# Abstracts

**Takafumi Amaba** (Ritsumeikan Univ, Shiga) *An Integration by Parts on Space of Loops*

We consider to construct a measure on space of loops in $\mathbb{C}$* (strictly speaking, a space of paths in a coefficient body), surrounding the origin by employing the utility of the (alternate) Loewner-Kufarev equation. We discuss about a simple integration by parts formula under the measure.

**David Applebaum** (Sheffield) *Stationary Random Fields on Unitary Duals of Compact Groups*

We give a new definition of stationarity for random fields on unitary duals, and show that this leads to a natural generalisation of the usual theory of stationary processes over the integers. The Cramer and Kolmogorov theorems both extend to this wider context. The spectral measure of the random field lives on the group itself, and every central measure is a spectral measure. Studying the associated field may give more insight into this important class of measures, which includes Gaussians and associated subordinated measures.

**Horatio Boedihardjo** (Oxford)* Iterated integrals of a rough path: Uniqueness*

In a controlled differential equation, the solution depends on the driving path only through its iterated integrals. We are interested in whether the iterated integrals of the paths essentially determine the paths. B. Hambly and T. Lyons gave a positive answer for paths with bounded variation in 2006, but the analogous problem for rough paths has been open since then. We propose a proof for this conjecture for a class of rough paths. Joint work with X. Geng, T. Lyons and D. Yang.

**Tom Cass** (Imperial) *Constrained rough paths*

I present some recent joint work with Bruce Driver and Christian Litterer on rough paths which are constrained to lie in a d-dimensional submanifold of a Euclidean space E. We will present a natural definition for this class of rough paths and then describe the (second) order geometric calculus which arises out of this definition. The talk will conclude with more advanced applications, including a rough version of Cartan’s development map and its stochastic version, which was developed by Eells and Elworthy and Malliavin.

**D. Crisan** (Imperial) *Kusuoka-Stroock gradient bounds for the solution of the filtering equation*

We obtain sharp gradient bounds for perturbed diffusion semigroups. In contrast with existing results, the perturbation is here random and the bounds obtained are pathwise. Our approach builds on the classical work of Kusuoka and Stroock and extends their program developed for the heat semi-group to solutions of stochastic partial differential equations. The work is motivated by and applied to nonlinear filtering. The analysis allows us to derive pathwise gradient bounds for the un-normalised conditional distribution of a partially observed signal. It uses a pathwise representation of the perturbed semigroup in the spirit of classical work by Ocone The estimates we derive have sharp small time asymptotics. This is joint work with C. Litterer (Ecole Polytechnique) and T. Lyons (Oxford).

**Masatoshi Fukushima** (Osaka) *Stochastic Komatu-Loewner evolutions and Brownian motion with darning*

A standard slit domain *D* is a planar domain obtained from the upper-half plane *H* by removing a finite number of mutually disjoint horizontal line segments called slits. The Brownian motion with darning (BMD) for the domain* D* is obtained from the absorbing Brownian motion on *H* by rendering each slit into a single point. BMD is an intrinsic and useful probabilistic tool to study conformal mappings among multiply connected planar domains. We make use of BMD to extend the Loewner differential equation and SLE (stochastic Loewner evolution, or Schramm-Loewner evolution) for *H* to a standard slit domain *D*. For the simply connected domain *H*, the Loewner equation is described by the complex Poisson kernel Ψ(z,ξ ), z ∈ *H*, ξ∈ ∂*H*, and the SLE is a family of random growing hulls *F _{t}* in

*H*driven by a constant multiple of the standard Brownian motion ξ(

*t*) on the

*x*-axis ∂

*H*. We replace

*H*by a standard slit domain

*D*. Then the Loewner equation is converted into the Komatu-Loewner equation described by the BMD-complex Poisson kernel Ψ

_{t}(

*z*,ξ ) and the driving process ξ(

*t*) is replaced by a joint process

*W*(

*t*) = (ξ (

*t*),

*s*(

*t*)) where

*s*(

*t*) represents a slit motion.

*W*(

*t*) is a solution of a SDE with the diffusion (resp. drift) coefficient α (resp.

