scholarly journals Mixing and average mixing times for general Markov processes

2020 ◽  
pp. 1-12
Author(s):  
Robert M. Anderson ◽  
Haosui Duanmu ◽  
Aaron Smith
Keyword(s):  
Markov Chains ◽  
Mixing Times ◽  
State Spaces ◽  
Multiplicative Constant ◽  
Strong Feller Property ◽  
Feller Property ◽  
Reversible Markov Chains ◽  
Finite State ◽  
Compact State ◽  
Strong Feller

Abstract Yuval Peres and Perla Sousi showed that the mixing times and average mixing times of reversible Markov chains on finite state spaces are equal up to some universal multiplicative constant. We use tools from nonstandard analysis to extend this result to reversible Markov chains on compact state spaces that satisfy the strong Feller property.

scholarly journals REFLECTED BROWNIAN MOTION ON SIMPLE NESTED FRACTALS

Fractals ◽  
2019 ◽  
Vol 27 (06) ◽  
pp. 1950104
Author(s):  
KAMIL KALETA ◽  
MARIUSZ OLSZEWSKI ◽  
KATARZYNA PIETRUSKA-PAŁUBA
Keyword(s):  
Brownian Motion ◽  
Transition Probability ◽  
Strong Feller Property ◽  
Feller Property ◽  
Reflected Diffusion ◽  
Geometric Conditions ◽  
Regularity Properties ◽  
Probability Densities ◽  
Free Brownian Motion ◽  
Strong Feller

For a large class of planar simple nested fractals, we prove the existence of the reflected diffusion on a complex of an arbitrary size. Such a process is obtained as a folding projection of the free Brownian motion from the unbounded fractal. We give sharp necessary geometric conditions for the fractal under which this projection can be well defined, and illustrate them by numerous examples. We then construct a proper version of the transition probability densities for the reflected process and we prove that it is a continuous, bounded and symmetric function which satisfies the Chapman–Kolmogorov equations. These provide us with further regularity properties of the reflected process such us Markov, Feller and strong Feller property.

Probabilistic approach to the strong Feller property

2000 ◽  
Vol 118 (2) ◽  
pp. 187-210 ◽  
Author(s):  
Bohdan Maslowski ◽  
Jan Seidler
Keyword(s):  
Probabilistic Approach ◽  
Strong Feller Property ◽  
Feller Property ◽  
Strong Feller

Geometric inequalities for the eigenvalues of concentrated Markov chains

2000 ◽  
Vol 37 (01) ◽  
pp. 15-28 ◽  
Author(s):  
Olivier François
Keyword(s):  
Markov Chains ◽  
Geometric Inequalities ◽  
State Spaces ◽  
Absolute Value ◽  
Minimization Problems ◽  
Isoperimetric Constant ◽  
Finite State ◽  
Eigenvalue Comparison ◽  
Second Eigenvalue ◽  
Second Largest Eigenvalue

This article describes new estimates for the second largest eigenvalue in absolute value of reversible and ergodic Markov chains on finite state spaces. These estimates apply when the stationary distribution assigns a probability higher than 0.702 to some given state of the chain. Geometric tools are used. The bounds mainly involve the isoperimetric constant of the chain, and hence generalize famous results obtained for the second eigenvalue. Comparison estimates are also established, using the isoperimetric constant of a reference chain. These results apply to the Metropolis-Hastings algorithm in order to solve minimization problems, when the probability of obtaining the solution from the algorithm can be chosen beforehand. For these dynamics, robust bounds are obtained at moderate levels of concentration.

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