Reflected (Degenerate) Diffusions and Stationary Measures

2020 ◽  
pp. 3-35
Author(s):  
Mauricio Duarte
Keyword(s):  
Stationary Measures ◽  
Degenerate Diffusions

Convergence to stationary measures in nonlinear Fokker—Planck—Kolmogorov equations

2018 ◽  
Vol 482 (4) ◽  
pp. 369-374
Author(s):  
V. Bogachev ◽  
◽  
M. Roeckner ◽  
S. Shaposhnikov ◽  
◽  
...  
Keyword(s):  
Kolmogorov Equations ◽  
Stationary Measures ◽  
Fokker Planck

scholarly journals Ergodic control of degenerate diffusions

1997 ◽  
Vol 15 (1) ◽  
pp. 1-17 ◽  
Author(s):  
Gopal K. Basak ◽  
Vevek S. Borkar ◽  
Mrinal K. Ghosh
Keyword(s):  
Ergodic Control ◽  
Degenerate Diffusions

scholarly journals Existence of ground states for aggregation-diffusion equations

2019 ◽  
Vol 17 (03) ◽  
pp. 393-423 ◽  
Author(s):  
J. A. Carrillo ◽  
M. G. Delgadino ◽  
F. S. Patacchini
Keyword(s):  
Free Energy ◽  
Ground States ◽  
Energy Functional ◽  
Multi Agent Systems ◽  
Linear Diffusion ◽  
Sharp Condition ◽  
Macroscopic Models ◽  
Energy Functionals ◽  
Continuous Description ◽  
Degenerate Diffusions

We analyze free energy functionals for macroscopic models of multi-agent systems interacting via pairwise attractive forces and localized repulsion. The repulsion at the level of the continuous description is modeled by pressure-related terms in the functional making it energetically favorable to spread, while the attraction is modeled through nonlocal forces. We give conditions on general entropies and interaction potentials for which neither ground states nor local minimizers exist. We show that these results are sharp for homogeneous functionals with entropies leading to degenerate diffusions while they are not sharp for fast diffusions. The particular relevant case of linear diffusion is totally clarified giving a sharp condition on the interaction potential under which the corresponding free energy functional has ground states or not.

The non-uniform stationary measure for discrete-time quantum walks in one dimension

2015 ◽  
Vol 15 (11&12) ◽  
pp. 1060-1075
Author(s):  
Norio Konno ◽  
Masato Takei
Keyword(s):  
Eigenvalue Problem ◽  
Discrete Time ◽  
Quantum Walks ◽  
Stationary Measure ◽  
Unitary Matrix ◽  
Uniform Measure ◽  
Simple Argument ◽  
Stationary Measures ◽  
Time Quantum ◽  
The One

We consider stationary measures of the one-dimensional discrete-time quantum walks (QWs) with two chiralities, which is defined by a 2 $\times$ 2 unitary matrix $U$. In our previous paper \cite{Konno2014}, we proved that any uniform measure becomes the stationary measure of the QW by solving the corresponding eigenvalue problem. This paper reports that non-uniform measures are also stationary measures of the QW except when $U$ is diagonal. For diagonal matrices, we show that any stationary measure is uniform. Moreover, we prove that any uniform measure becomes a stationary measure for more general QWs not by solving the eigenvalue problem but by a simple argument.

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