*b*) of ξ(

*t*) being a homogeneneous function of degree 0 (resp. —1), while

*s*(

*t*) has only the drift coefficient determined by the trace of Ψ

*to the slits. Given α and*

_{t}*b*as above, the associated growing hulls

*F*in

_{t}*D*is called the stochastic Komatu-Loewner evolution driven by the solution of the SDE with coefficients α,

*b*. It enjoys the local property when α = ±√6,

*b*= —

*b*for the BMD-domain constant

_{BMD}*b*indicating the discrepancy of D from H.

_{BMD}**Martin Hairer** (Warwick) *Regularity Structures*

**Yuzuru Inahama** (Nagoya) *Short time kernel asymptotics for rough differential equation driven by fractional Brownian motion*

We study a stochastic differential equation in the sense of rough path theory driven by fractional Brownian rough path with Hurst parameter *H*(1/3 < *Hle*1/2) under the ellipticity assumption at the starting point. In such a case, the law of the solution at a fixed time has a kernel, i.e., a density function with respect to Lebesgue measure. In this paper we prove a short time off-diagonal asymptotic expansion of the kernel under mild additional assumptions. Our main tool is Watanabe’s distributional Malliavin calculus.

**Neils Jacob** (Swansea) *Transition Functions of Levy Processes and Geometry*

**Naotaka Kajino** (Kobe University) *Continuity and estimates of the transition density of the Liouville Brownian motion*

The Liouville Brownian motion, recently introduced by Garban, Rhodes and Vargas, and in a weaker form also by Berestycki, is a diffusion process evolving in a planar random geometry induced by the Liouville measure *M*_{γ}, formally written as *M*_{γ}(*dz*) =$e^{γ X (z)- γ^2 E [X(z)^2]/2} dz$ , γ∈ (0, 2), for a (massive) Gaussian free field X. It is an Mγ-symmetric diffusion defined as the time-change of the standard two-dimensional Brownian motion by the positive continuous additive functional with Revuz measure *M*_{γ}.

In this talk I will present a detailed analysis of the heat kernel* p _{t}*(

*x*,

*y*). The results include its joint continuity, a locally uniform sub-Gaussian upper bound of the form

*p*(

_{t}*x*,

*y*) ≤

*C*

_{1}

*t*

^{-1}log(

*t*

^{−1}) exp(−

*C*

_{2}(|

*x*−

*y*|

^{β}/

*t*)

^{1/(β-1)}) for small

*t*and an on-diagonal lower bound of the form

*p*(

_{t}*x*,

*x*) ≥

*C*

_{3}

*t*

^{−1}(log(

*t*

^{−1}))

^{−η}for small

*t*for

*M*

_{γ}-a.e.

*x*with some concrete constant η > 0. As immediate corollaries it then turns out that the pointwise spectral dimension equals 2

*M*

_{γ}-a.e. and that the global spectral dimension is also 2.

This is a joint work with Sebastian Andres (Universität Bonn).

**Hiroshi Kawabi** (Okayama University) *Weak convergence of laws of non-symmetric random walks on crystal lattices*

In this talk, we discuss two kinds of weak convergence of laws of non-symmetric random walks on crystal lattices. We first show that the Brownian motion on the Euclidean space with the Albanese metric appears as the usual scaling limit of the random walk by subtracting the asymptotic direction. Next, we introduce a family of random walks which interpolates between the original (non-symmetric) random walk and the symmetric one. We then show that the Brownian motion with a constant drift on the Euclidean space also appears as the scaling limit. This talk is based on joint work with Satoshi Ishiwata (Yamagata University) and Motoko Kotani (Tohoku University).

**Wilfrid Kendall** (Warwick) *Shy Couplings, CAT(0) Spaces, and the Lion and Man*

Classical probabilistic coupling aims to construct realisations of two random processes (often Markov) on the same probability space, in such a manner that they have a high chance of meeting each other soon. However, as part of a wider programme aimed at acquiring better understanding of such couplings, suppose that one wishes to consider whether there can exist ”couplings” which actually do not meet at all; for which the relevant processes always stay at least some fixed positive distance away from each other? Recent work has established that such shy couplings cannot exist for reflected Brownian motion in the presence of convexity [1,2]; [5] considers a Riemannian context, and still more recent work [3,4] uncovers surprising relationships to the geometric theory of CAT(0) domains and to the classic Lion and Man problem of recreational mathematics.

