Kobe Workshop on Probabilistic
Potential Theory and Related Fields
Dates: 7 May 2019 (Tuesday) 09:00 – 8 May 2019 (Wednesday) 17:00
Venue: Room B428-430, Faculty of Science Building B, Kobe University
Organizer: Naotaka Kajino (Kobe University)

Schedule
May 7 (Tuesday)

May 8 (Wednesday)
09:00–10:00 Free Discussion
10:00–10:40 Ryokichi Tanaka
09:00–13:00 Free Discussion
10:50–11:30 Yuki Tokushige
11:40–12:30 Mathav Murugan
13:00–13:40 Masayoshi Takeda 12:30–13:30 (Lunch Break)
13:50–14:30 Daisuke Shiraishi 13:30–14:10 Kazumi Okamura
14:50–15:40Omer Angel
14:20–15:00 David Croydon
15:50–16:30
Yoshihiro Abe
15:20–16:00 Kouhei Matsuura
16:40–17:10 Ryosuke Shimuzu 16:10–16:50 Hiroaki Aikawa

Financial Supports
The organization of this workshop is partially supported by the following grants:• JSPS Grant-in-Aid for Scientific Research (B) Grant Number 17H02849「空間の構造と解析の関わりーフラクタルを出発点としてー」(Principal Investigator: Jun Kigami (Graduate School of Informatics, Kyoto University))• JSPS Grant-in-Aid for Scientific Research (B) Grant Number 18H01123「分野横断的視点によるフラクタル及びその上の確率過程の解析・幾何」(Principal Investigator: Naotaka Kajino (Graduate School of Science, Kobe University))1Program
7 May 2019 (Tuesday)• 9:00–13:00 Free Discussion• 13:00–13:40 Masayoshi Takeda (Kansai University)
Criticality theory and maximum principle for generalized Schrödinger forms• 13:50–14:30 Daisuke Shiraishi (Kyoto University)
Convergence of three-dimensional loop-erased random walk in the natural parametrization(Tea break for 20 minutes)• 14:50–15:40 Omer Angel (University of British Columbia)
Scaling limits of self interacting random walks• 15:50–16:30 Yoshihiro Abe (Chiba University)
Exceptional points of two-dimensional random walks at multiples of the cover time
• 16:40–17:10 Ryosuke Shimizu (Kyoto University)
Generalized resistance metrics on graphs

8 May 2019 (Wednesday)• 9:00–10:00 Free Discussion• 10:00–10:40 Ryokichi Tanaka (Tohoku University)
Cutoﬀ for product replacement on finite groups• 10:50–11:30 Yuki Tokushige (Kyoto University)
Jump processes on the Gromov boundary of a hyperbolic group: Probabilistic interpretation ofBourdon–Pajot’s Besov spaces• 11:40–12:30 Mathav Murugan (University of British Columbia)
Stability of the elliptic Harnack inequality(Lunch break for 60 minutes)• 13:30–14:10 Kazuki Okamura (Shinshu University)
Some results for range of random walk on graph with spectral dimension two• 14:20–15:00 David Croydon (Kyoto University)
Quenched and averaged tails of the heat kernel of the two-dimensional uniform spanning tree
(Tea break for 20 minutes)• 15:20–16:00 Kouhei Matsuura (Kyoto University)
Tightness property of reflecting Brownian motions on horn-shaped domains• 16:10–16:50 Hiroaki Aikawa (Chubu University)
Potential theoretic notions related to integrability of superharmonic functions and supertemperatures(End of Workshop)2

Abstracts
Criticality theory and maximum principle for generalized
Schrödinger forms
Masayoshi Takeda (Kansai University)
We define three function spaces related to a Schrödinger form and its semigroup: two of theseare spaces of excessive functions defined through the Schrödinger semigroup, one is the space of
weak subsolutions defined through the Schrödinger form. We define the maximum principle foreach space and prove the equivalence of three maximum principles. Moreover, we give a necessary
and suﬃcient condition for each maximum principle in terms of the principal eigenvalue of timechanged processes.

Convergence of three-dimensional loop-erased random
walk in the natural parametrization
Daisuke Shiraishi (Kyoto University)
In this talk, we consider loop-erased random walk (LERW) and its scaling limit in three dimensions, and prove that 3D LERW parametrized by renormalized length converges to its scaling
limit parametrized by some suitable measure with respect to the uniform convergence topology
in the lattice size scaling limit. This talk is based on joint work with Xinyi Li (University ofChicago).

Scaling limits of self interacting random walks
Omer Angel (University of British Columbia)
We consider several self interacting random walks in the plane. A central example is the random
walk in Z2 that moves vertically on its first visit to a site, and horizontally on subsequent moves.
It is conjectured that this random walk is recurrent. We prove that its scaling limit is a onedimensional Brownian motion. Joint with Mark Holmes and Alejandro Ramirez.

Exceptional points of two-dimensional random walks atmultiples of the cover time
Yoshihiro Abe (Chiba University)
I will talk about the local time of the continuous-time simple random walk on a large finite subsetof the two-dimensional lattice with the wired boundary condition. The running time of the walkis parametrized by the local time at the “boundary vertex” and approximated by a multiple ofthe cover time. The talk will focus on statistics of points where the local times are atypicallylarge, called thick points. This is based on joint work with Marek Biskup (UCLA).

