STOCHASTIC POROUS MEDIA EQUATION DRIVEN BY FRACTIONAL BROWNIAN MOTION

2013 ◽  
Vol 13 (04) ◽  
pp. 1350010 ◽  
Author(s):  
JAN BÁRTEK ◽  
MARÍA J. GARRIDO-ATIENZA ◽  
BOHDAN MASLOWSKI
Keyword(s):  
Porous Media ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Fractional Derivatives ◽  
Multiplicative Noise ◽  
Stochastic Equation ◽  
Random Dynamical System ◽  
Porous Media Equation ◽  
Media Equation ◽  
Long Time

The present work deals with stochastic porous media equation with multiplicative noise, driven by fractional Brownian motion B(H) with Hurst index H > 1/2. The stochastic integral with integrator B(H) is defined pathwise following the theory developed by Zähle [24], based on the so-called fractional derivatives. It is shown that there is a one-to-one correspondence between solutions to the stochastic equation and solutions to its deterministic counterpart. By means of this correspondence and exploiting properties of the deterministic porous media equation, the existence, uniqueness, regularity and long-time properties of the solution is established. We also prove that the solution forms a random dynamical system in an appropriate function space.

scholarly journals On a singular incompressible porous media equation

2012 ◽  
Vol 53 (11) ◽  
pp. 115602 ◽  
Author(s):  
Susan Friedlander ◽  
Francisco Gancedo ◽  
Weiran Sun ◽  
Vlad Vicol
Keyword(s):  
Porous Media ◽  
Porous Media Equation ◽  
Media Equation

Global random attractor of stochastic hydrodynamical equation for the Heisenberg paramagnet with multiplicative noise on ℝd

2015 ◽  
Vol 16 (01) ◽  
pp. 1650007 ◽  
Author(s):  
Yanfeng Guo ◽  
Chunxiao Guo ◽  
Yongqian Han
Keyword(s):  
Dynamical System ◽  
Multiplicative Noise ◽  
Stochastic Equation ◽  
Random Coefficients ◽  
Random Attractor ◽  
Random Dynamical System ◽  
Hydrodynamical Equation ◽  
Priori Estimates ◽  
Heisenberg Paramagnet ◽  
Truncation Function

The stochastic hydrodynamical equation for the Heisenberg paramagnet with multiplicative noise defined on the entire [Formula: see text] is mainly investigated. The global random attractor for the random dynamical system associated with the equation is obtained. The method is to transform the stochastic equation into the corresponding partial differential equations with random coefficients by Ornstein–Uhlenbeck process. The uniform priori estimates for far-field values of solutions have been studied via a truncation function, and then the asymptotic compactness of the random dynamical system is established.

FRACTIONAL BROWNIAN MOTION WITH STOCHASTIC VARIANCE: MODELING ABSOLUTE RETURNS IN STOCK MARKETS

2008 ◽  
Vol 19 (08) ◽  
pp. 1221-1242 ◽  
Author(s):  
H. E. ROMAN ◽  
M. PORTO
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Stock Markets ◽  
Distribution Functions ◽  
Conditional Heteroskedasticity ◽  
Arch Models ◽  
Variance Modeling ◽  
Probability Distribution Functions ◽  
Second Moments ◽  
Long Time

We discuss a model for simulating a long-time memory in time series characterized in addition by a stochastic variance. The model is based on a combination of fractional Brownian motion (FBM) concepts, for dealing with the long-time memory, with an autoregressive scheme with conditional heteroskedasticity (ARCH), responsible for the stochastic variance of the series, and is denoted as FBMARCH. Unlike well-known fractionally integrated autoregressive models, FBMARCH admits finite second moments. The resulting probability distribution functions have power-law tails with exponents similar to ARCH models. This idea is applied to the description of long-time autocorrelations of absolute returns ubiquitously observed in stock markets.

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