References:

1. Itai Benjamini, Krzysztof Burdzy, Zhen-Qing Chen Shy couplings. Probability Theory and Related Fields 137 (2007), nos. 3-4, 345-377. arxiv.org/abs/math/0509458

2. WSK. Brownian couplings, convexity, and shyness. Electronic Communications in Probability 14 (2009), Paper 7, 66-80. arxiv.org/abs/0809.4682

3. Maury Bramson, Krzysztof Burdzy, WSK. Shy Couplings, CAT(0) Spaces, and the Lion and Man. Annals of Probability (2013), 41:2, 744-784. arxiv.org/abs/1007.3199

4. Maury Bramson, Krzysztof Burdzy, WSK. Rubber Bands, Pursuit Games and Shy Couplings. Proceedings of the London Mathematical Society (to appear). arxiv.org/abs/1207.0597

5. Mihai Pascu, Ionel Popescu. Shy and Fixed-Distance Couplings of Brownian Motions on Manifolds. arxiv.org/abs/1210.7217

**H. Kunita** (Kyushu) *Stochastic flows and adjoint processes*

**Seiichiro Kusuoka** (Tohoku) *Hӧlder and Lipschitz continuity of the solutions to parabolic equations of non-divergence type*

We consider linear parabolic equations of the non-divergence type, and assume the ellipticity and the continuity on the coefficient of the second order derivatives and the boundedness on all the coefficients. Under the assumptions we show the H¨older continuity of the solution in the spatial component. Furthermore, adding an assumption on the continuity of the coefficient of the second order derivative, we have the Lipschitz continuity of the solution. In the proof, we use a probabilistic method, in particular the coupling method. As a corollary, under an additional assumption we obtain the Hӧlder and Lipschitz continuity of the fundamental solution in a component.

**Kazumasa Kuwada** (Tokyo Institute of Technology) *On the speed in transportation costs of heat distributions*

By regarding heat distributions as a curve in the space of probability measures, we consider its speed measured by some transportation costs. This is a work in progress and our main concern in this talk is heat distributions on (backward) Ricci flow. The speed can be expressed explicitly when heat distribution is identified with a gradient flow of the relative entropy. On Ricci flow, this interpretation does not seem to work well. Nevertheless we can show some results for a suitably chosen transportation costs. Indeed it extends some known results in optimal transportation on Ricci flow to noncompact case.

**Kazuhiro Kuwae** (Kumamoto) *Gaugeability and conditional gaugeability for generalized Feynman-Kac functionals*

I will talk on new family of (semi-)Green-tight measures of (extended) Kato class extending the class defined by Chen. These classes are stable under Girsanov transform in some sense in the framework of symmetric Markov processes. In terms of these classes, we can give a characterization of gaugeability or conditional gaugeability for generalized Feynman-Kac functionals with such classes of measures.

**Thierry Levy** (Paris 6 ) *Two dimensional Yang-Mills theory: a case study in non-perturbative gauge theory*

Two-dimensional Yang-Mills theory is one of the rare instances of a small but non-trivial piece of quantum field theory for which it is possible to rigorously ground Feynman’s formulation in terms of functional integrals. After recounting how functionals integrals with respect to the Yang-Mills measure arise from the quantisation of the electromagnetic field, I will present the construction of a stochastic process indexed by the set of loops traced on a compact surface and with values in a compact Lie group, and explain why this process deserves to be considered as a rigorous version of the Yang-Mills measure.

**H. Osada** (Kyushu University) *Infinite-dimensional stochastic differential equations arising from random matrix theory*

Interacting Brownian motions in infinite dimensions are stochastic dynamics describing infinitely many Brownian particles with interactions Ψ moving in $\mathbb{R}$^{d}. These dynamics are given by infinite-dimensional stochastic differential equations (ISDEs) of the form

$dX_t^i=dB_t^i-β/2 {\sum\limits_{j ≠ 1} ^{∞}}ΔΨ(X_t^i,X_t^j)dt $ $(i ∈ \mathbb{N})$

If Ψ(*x*, *y*) = - log |*x* -*y*|, then the ISDEs are related random point fields arising from random matrices such as Sine_{β} 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 σ-fields, Itô scheme, and Dirichlet form theory.

