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1. CJM 2013 (vol 66 pp. 641)

Grigor'yan, Alexander; Hu, Jiaxin
 Heat Kernels and Green Functions on Metric Measure Spaces We prove that, in a setting of local Dirichlet forms on metric measure spaces, a two-sided sub-Gaussian estimate of the heat kernel is equivalent to the conjunction of the volume doubling propety, the elliptic Harnack inequality and a certain estimate of the capacity between concentric balls. The main technical tool is the equivalence between the capacity estimate and the estimate of a mean exit time in a ball, that uses two-sided estimates of a Green function in a ball. Keywords:Dirichlet form, heat kernel, Green function, capacityCategories:35K08, 28A80, 31B05, 35J08, 46E35, 47D07

2. CJM 2011 (vol 64 pp. 869)

Hu, Ze-Chun; Sun, Wei
 Balayage of Semi-Dirichlet Forms In this paper we study the balayage of semi-Dirichlet forms. We present new results on balayaged functions and balayaged measures of semi-Dirichlet forms. Some of the results are new even in the Dirichlet forms setting. Keywords:balayage, semi-Dirichlet form, potential theoryCategories:31C25, 60J45

3. CJM 2009 (vol 61 pp. 534)

Chen, Chuan-Zhong; Sun, Wei
 Girsanov Transformations for Non-Symmetric Diffusions Let $X$ be a diffusion process, which is assumed to be associated with a (non-symmetric) strongly local Dirichlet form $(\mathcal{E},\mathcal{D}(\mathcal{E}))$ on $L^2(E;m)$. For $u\in{\mathcal{D}}({\mathcal{E}})_e$, the extended Dirichlet space, we investigate some properties of the Girsanov transformed process $Y$ of $X$. First, let $\widehat{X}$ be the dual process of $X$ and $\widehat{Y}$ the Girsanov transformed process of $\widehat{X}$. We give a necessary and sufficient condition for $(Y,\widehat{Y})$ to be in duality with respect to the measure $e^{2u}m$. We also construct a counterexample, which shows that this condition may not be satisfied and hence $(Y,\widehat{Y})$ may not be dual processes. Then we present a sufficient condition under which $Y$ is associated with a semi-Dirichlet form. Moreover, we give an explicit representation of the semi-Dirichlet form. Keywords:Diffusion, non-symmetric Dirichlet form, Girsanov transformation, $h$-transformation, perturbation of Dirichlet form, generalized Feynman-Kac semigroupCategories:60J45, 31C25, 60J57

4. CJM 2008 (vol 60 pp. 822)

Kuwae, Kazuhiro
 Maximum Principles for Subharmonic Functions Via Local Semi-Dirichlet Forms Maximum principles for subharmonic functions in the framework of quasi-regular local semi-Dirichlet forms admitting lower bounds are presented. As applications, we give weak and strong maximum principles for (local) subsolutions of a second order elliptic differential operator on the domain of Euclidean space under conditions on coefficients, which partially generalize the results by Stampacchia. Keywords:positivity preserving form, semi-Dirichlet form, Dirichlet form, subharmonic functions, superharmonic functions, harmonic functions, weak maximum principle, strong maximum principle, irreducibility, absolute continuity conditionCategories:31C25, 35B50, 60J45, 35J, 53C, 58

5. CJM 2008 (vol 60 pp. 457)

Teplyaev, Alexander
 Harmonic Coordinates on Fractals with Finitely Ramified Cell Structure We define sets with finitely ramified cell structure, which are generalizations of post-crit8cally finite self-similar sets introduced by Kigami and of fractafolds introduced by Strichartz. In general, we do not assume even local self-similarity, and allow countably many cells connected at each junction point. In particular, we consider post-critically infinite fractals. We prove that if Kigami's resistance form satisfies certain assumptions, then there exists a weak Riemannian metric such that the energy can be expressed as the integral of the norm squared of a weak gradient with respect to an energy measure. Furthermore, we prove that if such a set can be homeomorphically represented in harmonic coordinates, then for smooth functions the weak gradient can be replaced by the usual gradient. We also prove a simple formula for the energy measure Laplacian in harmonic coordinates. Keywords:fractals, self-similarity, energy, resistance, Dirichlet forms, diffusions, quantum graphs, generalized Riemannian metricCategories:28A80, 31C25, 53B99, 58J65, 60J60, 60G18