Titlebook: Dirichlet Forms and Related Topics; In Honor of Masatosh Zhen-Qing Chen,Masayoshi Takeda,Toshihiro Uemura Conference proceedings 2022 The E

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On Non-negative Solutions to Space-Time Partial Differential Equations of Higher Order,We discuss when certain higher order partial differential operators in space and time admit non-negative solutions which have a semigroup representation as well as a representation by some associated Markov process.

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Two-Sided Heat Kernel Estimates for Symmetric Diffusion Processes with Jumps: Recent Results, processes having both diffusive and jumping components in metric measure spaces. We summarize some of the main results obtained recently in [., .] and provide several examples. We also discuss new ideas of the proof for the off-diagonal upper bounds of transition densities which are based on a generalized Davies’ method developed in [.].

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,Monotonicity Properties of Regenerative Sets and Lorden’s Inequality,esult in the context of regenerative sets, and remove the factor of 2 when the regenerative set enjoys a certain monotonicity property. This property occurs precisely when the Lévy exponent of the associated subordinator is a . function. Several equivalent stochastic monotonicity properties of such a regenerative set are demonstrated.

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,On , Liouville Theorems for Dirichlet Forms, domain of the . generator. Secondly we prove analogues of Yau’s and Karp’s Liouville theorems for weakly harmonic functions. Both say that weakly harmonic functions which satisfy certain . growth criteria must be constant. As consequence we give an integral criterion for recurrence.

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,Scattering Lengths for Additive Functionals and Their Semi-classical Asymptotics,sarily continuous. After giving a systematic presentation of the fundamentals of the scattering length, we study the problems of semi-classical asymptotics for scattering length under relativistic stable processes, which extend previous results for the case of positive continuous additive functionals.

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