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Construction of diffusion processes on fractals, dsets, and general metric measure spaces
, 2003
"... We give a su#cient condition to construct nontrivial symmetric di#usion processes on a locally compact metric measure space (M, #, ). These processes are associated with local regular Dirichlet forms which are obtained as continuous parts of #limits for approximating nonlocal Dirichlet forms. Fo ..."
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Cited by 10 (3 self)
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We give a su#cient condition to construct nontrivial symmetric di#usion processes on a locally compact metric measure space (M, #, ). These processes are associated with local regular Dirichlet forms which are obtained as continuous parts of #limits for approximating nonlocal Dirichlet forms. For various fractals, we can use existing estimates to verify our assumptions. This shows that our general method of constructing di#usions can be applied to these fractals.
Jump processes and nonlinear fractional heat equations on fractals
 Math. Nachr
"... Abstract. 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 Dirichl ..."
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Cited by 3 (3 self)
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Abstract. 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
"... Abstract. 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 assum ..."
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Cited by 1 (0 self)
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Abstract. 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. Contents
unknown title
, 2005
"... www.elsevier.com/locate/jfa Equivalence conditions for ondiagonal upper bounds of heat kernels on selfsimilar spaces ✩ ..."
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www.elsevier.com/locate/jfa Equivalence conditions for ondiagonal upper bounds of heat kernels on selfsimilar spaces ✩
Obtaining Upper . . . From Lower Bounds
, 2007
"... We show that a neardiagonal lower bound of the heat kernel of a Dirichlet form on a metric measure space with a regular measure implies an ondiagonal upper bound. If in addition the Dirichlet form is local and regular, then we obtain a full offdiagonal upper bound of the heat kernel provided the ..."
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We show that a neardiagonal lower bound of the heat kernel of a Dirichlet form on a metric measure space with a regular measure implies an ondiagonal upper bound. If in addition the Dirichlet form is local and regular, then we obtain a full offdiagonal upper bound of the heat kernel provided the Dirichlet heat kernel on any ball satisfies a neardiagonal lower estimate. This reveals a new phenomenon in the relationship between the lower and upper bounds of the heat kernel.
Probabilistic characterisation of BesovLipschitz spaces on metric measure spaces ∗
, 810
"... We give a probabilistic characterisation of the BesovLipschitz spaces Lip(α, p, q)(X) on domains which support a Markovian kernel with appropriate exponential bounds. This extends former results of [11, 14, 15, 7] which were valid for α = dw 2, p = 2, q = ∞, where dw is the walk dimension of the sp ..."
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We give a probabilistic characterisation of the BesovLipschitz spaces Lip(α, p, q)(X) on domains which support a Markovian kernel with appropriate exponential bounds. This extends former results of [11, 14, 15, 7] which were valid for α = dw 2, p = 2, q = ∞, where dw is the walk dimension of the space X. 1
Inhomogeneous parabolic equations on unbounded metric measure spaces
, 2011
"... We study the inhomogeneous semilinear parabolic equation ut =∆u+ u p + f(x), with source term f independent of time and subject to f(x) � 0 and with u(0,x)=ϕ(x) � 0, for the very general setting of a metric measure space. By establishing Harnacktype inequalities in time t and some powerful estima ..."
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We study the inhomogeneous semilinear parabolic equation ut =∆u+ u p + f(x), with source term f independent of time and subject to f(x) � 0 and with u(0,x)=ϕ(x) � 0, for the very general setting of a metric measure space. By establishing Harnacktype inequalities in time t and some powerful estimates, we give sufficient conditions for nonexistence, local existence and global existence of weak solutions, depending on the value of p relative to a critical exponent. 1.