Global ETD Search
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Showing 1 to 12 of 12 for “"Harnack inequality"”.
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Harnack inequality for nonlocal operators
Harnack inequalities are a fundamental property in both probability theory and analysis.The scale-variant Harnack inequalities play an important role in studying various properties, such as the regularity of harmonic functions in probability and analysis. This thesis focuses on scale-invariant …
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Fine properties of stochastic evolution equations and their applications
… noise. The main tool is the dimension-free Harnack inequality, which is established by using a coupling method and Girsanov transformation techniques. Subsequently, the Harnack inequality is used to derive the ergodicity, compactness and contractivity (e.g. hyperboundedness or …
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Potential Theory for Subordinate Killed Brownian Motion in Some Unbounded Domains
… harmonic functions, we also show that the Harnack inequality and boundary Harnack principle hold for all the positive harmonic functions of the subordinate killed Brownian motion in the unbounded domains mentioned above.
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Convergence of complete Ricci-βat manifolds
… second part of this thesis, we prove a matrix Harnack inequality for the Laplace equation on manifolds with suitable curvature and volume growth assumptions, which is a pointwise estimate for the integrand of the aforementioned functional. This result provides an elliptic analogue of matrix …
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Potential Theory of Generalized Hyperbolic Processes
… jumping function of Yt near zero. We prove that Harnack inequality is valid for nonnegative harmonic functions of Yt. In chapter 3, we show that the rotationally invariant generalized hyperbolic processes in a bounded C1,1 open set D can be obtained from rotationally invariant Cauchy processes in …
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Brownian Motion on Spaces with Varying Dimension
… type. However, we show that the parabolic Harnack inequality fails for such process and the measure on the underlying state space does not satisfy volume doubling property. Brownian motion on some other state spaces with varying dimensions are also studied in this thesis. For instance, we …
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Aspects Of The Ricci Flow
… flow (RF), from preserved curvature conditions, Harnack estimates, long-time existence results, to gradient Ricci solitons. Recently, Wilking [98] proved a theorem giving a simple criterion to check if a curvature condition is preserved along the RF. Using his approach, we show another criterion …
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Potential Theory for Subordinate Brownian Motion by Tempered Stable Subordinator
… some extra conditions, we can establish the Harnack inequality and boundary Harnack principle for all processes in this class in a unified way. Then the nonnegative harmonic functions of the symmetric alpha-stable processes and relativistic alpha-stable process have these properties as …
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Uniform positive recurrence and long term behavior of Markov decision processes, with applications in sensor scheduling
… examine and exploit the previously underutilized Harnack inequality for discrete Markov chains; one aim of this work was to discover how much can be accomplished for models with this property. Convergence of the value iteration algorithm is typically treated in the literature under blanket …
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Monotonicity Formulas for Diffusion Operators on Manifolds and Carnot Groups, Heat Kernel Asymptotics and Wiener's Criterion on Heisenberg-type Groups
… diffusion operators, and the curvature dimension inequality.</p> <p>Chapter 2 incorporates excerpts from a paper by N. Garofalo and the author, [42]. In it, we propose a generalization of Almgren’s frequency function <em>N</em> : (0, 1) → R for solutions to the sub-elliptic Laplace equation …
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Heat kernel estimates on glued spaces
In this thesis, we prove heat kernel estimates in two main contexts: (1) manifolds with ends with mixed Dirichlet and Neumann boundary condition and (2) infinite (countable) graphs satisfying certain properties, which we call book-like graphs. In both of these settings, we start with ''sufficiently …