2025/03/20 by Peng, Xuhui, Su, Houqi · 1 citation
#35L65 #35R60 #37A50 #60H15 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2503.15828
This paper establishes the ergodicity in H^\nn,\nn=\lfloor(d)/(2)+1\rfloor of the viscous scalar conservation laws on torus \mTd with general polynomial flux and a degenerate noise. The noise could appear in as few as several directions. We introduce a localized framework that restricts attention to trajectories with controlled energy growth, circumventing the limitations of traditional contraction-based approaches. This localized method allows for a demonstration of e-property and consequently proves the uniqueness of invariant measure under a Hörmander-type condition. Furthermore, we characterize the absolute continuity of the invariant measure's projections onto any finite-dimensional subspaces under requirement on a new algebraically non-degenerate condition for the flux.