannian manifolds and convergence of induced Dirichlet forms
Abstract: In this talk, we consider a transient open Riemannian manifold, and discuss theconvergence of the Dirichlet forms induced on the boundaries of smooth compact subdomains
as they exhaust the manifold, and the limit form on a certain ideal boundary of the manifold.

Young-Heon Kim (University of British Colombia)
Optimal martingale transport in general dimensions I, II
Abstract: We discuss the optimal solutions to a transport problem where mass has to move
under martingale constraint; this constraint forces the transport to split the mass. This
problem was originated from mathematical finance, e.g. option pricing. There have been
intensive studies on the one-dimensional case, but, rarely in higher dimensions. We present
structural results in general dimensions. This is a joint work with Nassif Ghoussoub andTongseok Lim.

Paul Woon Yin Lee (Chinese University of Hong Kong)
The Harnack inequality for the Jordan-Kinderlehrer-Otto/minimizing movement schemeof the heat equation
Abstract: The Jordan-Kinderlehrer-Otto or the minimizing movement scheme on the Wasserstein space is a scheme to produce discrete time approximations to Wasserstein gradient flows.
Local L1 convergence of these approximations was shown by Jordan-Kinderlehrer-Otto in
the case of Fokker-Planck equation. In this talk, I will discuss how this can be improved (atleast in the case of the heat equation). We also discuss a version of the Harnack inequality
satisfied by these discrete time approximations.
Ricci curvature type lower bounds on Sasakian manifolds
Abstract: Measure contraction property is a synthetic Ricci curvature lower bound for metric measure spaces introduced by Sturm and Ohta. We consider Sasakian manifolds with
non-negative Tanaka-Webster Ricci curvature equipped with a one-parameter family of Riemannian metrics which degenerate to a subriemannian one. We show that each of these
Riemannian metric or the subriemannian one, when equipped with a natural measure, satisfies the measure contraction property. We will also discuss the consequences following fromthis condition.
2Jun Masamune (Hokkaido University)
On the L1 Liouville property of a manifold with ends
Abstract: In this talk we will learn how to construct nontrivial integrable harmonic functions
on a complete non-compact Riemannian manifold. Joint with A. Grigor’yan and M. Murata.
Laurent Saloﬀ-Coste (Cornell University)
The isoperimetric profile for random walks on finitely generated groups I–III
Abstract: I will discuss techniques to bound the isoperimetric profile of symmetric measures
on finitely generated groups, both for finitely supported measures and for more spread-out
measures. I will also discuss the relation between the isoperimetric profile, the probabilityof return (or heat kernel decay) and other parameters.
Tomoyuki Shirai (Kyushu University)
Persistent homology of random complexes built over point processes
Abstract: Persistent homology appeared around 2000 as an algebraic method which measures
topological features of objects or point cloud data. Recently, much attention has been paid to
it in the context of Topological Data Analysis (TDA). Persistent homology describes, roughly
speaking, the birth and death of topological feature (connected components, holes, voids, andso on) of an increasing sequence of topological objects. Connected components of random
objects have been studied for long time in probability theory, especially, in percolation theory
and random geometric graph theory. Persistent homology theory sheds new light on such
topics in several ways. In this talk, we would like to discuss mainly some topics on persistenthomology for random complexes built over point cloud data.
Daisuke Shiraishi (Kyoto University)
Geometry of Brownian motionAbstract: In this talk, we will consider the nature of self-intersections of the Brownian
motion in Rd . It is well-known that the Brownian motion is a simple path if d ≥ 4, while
it has loops when d ≤ 3. What are loops of the Brownian motion? How are those loops
distributed in space? These are questions we want to address. In the talk, we give an explicit
representation of such loops for d ≤ 3 by establishing a decomposition of the Brownian path
into independent simple path and a set of loops. It turns out that the simple path and the setof loops can be described by the scaling limit of the loop-erased random walk and a Poisson
point process on a path space, respectively. We will also explain a relation between ourresults and Itô’s excursion theory. This is joint work with Artem Sapozhnikov (Universityof Leipzig).
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τ -Information geometry endowed with a contact structure
Masaru Tanaka
Department of Applied Mathematics, Faculty of Science
Fukuoka University
τ -Information geometry is an information geometry based on two aﬃne spaces: one is an aﬃne spaceτ =s

P constructed by finite positive measures which are mutually absolutely continuous and defined onτ =1−ssome measurable space, and the other is an aﬃne space

Pgiven by taking “Hölder conjugate” withrespect to a parameter τ which controls an aﬃne property of the space of finite positive measures P. τ Information geometry yields the same results from Amari-style information geometry, but it sometimes
gives them diﬀerent interpretations. The most prominent feature of τ -Information geometry is thatτ =s

the magnitude of a measure is changed depending on a translation in the aﬃne spaces P andτ =1−sP ,and conversely a scale transformation induces a translation in the aﬃne spaces. Here we pay attention
to an entropy. In the context of τ -Information geometry, the entropy is given in the form of Tsallislike non-extensive entropy. However, in general, an entropy is aﬀected by a normalization/scale, aconformal entropy is naturally defined and used. Considering the change of a normalization caused
by a translation as the change of a conformal entropy, we can restore the Tsallis-like non-extensive
conformal entropy to being an extensive conformal entropy. This is quite important because an entropy
should be extensive in any physical contexts. The principle that an entropy should be extensive leads to
the AdS/CFT correspondence or the holography principle, which makes complex and diﬃcult situations
occurring to calculate physical quantities simple and easy situations by introducing an extra codimension.
So, we introduce a coordinate for a scale transformation as an extra codimension, and consider the()(2r + 1)-dimensional space of natural coordinates θ1 , . . . , θr for characterizing a distribution, their dual( )coordinates (η1 , . . . , ηr ), and the scale coordinate θ0 . Then, if all of the coordinates are independentof each other, a contact structure is naturally arises. Furthermore, the corresponding Heisenberg groupis also available so that the group structure of the Heisenberg group gives a Legendre transformation forpotential functions.
In this talk, it is shown that how the contact structure appears in the context of τ -Information geometry
without regard to mathematical strictness too much

Yoshihiro Tonegawa (Tokyo Institute of Technology)
Long-time existence of mean curvature flow of grain boundaries
Abstract: Given an arbitrary closed countably rectifiable set of codimension one in the
Euclidean space, we prove a time-global existence theorem for a mean curvature flow startingfrom this set. The flow can go through topological changes and is continuous in a certain
sense to be clarified in the talk. This is a joint work with Lami Kim.

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