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34
Heat kernel estimates for stablelike processes on dsets, Stochastic Process
 Appl
"... The notion of dset arises in the theory of function spaces and in fractal geometry. Geometrically selfsimilar sets are typical examples of dsets. In this paper stablelike processes on dsets are investigated, which include reflected stable processes in Euclidean domains as a special case. More ..."
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Cited by 146 (46 self)
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The notion of dset arises in the theory of function spaces and in fractal geometry. Geometrically selfsimilar sets are typical examples of dsets. In this paper stablelike processes on dsets are investigated, which include reflected stable processes in Euclidean domains as a special case. More precisely, we establish parabolic Harnack principle and derive sharp twosided heat kernel estimate for such stablelike processes. Results on the exact Hausdorff dimensions for the range of stablelike processes are also obtained.
Characterization of subGaussian heat kernel . . .
, 2000
"... SubGaussian estimates for random walks are typical of fractal graphs. We characterize them in the strongly recurrent case, in terms of resistance estimates only, without assuming elliptic Harnack inequalities. ..."
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Cited by 45 (8 self)
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SubGaussian estimates for random walks are typical of fractal graphs. We characterize them in the strongly recurrent case, in terms of resistance estimates only, without assuming elliptic Harnack inequalities.
Construction of diffusion processes on fractals, dsets, and general metric measure spaces
 J. Math. Kyoto Univ
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Heat kernels and sets with fractal structure. In Heat kernels and analysis on manifolds, graphs, and metric spaces
 of Contemp. Math
, 2003
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Bounds on volume growth of geodesic balls under Ricci flow. arXiv:1107.4262. Xiuxiong chen
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WEAK UNCERTAINTY PRINCIPLE FOR FRACTALS, GRAPHS AND METRIC MEASURE SPACES
, 2007
"... We develop a new approach to formulate and prove the weak uncertainty inequality which was recently introduced by Okoudjou and Strichartz. We assume either an appropriate measure growth condition with respect to the effective resistance metric, or, in the absence of such a metric, we assume the Poin ..."
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Cited by 7 (1 self)
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We develop a new approach to formulate and prove the weak uncertainty inequality which was recently introduced by Okoudjou and Strichartz. We assume either an appropriate measure growth condition with respect to the effective resistance metric, or, in the absence of such a metric, we assume the Poincaré inequality and reverse volume doubling property. We also consider the weak uncertainty inequality in the context of Nashtype inequalities. Our results can be applied to a wide variety of metric measure spaces, including graphs, fractals and manifolds.
Jump processes and nonlinear fractional heat equations on fractals
, 2003
"... Jump processes on metricmeasure spaces are investigated by using heat kernels. It is shown that the heat kernel corresponding to a σstable type process on a metricmeasure space decays at a polynomial rate rather than at an exponential rate as a Brownian motion. The domain of the Dirichlet form ..."
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Cited by 6 (6 self)
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Jump processes on metricmeasure spaces are investigated by using heat kernels. It is shown that the heat kernel corresponding to a σstable type process on a metricmeasure space decays at a polynomial rate rather than at an exponential rate as a Brownian motion. The domain of the Dirichlet form associated with the jump process is a SobolevSlobodeckij space, and the embedding theorems for this space are derived by using the heat kernel technique. As an application, we investigate nonlinear fractional heat equations of the form ∂u ∂t (t, x) = −(−∆)σu(t, x) + u(t, x) p with nonnegative initial values on a metricmeasure space F, and show the nonexistence of nonnegative global solution if 1 < p ≤ 1 + σβ, where α is the Hausdorff dimension of α F whilst β is the walk dimension of F.
GENERALIZED BESSEL AND RIESZ POTENTIALS ON METRIC MEASURE SPACES
"... Abstract. We introduce generalized Bessel and Riesz potentials on metric measure spaces and the corresponding potential spaces. Estimates of the Bessel and Riesz kernels are given which reflect the intrinsic structure of the spaces. Finally, we state the relationship between Bessel (or Riesz) operat ..."
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Cited by 5 (2 self)
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Abstract. We introduce generalized Bessel and Riesz potentials on metric measure spaces and the corresponding potential spaces. Estimates of the Bessel and Riesz kernels are given which reflect the intrinsic structure of the spaces. Finally, we state the relationship between Bessel (or Riesz) operators and subordinate semigroups. Contents