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This book presents the latest developments in the theory of gradient flows in random walk spaces. A broad framework is established for a wide variety of partial differential equations on nonlocal models and weighted graphs. Within this framework, specific gradient flows that are studied include the heat flow, the total variational flow, and evolution problems of Leray-Lions type with different types of boundary conditions. With many timely applications, this book will serve as an invaluable addition to the literature in this active area of research.
Variational and Diffusion Problems in Random Walk Spaces will be of interest to researchers at the interface between analysis, geometry, and probability, as well as to graduate students interested in exploring these areas.
Preface
Random Walks
Markov Chains
φ-Essential Irreducibility
Random Walk Spaces
Examples
The Nonlocal Gradient, Divergence and Laplace Operators
The Nonlocal Boundary, Perimeter and Mean Curvature
Poincaré-Type Inequalities
Global Poincaré-Type Inequalities
Poincaré-Type Inequalities on Subsets
The Heat Flow in Random Walk Spaces
The m-Heat Flow
Infinite Speed of Propagation
Asymptotic Behaviour
Ollivier-Ricci Curvature
The Bakry-Émery Curvature-Dimension Condition
Logarithmic-Sobolev Inequalities
Transport Inequalities
The m-Heat Content
Probabilistic Interpretation
The Spectral m-Heat Content
The Total Variation Flow in Random Walk Spaces
The m-Total Variation
The m--Laplacian and m-The Total Variation Flow
Asymptotic Behaviour
m-Cheeger and m-Calibrable Sets
The Eigenvalue Problem for - m
Isoperimetric Inequality
The m-Cheeger Constant
The m-Cheeger Constant and the m-Eigenvalues of -m
ROF-Models in Random Walk Spaces
The m-ROF Model with L-Fidelity Term
The Gradient Descent Method
The m-ROF-Model with L-Fidelity Term
The Geometric Problem
Regularity of Solutions in Terms of the NonlocalCurvature
Thresholding Parameters
The Gradient Descent Method
Least Gradient Functions in Random Walk Spaces
The Nonlocal Least Gradient Problem
Nonlocal Median Value Property
Nonlocal Poincaré Inequality
Doubly Nonlinear Nonlocal Stationary Problems of Leray-Lions Type with Nonlinear Boundary Conditions
Nonlocal Diffusion Operators of Leray-Lions Type and Nonlocal Neumann Boundary Operators
Nonlocal Stationary Problems with Neumann Boundary Conditions of Gunzburger-Lehoucq Type
Existence of Solutions of an Approximate Problem
Some Estimates on the Solutions of the Approximate Problems
Monotonicity of the Solutions of the ApproximateProblems
An Lp-Estimate for the Solutions of the Approximate Problems
Proof of the Existence Result
Neumann Boundary Conditions of Dipierro-Ros-Oton-Valdinoci Type
Doubly Nonlinear Nonlocal Diffusion Problems of Leray-Lions Type with Nonlinear Boundary Conditions
Evolution Problems with Neumann Boundary Conditions of Gunzburger-Lehoucq Type
Nonlinear Dynamical Boundary Conditions
The Evolution Problem for a Nonlocal Dirichlet-to-Neumann Operator
Doubly Nonlinear Boundary Conditions
Nonhomogeneous Boundary Conditions
Evolution Problems Under Neumann Boundary Conditions of Dipierro-Ros-Oton-Valdinoci Type
Nonlinear Semigroups
Introduction
Abstract Cauchy Problems
Mild Solutions
Accretive Operators
Existence and Uniqueness Theorem
Regularity of the Mild Solution
Completely Accretive Operators
Yosida Approximation of Maximal Monotone Graphs in RR
Bibliography
Index
Index of notations