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31
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 108 (42 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 32 (4 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
 Contemporary Math
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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 7 (7 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.
ESTIMATES OF HEAT KERNELS FOR NONLOCAL REGULAR DIRICHLET FORMS
"... Abstract. In this paper we present new heat kernel upper bounds for a certain class of nonlocal regular Dirichlet forms on metric measure spaces, including fractal spaces. We use a new purely analytic method where one of the main tools is the parabolic maximum principle. We deduce offdiagonal uppe ..."
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Cited by 3 (1 self)
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Abstract. In this paper we present new heat kernel upper bounds for a certain class of nonlocal regular Dirichlet forms on metric measure spaces, including fractal spaces. We use a new purely analytic method where one of the main tools is the parabolic maximum principle. We deduce offdiagonal upper bound of the heat kernel from the ondiagonal one under the volume regularity hypothesis, restriction of the jump kernel and the survival hypothesis. As an application, we obtain twosided estimates of heat kernels for nonlocal regular Dirichlet forms with finite effective resistance, including settings with the walk dimension greater than 2. Contents
Domains of Dirichlet forms and effective resistance estimates on p.c.f. fractals
 Studia Math
"... Abstract. In this paper we consider postcritically finite selfsimilar fractals with regular harmonic structures. We first obtain effective resistance estimates in terms of the Euclidean metric, which particularly imply the embedding theorem for the domains of the Dirichlet forms associated with th ..."
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Cited by 3 (2 self)
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Abstract. In this paper we consider postcritically finite selfsimilar fractals with regular harmonic structures. We first obtain effective resistance estimates in terms of the Euclidean metric, which particularly imply the embedding theorem for the domains of the Dirichlet forms associated with the harmonic structures. We then characterize the domains of the Dirichlet forms. 1.
On some second order transmission problems
 Arab. J. Sci. Eng. Sect. C Theme Issues
"... In this paper we focus our attention on a model problem, considered in [1]. This is a secondorder transmission problem with a “flat ” smooth layer, formally stated as ..."
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Cited by 3 (1 self)
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In this paper we focus our attention on a model problem, considered in [1]. This is a secondorder transmission problem with a “flat ” smooth layer, formally stated as