Abstract: We implement a version of conformal field theory in a multiply connected domain and relate it to the theory of SLE. We present the generalized Eguchi-Ooguri equations and use them to derive the explicit form of Ward's equations, which describe the insertion of a stress tensor in terms of Lie derivatives and differential operators that depend on the Teichmüller modular parameters. Furthermore, we provide a conformal field-theoretic realization of SLE in a multiply connected domain and construct a class of martingale observables for this SLE process. This is joint work with Tom Alberts and Sung-Soo Byun.

Abstract: Sharp convergence rates in Wasserstein distance are derived for empirical measures of diffusion processes on Riemannian manifolds, and the renormalization limits are explicitly formulated by using eigenvalues and eigenfunctions of the associated elliptic operator. For explosive diffusion processes, the convergence is described by conditional expectations.   

Abstract: Stochastic analysis and metric measure geometry, especially the concentration of measure phenomenon, Gromov's pyramids, etc., have been widely studied. On the other hand, there are not many studies that apply the research in each to the other. Gromov's pyramids are based on the concentration of measure phenomenon, and in probabilistic terms, it can be said to be ``metric measure geometry based on the asymptotic behavior of a sequence of random variables''.  In this sense, we believe that this is a field applied to stochastic analysis in the future. In this talk, we will consider a generalization of the results for the $n$-dimensional Cauchy distribution, which were previously obtained using the general theory of  "concentration of cones''. In particular, we will discuss the changes in the phenomenon when the conventional results for $\ell^2$ distance are replaced with $\ell^{\beta}$ distance. We also discuss a relation between the results and the ergodic theory.

This talk is based on a joint work with Daisuke Kazukawa(Kyushu) and Ayato Mitsuishi(Fukuoka). 

Abstract: We investigate the long-time behavior of stochastic heat equations perturbed by space-time white noise on one-dimensional domains such as a torus and the real line. The long-time behavior depends on the spatial domain and also initial functions. We show how the spatial domain and the initial function influence the long-time behavior of the solution. This is based on joint works with Davar Khoshnevisan and Carl Mueller.

Abstract: The irreducibility is fundamental for the study of ergodicity of stochastic dynamical systems. In the literature, there are very few results on the irreducibility of stochastic partial differential equations (SPDEs) and stochastic differential equations (SDEs) driven by pure jump noise. The existing methods on this topic are basically along the same lines as that for the Gaussian case. They heavily rely on the fact that the driving noises are additive type and more or less in the class of stable processes. The use of such methods to deal with the case of other types of additive pure jump noises appears to be unclear, let alone the case of multiplicative noises.

We develop a new, effective method to obtain the irreducibility of SPDEs and SDEs driven by multiplicative pure jump noise. The conditions placed on the coefficients and the driving noise are very mild. This leads to not only significantly improving all of the results in the literature, but also to new irreducibility results of a much larger class of equations driven by pure jump noise with much weaker requirements than those treatable by the known methods. As a result, we are able to apply the main results to SPDEs with locally monotone coefficients, SPDEs/SDEs with singular coefficients, nonlinear Schrodinger equations, Euler equations etc. We emphasize that the driving noises could be compound Poisson processes, even allowed to be infinite dimensional.