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In this paper, we establish the coincidence of two classes of L p -Kato class measures in the framework of symmetric Markov processes admitting upper and lower estimates of heat kernel under mild conditions. One class of L p -Kato class measures is defined by the pth power of positive order resolvent kernel, another is defined in terms of the pth power of Green kernel depending on some exponents related to the heat kernel estimates. We also prove that functions u such that sup x ∈ E ∫ B 1 (x) | u | q d m < ∞ are of L p -Kato class if q is greater than a constant related to p and the constants appeared in the upper and lower estimates of the heat kernel. These are complete extensions of some results by Aizenman–Simon and the recent results by the second named author in the framework of Brownian motions on Euclidean space. We further give necessary and sufficient conditions for a Radon measure with Ahlfors regularity to belong to L p -Kato class. Our results can be applicable to many examples, for instance, symmetric (relativistic) stable processes, jump processes on d-sets, Brownian motions on Riemannian manifolds, diffusions on fractals and so on. [ABSTRACT FROM AUTHOR]

Under mild conditions on measures used in the perturbation, we establish the L p -independence of spectral radius for generalized Feynman–Kac semigroups without assuming the irreducibility and the boundedness of the function appeared in the continuous additive functionals locally of zero energy in the framework of symmetric Markov processes. These results are obtained by using the gaugeability approach developed by the first named author as well as the recent progress on the irreducible decomposition for Markov processes proved by the third author and on the analytic characterizations of gaugeability for generalized Feynman–Kac functionals developed by the second and third authors. [ABSTRACT FROM AUTHOR]

We give a general analytic characterization of the gaugeability for generalized Feynman-Kac functionals involving continous additive functionals locally of zero energy generalizing the earlier work on the analytic characterization of the gaugeability of Feynman-Kac functional for absorbing Brownian motion on a bounded domain given by Aizenman-Simon (Commun Pure Appl Math 35(2), 209-273, 1982). The result also extends the previous known results on the analytic characterization of the gaugeability for generalized Feynman-Kac functionals in the framework of symmetric Markov processes satisfying irreducibility condition and absolute continuity condition. [ABSTRACT FROM AUTHOR]

We prove a Jensen's inequality on $$p$$-uniformly convex space in terms of $$p$$-barycenters of probability measures with $$(p-1)$$-th moment with $$p\in ]1,\infty [$$ under a geometric condition, which extends the results in Kuwae (Jensen's inequality over CAT $$(\kappa )$$-space with small diameter. In: Proceedings of Potential Theory and Stochastics, Albac Romania, pp. 173-182. Theta Series in Advanced Mathematics, vol. 14. Theta, Bucharest, ) , Eells and Fuglede (Harmonic maps between Riemannian polyhedra. In: Cambridge Tracts in Mathematics, vol. 142. Cambridge University Press, Cambridge, ) and Sturm (Probability measures on metric spaces of nonpositive curvature. Probability measures on metric spaces of nonpositive curvature. In: Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002), pp. 357-390. Contemporary Mathematics, vol. 338. American Mathematical Society, Providence, ). As an application, we give a Liouville's theorem for harmonic maps described by Markov chains into $$2$$-uniformly convex space satisfying such a geometric condition. An alternative proof of the Jensen's inequality over Banach spaces is also presented. [ABSTRACT FROM AUTHOR]

We give a stochastic proof of an extension of E. Calabi's strong maximum principle under some geometric conditions in the framework of strong Feller diffusion processes associated to local regular semi-Dirichlet forms with lower bounds. As a corollary, our notion of subharmonicity implies a notion of viscosity subsolution in a stochastic sense. We can apply our result to singular geometric object like Alexandrov space, limit space under spectral distance of Riemannian manifolds with uniform lower Ricci curvature bound and so on. [ABSTRACT FROM AUTHOR]

Reflected Dirichlet space for quasi-regular Dirichlet forms is presented in this paper. We show the closedness of the (active) reflected Dirichlet forms without using the first definition of reflected Dirichlet space by Silverstein and the characterization by Chen. As an application of the closedness, the closability of distorted forms are discussed. We also show the maximality of (active) reflected Dirichlet space in the class of Silverstein's extensions and consider the uniqueness problem. Only the techniques of the transfer method and the change of underlying measures are used. [ABSTRACT FROM AUTHOR]

We prove the compactness of the imbedding of the Sobolev space $W^{1,2}_0(\Omega)$ into $L^2(\Omega)$ for any relatively compact open subset $\Omega$ of an Alexandrov space. As a corollary, the generator induced from the Dirichlet (energy) form has discrete spectrum. The generator can be approximated by the Laplacian induced from the DC-structure on the Alexandrov space. We also prove the existence of the locally Hölder continuous heat kernel. [ABSTRACT FROM AUTHOR]

We construct Sobolev spaces and energy functionals over maps between metric spaces under the strong measure contraction property of Bishop–Gromov type, which is a generalized notion of Ricci curvature bounded below. We also present the notion of generalized measure contraction property, which gives a characterization of energies by approximating energies of Sturm type over Lipschitz maps. [ABSTRACT FROM AUTHOR]

Summary. In this paper, we show the convergence of forms in the sense of Mosco associated with the part form on relatively compact open set of Dirichlet forms with locally uniform ellipticity and the locally uniform boundedness of ground states under regular Dirichlet space setting. We also get the same assertion under Dirichlet space in infinite dimensional setting. As a result of this, we get the weak convergence under some conditions on initial distributions and the growth order of the volume of the balls defined by (modified) pseudo metric used in K. Th. Sturm. [ABSTRACT FROM AUTHOR]