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Generalized resistance metrics on graphs
Ryosuke Shimizu (Kyoto University)Albin, Clemens, Fernando and Poggi-Corradini (2018) defined a new metric associated with penergy, where the metric is the resistance metric if p = 2. In their work, a duality theorem formoduli and a connection between moduli and conductances played important roles. On the otherhand, a similar duality problem has been studied by Nakamura and Yamasaki (1976). In thistalk, we will apply Nakamura–Yamasaki’s result to extend the result by Albin et al. to the caseof infinite graphs.

Cutoﬀ for product replacement on finite groups
Ryokichi Tanaka (Tohoku University)
We analyze a Markov chain, known as the product replacement chain, on the set of generatingn-tuples of a fixed finite group G. Based on joint work with Yuval Peres and Alex Zhai, we showthat as n → ∞, the total-variation mixing time of the chain has a cutoﬀ at time (3/2)n log n withwindow of order n. This generalizes a result of Ben-Hamou and Peres (who established the resultfor G = Z/2) and confirms a conjecture of Diaconis and Saloﬀ-Coste that for an arbitrary but
fixed finite group, the mixing time of the product replacement chain is O(n log n).

Jump processes on the Gromov boundary of a hyperbolic
group: Probabilistic interpretation of Bourdon–Pajot’s
Besov spaces
Yuki Tokushige (Kyoto University)
In this talk, we will discuss how Besov spaces on a compact metric space introduced by BourdonPajot arise in the context of random walks on a hyperbolic group and the theory of Markov
processes. Specifically, under an assumption on the Ahlfors-regular conformal dimension, we will
prove the regularity of Dirichlet forms defined by those Besov spaces, and give to them a concrete
probabilistic description by means of random walks. This talk is based on a joint work with P.
Mathieu (Aix-Marseille Université).

Stability of the elliptic Harnack inequality
Mathav Murugan (University of British Columbia)
Harnack inequalities have proved to be a powerful tool in PDE (regularity estimates), geometry (geometric flows) and probability (heat kernel estimates). In the early 1990s Grigor’yan and
Saloﬀ-Coste gave a characterization of the parabolic Harnack inequality (PHI). This characterization implies that PHI is stable under perturbations (rough isometries). In this talk, I will
provide an introduction to Harnack inequalities and discuss the stability of the elliptic Harnack
inequality. The proof uses a quasisymmetric transformation of the space and suggests a variantof the quasisymmetric uniformization problem. This is joint work with Martin Barlow.

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Some results for range of random walk on graph with
spectral dimension two
Kazuki Okamura (Shinshu University)
We consider the range of the simple random walk on graphs with spectral dimension two. We
give a form of strong law of large numbers under a certain uniform condition, which is satisfied by
not only the square integer lattice but also a class of fractal graphs. Our results imply the stronglaw of large numbers on the square integer lattice established by Dvoretzky and Erdos. Our proof
does not depend on spacial homogeneity of space and gives a new proof of the strong law of large
numbers on the lattice. As an application, we establish a form of law of the iterated logarithms
for lamplighter random walks in the case that the spectral dimension of the underlying graph istwo. If time is permitted, we will discuss an extension to the framework of stable random walks.

Quenched and averaged tails of the heat kernel of thetwo-dimensional uniform spanning tree
David Croydon (Kyoto University)
I will describe ongoing joint work (with M. T. Barlow and T. Kumagai) in which a detailed
investigation of the heat kernel associated with the two-dimensional uniform spanning tree is conducted. This demonstrates the occurrence of log-logarithmic fluctuations around the leading orderpolynomial behaviour for the on-diagonal part of the quenched heat kernel. Moreover, two-sidedestimates are given for the averaged heat kernel. Notably, these demonstrate a discrepancy between the exponents that appear in the oﬀ-diagonal parts of the quenched and averaged versionsof the heat kernel, and also yield precise asymptotics for the average distance travelled by theassociated random walk. Finally, we derive various scaling limits for the heat kernel, the implications of which include enabling us to sharpen the known asymptotics regarding the on-diagonalpart of the averaged heat kernel.

Tightness property of reflecting Brownian motions onhorn-shaped domains
Kouhei Matsuura (Kyoto University)
We consider a tightness property for symmetric Hunt processes. One dimensional diﬀusions possess the tightness property if and only if they have no natural boundaries. Therefore, symmetric
Hunt processes with tightness property are similar to one-dimensional diﬀusions having no natural boundaries. In this talk, we give suﬃcient conditions for reflecting Brownian motions on
unbounded domains to possess the tightness property. We also discuss their spectral and ergodic
properties.

5Potential theoretic notions related to integrability ofsuperharmonic functions and supertemperatures
Hiroaki Aikawa (Chubu University)
Ever since Armitage proved that every nonnegative superharmonic function in a bounded domain
D ⊂ Rn of bounded curvature is Lp -integrable on the whole D for 0 < p < n/(n − 1), the global
integrability of nonnegative supersolutions has attracted many mathematicians. Illustrating related potential theoretic notions, we show how this problem has been settled and extended to theparabolic case.

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