Conference schedule
30 th/AUG.
Gaugeability with Applications to Symmetric $\alpha$-Stable Processes
see Abstract : dvi-file or pdf-file
Stochastic differential equations with non-Lipschitz coefficients.
Semi-self-similar Levy processes on the $p$-adics
The field of $p$-adic numbers is totally disconnected and any stochastic process thereon is of purely jump type. We will discuss on properties of (non-symmetric) semi-self -similar Levy processes on the $p$-adics and limit theorems for them.
Supprot theorem of jump type process and related topics
On $L^p$-semigroups: Poincar\'e inequallity and transience (joint work with N. Jacob)
Criticality of Schrodinger type operators and differentiability of spectral functions
Let $\mu$ be a signed Radon measure in the Kato class and ${\cal H}^{\mu}$ a Schrodinger type operators, ${\cal H}^{\mu}=(-\Delta)^{\alpha/2} -\mu$. When ${\cal H}^{\mu}$ is critical, we give a new method for construction an ${\cal H}^{\mu}$-harmonic function. Using an $h$-transform defined by the harmonic function, we show the differentiability of spectral functions for symmetric $\alpha$-stable processes.
Principal eigenvalues for time changed processes of one-dimensional $\alpha$-stable processes
In this talk, we calculate principal eigenvalues for time changed processes of absorbing $\alpha$-stable processes in one dimension. As its application, we give a necessary ans sufficient condition for some measures being gaugeable.
31 st/AUG.
Heat Kernel estimates for jump-type processes on $d$-sets
Potential theory of geometric stable processes
Perturbation of symmetric Markov processes (joint work with Z.-Q.Chen, P.J. Fitzsimmons and T.-S. Zhang)
We provide a path-space integral representation of the semigroup associated with the quadratic form obtained by lower order perturbation of a symmetric Dirichlet form including jump and killing terms. The representation is a combination of Feynman-Kac and Girsanov formula, and extends the all previous known results in the framework of symmetric Markov processes. We also provide a Feynman-Kac formula for Nakao's divergence-like CAF.
$(r,p)$-capacities associated with Dirichlet spaces on a local field.
On a local field, one can find stochastic process whose infinitesimal generator is described as some r-th order derivative. Even non-linear potential theoretic features could be discussed based on Dirichlet space theory.
Integrability of exit times (joint work with Rodrig Banuelos)
see Abstract : dvi-file or pdf-file
Feller properties of some classes of jump-type symmetric Markov processes
We show the Feller property for a jump-type symmetric Markov process. In order to show them, we first give the generator of the corresponding Dirichlet form of the process in a precise form and estimate the exit times from balls of the process. Following the ideas of Bass-Levin (see also Bass-Kassman, Song-Vondrachek...) , we show the Holder continuities of the harmonic funcitions. Then using this reguralities, we are able to show the Feller property.
1 st/SEP.
Poisson point processes attached to symmetric Markov processes (joint work with Hiroshi Tanaka)
For a symmetric Markov process accessible to a specific one point of the state space, we give its decomposition and construction in terms of the absorbed process and Poisson point process of excursions at the point, completing Ito's programme under symmetry.
Skew-Brownian flow: Ray-Knight-type theorem and lenses (joint work with M. Barlow, H. Kaspi, A. Mandelbaum and Z. Chen)
Skew-Brownian flow is a family of skew-Brownian motions driven by the same white noise. A version of the classical Ray-Knight theorem holds for this flow. The family of local times for the flow is a Markov process with jumps and no diffusion part. Lenses are parts of the flow where two processes in the flow start from the same point, diverge, and then coalesce. An excursion theory for a family of lenses can be constructed. Several Markov processes with no diffusion part arise naturally in this context.
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