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20
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 4 (4 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.
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 3 (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.
POTENTIAL SPACES ON FRACTALS
"... Abstract. We introduce potential spaces on fractal metric spaces, investigate their embedding theorems, and derive various Besov spaces. Our starting point is that there exists a local, stochastically complete heat kernel satisfying a twosided estimate on the fractal considered. The results of this ..."
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Cited by 1 (0 self)
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Abstract. We introduce potential spaces on fractal metric spaces, investigate their embedding theorems, and derive various Besov spaces. Our starting point is that there exists a local, stochastically complete heat kernel satisfying a twosided estimate on the fractal considered. The results of this paper are among the marvelous consequences of the heat kernel on the fractal. 1.
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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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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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