scholarly journals Harnack inequality and strong Feller property for stochastic fast-diffusion equations

2008 ◽  
Vol 342 (1) ◽  
pp. 651-662 ◽  
Author(s):  
Wei Liu ◽  
Feng-Yu Wang
Keyword(s):  
Harnack Inequality ◽  
Diffusion Equations ◽  
Fast Diffusion ◽  
Strong Feller Property ◽  
Feller Property ◽  
Strong Feller ◽  
Fast Diffusion Equations

scholarly journals LOG-HARNACK INEQUALITY FOR STOCHASTIC DIFFERENTIAL EQUATIONS IN HILBERT SPACES AND ITS CONSEQUENCES

2010 ◽  
Vol 13 (01) ◽  
pp. 27-37 ◽  
Author(s):  
MICHAEL RÖCKNER ◽  
FENG-YU WANG
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Cost Function ◽  
Harnack Inequality ◽  
Hilbert Spaces ◽  
Additive Noise ◽  
Strong Feller Property ◽  
Feller Property ◽  
Logarithmic Type ◽  
Strong Feller

A logarithmic type Harnack inequality is established for the semigroup of solutions to a stochastic differential equation in Hilbert spaces with non-additive noise. As applications, the strong Feller property as well as the entropy-cost inequality for the semigroup are derived with respect to the corresponding distance (cost function).

Poincaré inequalities for linearizations of very fast diffusion equations

Nonlinearity ◽  
2002 ◽  
Vol 15 (3) ◽  
pp. 565-580 ◽  
Author(s):  
J A Carrillo ◽  
C Lederman ◽  
P A Markowich ◽  
G Toscani
Keyword(s):  
Diffusion Equations ◽  
Fast Diffusion ◽  
Poincaré Inequalities ◽  
Poincare Inequalities ◽  
Fast Diffusion Equations

Extinction behaviour for fast diffusion equations with absorption

2001 ◽  
Vol 43 (8) ◽  
pp. 943-985 ◽  
Author(s):  
Raúl Ferreira ◽  
Juan Luis Vazquez
Keyword(s):  
Diffusion Equations ◽  
Fast Diffusion ◽  
Fast Diffusion Equations

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.

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