Yasushi Ishikawa (Ehime University, Japan)

• Malliavin calculus on the Wiener-Poisson space and its application to canonical SDE (joint work with H. Kunita)

Moritz Kassmann (Universita"t Bonn, Institut fu"r Angewandte Mathematik, Germany)

• The Cauchy problem and the martingale problem for integro-differential operators with non-smooth kernels (joint with H. Abels)

Abstract: We consider the linear integro-differential operator $L$ defined by

\[ Lu(x) =\int_{\Rn} \left(u(x+y)-u(x)-\mathbbm{1}_{[1,2]}(\alpha) \mathbbm{1}_{\{|y|\leq 2\}}(y)y\cdot \nabla u(x) \right)k(x,y) \sd y \,. \]

Here the kernel $k(x,y)$ behaves like $|y|^{-d-\alpha}$, $\alpha \in (0,2)$, for small $y$ and is H\"older-continuous in the first variable, precise definitions are given below. The aim of this work is twofold. On one hand, we study the unique solvability of the Cauchy problem corresponding to $L$. On the other hand, we study the martingale problem for $L$. The analytic results obtained for the deterministic parabolic equation guarantee that the martingale problem is well-posed. Our strategy follows the classical path of Stroock-Varadhan. The assumptions allow for cases that have not been dealt with so far.

• Markov chain approximations for symmetric jump processes (joint with R. Husseini)

Abstract: A central limit theorem for Markov chain approximations of symmetric jump processes is established. The assumptions on the chain are quite general and cover cases where the limit process is degenerate in some sense.

Panki Kim (Seoul National University, South Korea)

• On censored stable process

Abstract: For $\alpha$ in (0, 2), a censored $\alpha$-stable process Y in an open set D is a process obtained from a symmetric $\alpha$-stable L\'evy process by restricting its L\'evy measure to D. It is recently known that when D is a bounded Lipschitz open set, Y is transient if and only if $\alpha>1$. In this talk, we will discuss a sharp two-sided estimates for Green functions of censored $\alpha$-stable process Y in a bounded $C^{1,1}$ open set D for $1<$\alpha$<2. This is a joint work with Zhen-Qing Chen.

• Intrinsic Ultracontractivity for Non-symmetric Levy Processes

Abstract: Recently the concept of intrinsic ultracontractivity to non-symmetric semigroups has been introduced by Kim and Song. In this talk, we study the intrinsic ultracontractivity for non-symmetric discontinuous Levy processes. This is a joint work with Renming Song.

Takashi Komatsu (Osaka City University, Japan)

• On the martingale problem for pseudo-differential operators of variable order

Kazuhiro Kuwae (Kumamoto University, Japan)

• On Stratonovich Type Integrals and Lyons-Zheng Decompositions over Symmetric Markov Processes

Abstract: I will talk about the Fisk-Stratonovich type integrals by Dirichlet processes appeared in Fukushima's decomposition in the framework of symmetric Markov processes. Such integrals were firstly defined by Nakao and also by Lyons-Zhang in a different way based on the time reverse operator if the underlying process is a diffusion with no killing inside. We generalize integrals including jump and killing parts, whose definitions are different from what were discussed by Meyer, Protter even if in the framework of semi-martingales. I give a representation of the Fisk-Stratonovich type integrals by the discontinuous part of such Dirichlet processes as an integrator. As a corollary, under the law for quasi-everywhere starting points, we establish an extension of Fukushima's decomposition for the function locally in the domain of forms, whose jump part can be expressed as a convergence of the sum of jumps. We also show that our Fisk-Stratonovich type integrals allow Lyons-Zheng's type decompositions under the law for quasi-everywhere starting point, which strengthen the relation between two definitions on the Fisk-Stratonovich integrals by Nakao and by Lyons-Zhang for diffusions without killing inside.

Yoichi Oshima (Kumamoto University, Japan)

• On the recurrence of certain time-inhomogeneous diffusion processes

Abstract: The purpose of this talk is to give a general criterion for the recurrence of time-inhomogeneous diffusion processes which are given by a transformation by a multiplicative functional from recurrent diffusion processes. Also an application to the well studied case will be concerned.

Yuichi Shiozawa (Tohoku University, Japan)

• Limit theorems for branching Markov processes: (joint work with Zhen-Qing Chen (University of Washington))

Abstract: We establish limit theorems for branching symmetric Hunt processes in terms of the principal eigenvalue and the ground state for some associated Schr\"odinger operator. Here the branching rate and the branching mechanism are state-dependent. In particular, the branching rate can be a measure belonging to a certain Kato class and is allowed to be singular with respect to the symmetrizing measure for the underlying Hunt process. The limit theorems are established under the assumption that the associated Schr\"odinger operator has a spectral gap.

Atsushi Takeuchi (Osaka City University, Japan)

• Absolute continuity for stochastic functional differential equations with jumps

Abstract: We consider stochastic functional differential equations driven by L\'evy processes, and study the absolute continuity for marginal distributions of the solution. It will be shown that there exists a probability density for the solution under non-degerenrate conditions on the coefficients of the equation. The Malliavin calculus on the Poisson space plays an important role in our framework.

Kaneharu Tsuchida (Tohoku University, Japan)

• Differentiability of spectral functions for relativistic $\alpha$-stable processes with application to large deviations

Abstract: We consider the relativistic $\alpha$-stable process, a pure jump Markov process generated by $\H^{\alpha} = (-\Delta + m^{2/\alpha})^{\alpha/2}-m$. Let $-C(\lambda)$ be the bottom of spectrum of Schr\"odinger type operator $\H^{\lambda \mu} = \H^{\alpha} - \lambda \mu$, where $\mu$ is a signed Kato measure. We prove the differentiability of $C(\lambda)$. As an application of it, we establish a large deviation principle for additive functional $A_t^{\mu}$ corresponding with the measure $\mu$.

Takahiro Tsuchiya (Ritsumeikan University, Japan)

• Term Structure Models Generated by Symmetric $\alpha$-Stable Processes

Abstract: This presentation proposes term structure models with jumps via state price density approach. By generalizing Gaussian HJM models, a generalization of Shirakawa model is obtained, while term structure models generated by multi-dimensional symmetric $\alpha$-stable processes are obtained by generalizing quadratic Gaussian models.

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