確率論サマースクール2014

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講義ノート配布中！

Weber氏の講義ノートが整理されました。(2015/8/18)

講義内容の詳細が記された講義ノートを配布しています。
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時間割

Young forumは参加している大学院生・若手研究者の方を中心に、現在勉強していること、これから勉強したいこと、現在研究中の課題などについて数分～十数分程度で紹介をしてもらう場です。貴重な助言をもらえることもありますので、後込みせずに発表してください。
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講義内容

ラフパス理論の発生からHairer理論の誕生までの概説
〔稲浜譲〕

講義の予定

1. Lyons' original rough path theory
2. Gubinelli's version of RP theory (i.e., controlled path theory)
3. Hairer's SPDE theory (as an extended version of 2, not as a special case of 4
4. regularity structure theory

Stochastic PDEs, Regularity structures, and interacting particle systems
〔Hendrik Weber〕

In this series of lectures I will report on recent progress in the theory of stochastic PDEs. My main aim is to explain the theory of "Regularity structures" developed recently by M. Hairer. This theory gives a way to study well-posedness for a class of stochastic PDEs that could not be treated previously. Prominent examples include the KPZ equation as well as the dynamic $\Phi^4_3$ model. Such equations were treated previously as perturbative expansions. Roughly speaking the theory of regularity structures provides a way to truncate this expansion after finitely many terms and to solve a fixed point problem for the "remainder". The key ingredient is a new notion of "regularity" which is based on the terms of this expansion. I will also discuss how this solution theory can be applied to study scaling limits of interacting particle systems. If time permits I will also mention an application in large deviation theory.
講義の予定

1. Introduction of main examples (KPZ, $\Phi^4_3$, Motivation from statistical mechanics) Scaling and regularity (Regularity for linear stochastic PDEs, the notion of "subcriticality") The need for infinite Renormalisation constants
2. A formal expansion of the solution - which terms matter? Review of Gaussian Analysis: Feynman diagrams
3. Main vocabulary: Regularity structure, Models, controlled distributions How to construct the regularity structure for a given equation? Reconstruction Theorem and multiplication
4. Setting up and solving a fixed point argument in a space of controlled distributions. (If time permits - comparison with the theory of "modelled distributions" put forward by M. Gubinelli)
5. Application: Near critical Kac-Ising model converges to $\Phi^4$ model (at least in one and two dimensions). (if time permits - Large deviations)

Infinite-dimensional stochastic differential equations arising from random matrix theory
〔長田博文〕

Interacting Brownian motions in infinite dimensions are stochastic dynamics describing infinitely many Brownian particles with interactions $\Psi$ moving in $\mathbb{R}^d$. These dynamics are given by infinite-dimensional stochastic differential equations (ISDEs) of the form \begin{align*}& dX_t^i = dB_t^i -\frac{\beta}{2} \sum_{j\not=i}^{\infty} \nabla \Psi (X_t^i,X_t^j) dt \quad (i\in\mathbb{N}) .\end{align*} If $\Psi (x,y) = -\log |x-y|$, then the ISDEs are related random point fields arising from random matrices such as $\mathrm{Sine}_{\beta}$ RPFs and Ginibre RPF. We prove the existence and uniqueness of strong solutions of the ISDEs. We present a new method to construct unique strong solutions of the ISDEs. Our method is based on the analysis of tail $\sigma$-fields, It\^o scheme, and Dirichlet form theory. As an application, we solve the ISDEs arising from random matrix theory such as Sine$_{\beta} $, Airy$_{\beta} $, Bessel$ _2$, Ginibre interacting Brownian motions. We also prove the uniqueness of quasi-regular Dirichlet forms from our strong uniqueness result of the solutions of the ISDEs. Moreover, we prove the convergence of finite-particle approximations of the solutions of the ISDEs. This series of lectures is an extended version of the lectures in Warwick (2014/Sep). We add some preparation on random point fields, and detailed proof or explanation to basic notions newly developed here. This talk is based on the joint work with H. Tanemura (Chiba Univ.) and Y. Kawamoto (Kyushu univ.). The plan of the talk is:

1. Examples, results, and strategy of the proof.
2. Preparation from random point fields.
3. Weak solutions: Quasi-Gibbs measures and logarithmic derivatives.
4. Strong solutions and path wise uniqueness: IFC solutions and analysis of tail $ \sigma $-fields. A note on random point fields and slides in Warwick are avilable on the home page of the summer school. The lectures are based on the papers in [42, 44, 46, 47, 48, 49, 50, 55, 56, 57] in the reference of the attached note.

1. 1406.3913 math.PR
   Hirofumi Osada, Tomoyuki Shirai:
   Absolute continuity and singularity of Palm measures of the Ginibre point process
2. 1405.6123 math-ph
   Hirofumi Osada:
   Dynamical rigidity of stochastic Coulomb systems in infinite-dimensions
3. 1405.4304 math.PR
   Hirofumi Osada, Hideki Tanemura:
   Cores of Dirichlet forms related to random matrix theory
4. 1405.0523 math.PR
   Ryuichi Honda, Hirofumi Osada:
   Infinite-dimensional stochastic differential equations related to Bessel random point fields
5. 1209.0609 math.PR
   Hirofumi Osada:
   Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials II: Airy random point field
6. 1004.0301 math.PR
   Hirofumi Osada:
   Infinite-dimensional stochastic differential equations related to random matrices
7. 0905.3973 math.PR
   Hirofumi Osada:
   Tagged particle processes and their non-explosion criteria
8. 0902.3561 math.PR
   Hirofumi Osada:
   Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials

The following will be uploaded:

9. Hirofumi Osada, Hideki Tanemura:
   Infinite-dimensional stochastic differential equations arising from Airy random point fields
10. Hirofumi Osada, Hideki Tanemura:
    Infinite-dimensional stochastic differential equations and tail $\sigma$-fields
11. Hirofumi Osada:
    Palm resolution and restore density formulae of the Ginibre point process
12. Hirofumi Osada:
    DLR equations for Coulomb point processes
13. Hirofumi Osada:
    Ginibre interacting Brownian motions in infinite dimensions are sub-diffusive

Young Forum

1. 徐路（東大数理）
   An invariance pronciple for stochastic heat equations with periodic coefficients
2. 李嘉衣（東大数理）
   The generation and motion of interface for one-dimensional stochastic Allen-Cahn equations
3. 田井みなみ（東大数理）
   Invariant measure of the Allen-Cahn equations
4. 星野壮登（東大数理）
   Fractional KPZ equation
5. 鈴木康平（京大理）
   A complete metric among pairs of compact metric spaces and probability measures on paths spaces
6. 阿部圭宏（京大理）
   On local times of 2-dimensional random walks
7. 鈴木裕行（中央大理工）
   On Loewner equations
8. 久保田直樹（日大理工）
   On the chemical distance for supercritical percolation clusters
9. 岡田いず海（東工大理）
   Large deviation results for Brownian motions
10. 中村ちから（京大理）
    The rate of escape of random walks on groups
11. 永沼伸顕（東北大理）
    Exact convergence rate of the Wong-Zakai approximation for RDE driven by Gaussian rough paths
12. 江崎翔太（千葉大理）
    Dirichlet form approach to infinite particle systems of jump type with singular interaction
13. 河本陽介（九大数理）
    On SDE gaps
14. 野場啓（京大理）
    General theory of one-dimensional Levy processes
15. 伊藤悠（京大情報）
    Differential equations driven by rough paths: an approach via fractional calculus