Invariant measure of stochastic fractional Burgers equation with degenerate noise on a bounded interval

Yan Wang; Guanggan Chen

doi:10.3934/cpaa.2019140

Article Contents

2019, Volume 18, Issue 6: 3121-3135. Doi: 10.3934/cpaa.2019140

This issue Previous Article Molecular decomposition and a class of Fourier multipliers for bi-parameter modulation spaces Next Article Pointwise gradient estimates for subquadratic elliptic systems with discontinuous coefficients

Invariant measure of stochastic fractional Burgers equation with degenerate noise on a bounded interval

Yan Wang and
Guanggan Chen^,

School of Mathematical Science, and V.C. & V.R. Key Lab, Sichuan Normal University, Chengdu 610068, China

^* Corresponding author
^* Corresponding author

Received: October 31, 2018

Revised: December 31, 2018

Published: May 2019

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Abstract

This work is concerned with the invariant measure of a stochastic fractional Burgers equation with degenerate noise on one dimensional bounded domain. Due to the disturbance and influence of the fractional Laplacian operator on a bounded interval interacting with the degenerate noise, the study of the system becomes more complicated. In order to get over the difficulties caused by the fractional Laplacian operator, the usual Hilbert space does not fit the system, we introduce an appropriate weighted space to study it. Meanwhile, we apply the asymptotically strong Feller property instead of the usually strong Feller property to overcome the trouble caused by the degenerate noise, the corresponding Malliavin operator is not invertible. We finally derive the uniqueness of the invariant measure which further implies the ergodicity of the stochastic system.

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Mathematics Subject Classification: Primary: 37L55, 37L40, 37A25; Secondary: 60A10, 35Q53.

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