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

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.

Exponential mixing of 2D stochastic Ginzburg–Landau–Newell equations driven by degenerate noise

2019 ◽  
Vol 20 (01) ◽  
pp. 2050004
Author(s):  
Yan Zheng ◽  
Jianhua Huang
Keyword(s):  
Dynamical System ◽  
Weak Form ◽  
Markov Semigroup ◽  
Strong Type ◽  
Stochastic Forcing ◽  
Ginzburg Landau ◽  
Strong Feller Property ◽  
Feller Property ◽  
Degenerate Noise ◽  
Strong Feller

The current paper is devoted to stochastic Ginzburg–Landau–Newell equation with degenerate stochastic forcing. First, we establish a type of gradient inequality, which is also essential to proving asymptotic strong Feller property. Then, we prove that the corresponding dynamical system possesses a strong type of Lyapunov structure and is of a relatively weak form of irreducibility. Finally, we prove that the corresponding Markov semigroup possesses a unique, exponentially mixing invariant measure.

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