Journal Article10.1002/ZAMM.200410229
Lyapunov functions for positive linear evolution problems
TL;DR: In this paper, the existence of time monotone Lyapunov functions is not a consequence of any physical properties, but of the positivity and norm conservation of the equation.
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Abstract: We rigorously investigate the time monotonicity of Lyapunov functions for general positive linear evolution problems, including degenerate problems. This can be done by considering the problem in the convex set of probability measures and finding a general inequality for such Radon measures and Markov operators. For linear evolution problems (with discrete or continuous time), the existence of time monotone Lyapunov functions is not a consequence of any physical properties, but of the positivity and norm conservation of the equation. In some special cases the structure of such equations is given. Moreover, we describe completely the case of time constant Lyapunov functions - a property of deterministic dynamical systems.
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Citations
Modeling of drift-diffusion systems
TL;DR: In this article, the authors derived drift-diffusion systems describing transport processes starting from free energy and equilibrium solutions by a unique method, including several statistics, heterostructures and cross diffusion.
Modeling of drift-diffusion systems
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- 28 Jan 2009
TL;DR: In this article, the authors derived drift-diffusion systems describing transport processes starting from free energy and equilibrium solutions by a unique method, including several statistics, heterostructures and cross diffusion.
4
Positivity and time behavior of a linear reaction–diffusion system, non‐local in space and time
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Memory equations as reduced Markov processes
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TL;DR: In this article, a Markov process is used to model a linear memory differential equation in one dimension, where the evolution depends on the whole history, and the model can be viewed as a projection of the Markov processes living in a higher dimensional space.
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One-parameter Semigroups of Positive Operators
Wolfgang Arendt,Annette Grabosch,Günther Greiner,Ulrich Moustakas,Rainer Nagel,Ulf Schlotterbeck,Ulrich Groh,Heinrich P. Lotz,Frank Neubrander +8 more
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TL;DR: In this article, Spectral theory of positive semigroups on Co(X) and asymptotics of positive operators on Co (X) were presented. But the results on the spectral properties of the positive operators were not considered.
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•Journal Article
Fractional calculus and stable probability distributions
TL;DR: In this article, a random walk model is proposed for space fractional diffusion, where the fundamental solutions of these generalized diffusion equations are shown to provide certain probability density functions, in space or time, which are related to the relevant class of stable distributions.
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