As an application, we solve the ISDEs arising from random matrix theory such as Sine_{β}, Airy_{β}, Besse_{l2}, 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 talk is based on the joint work with H. Tanemura (Chiba Univ) and Y. Kawamoto (Kyushu University)

The plan of the talk:

- 1st Talk: Examples, results, and strategy of the proof.
- 2nd Talk: Weak solutions: Quasi-Gibbs measures and logarithmic derivatives.
- 3rd Talk: Strong solutions and path wise uniqueness: IFC solutions and analysis of tail σ-fields.

**Michela Ottobre** (Imperial) *Diffusion limit for Random Walk Metropolis algorithm out of stationarity*

The Random Walk Metropolis algorithm is a Monte Carlo-Markov Chain method which creates a Markov chain which is reversible with respect to a given target distribution with Lebesgue density on *R ^{N}*; it can hence be used to approximately sample the target distribution. When N is large a key question is to determine the computational complexity of the algorithm as a function of

*N*. One approach to this question, which we adopt here, is to derive diffusion limits for the algorithm. We study the situation where the algorithm is started out of stationarity and the target measure is in non- product form. We thereby significantly extend previous works which consider either only measures of product form, when the Markov chain is started out of stationarity, or measures defined via a density with respect to a Gaussian, when the Markov chain is started in stationarity.

**Tomoyuki Shirai** (Kyushu) *Absolute continuity and singularity for the Ginibre point process and its Palm measures*

Ginibre point process arises from random matrix theory and it is one of the most important examples of determinantal point process. In this talk, we discuss dichotomy between absolute continuity and singularity for Ginibre point process and its Palm measures.

**Setsuo Taniguchi** (Kyushu University) *Diffusion processes on CR-manifolds
*The diffusion process associated with the sublaplacian on a strictly pseudo-convex CR manifold is constructed in the globally geometric manner with the help of the Tanaka-Webster connection, and is applied to studies of the related heat kernel and Dirichlet problem.

**Yuxin Yang** (Imperial) *The Clark-Ocone approach to Hodge theory by examples*

**Tusheng Zhang** (Manchester University) *Smoothness of solutions of SDEs with singular coefficients*

In this talk I will present results on Malliavin smoothness and Sobolev differentiability of solutions of SDEs with measurable coefficients.

**H. Zhou** (Loughborough) *Random periodic solutions*

**Short talks**

**Christian Fonseca-Mora**(Sheffield)*Stochastic Partial Differential Equations with Lévy Noise in Duals of Nuclear Spaces***Carina Geldhauser**(Bonn)*Existence of solutions to an SPDE with long-range interactions***Nobuaki Naganuma**(Tohoku)*Exact convergence rate of the Wong-Zakai approximation to RDEs driven by Gaussian rough paths***Kiyotaki Suzaki**(Osaka)*An SDE approach to leafwise diffusions on foliated spaces and its applications***Ryoichi Suzuki**(Keio)*Explicit representations of locally risk-minimizing hedging strategy for Lévy markets***James Thompson**(Warwick)*An Asymptotic Relation for the Integrated Heat Kernel***Maria Veretennikova**(Warwick)*Controlled fractional dynamics***Yue Wu**(Loughborough)*Random periodic solutions of SDEs with linear multiplicative noise*

**Lewis Bray and James Harris**(Swansea)*Transition Densities of Lévy Processes and Geometry***Oxana Manita**(Moscow)*Well-posedness of the Cauchy problem for nonlinear Kolmogorov equations***Eamon McMurray**(Imperial)*Smoothing properties ofMcKean-Vlasov SDEs via Malliavin Calculus***Kenneth Uda**(Loughborough)*Existence of Random Periodic Curves***Xince Wang**(Loughborough)*Probabilistic Representation of Weak Solutions of Quasilinear Parabolic Partial Differential Equations*