Invited Session 05：奇异随机偏微分方程
41
with parameters ? ∈ (0,1/2) and ? ∈ (−
1
2
+ ?,
1
2
+ ?).
We study the sample path properties of GFBM X and establish the exact uniform
modulus of continuity, small ball probabilities, and Chung's laws of iterated logarithm. Our
results show that the local regularity properties away from the origin and fractal properties
of GFBM ? are determined by the index ? +
1
2
, instead of the self-similarity index ?. This
talk is based on joint works with Prof. Yimin Xiao.
A class of supercritical/critical singular stochastic PDEs:
existence, non-uniqueness, non-Gaussianity, non-unique ergodicity
Xiangchan Zhu, 朱湘禅
AMSS, Chinese Academy of Sciences zhuxiangchan@126.com
Abstract
We study the surface quasi-geostrophic equation with an irregular spatial perturbation
??? + ? ∙ ?? = −?(−∆)
?/2? + ?， ? = ?
⊥(−∆)
−1?,
on [0, ∞) × ?
2
, with ? ∈ [0,1] , ? ∈ [0,3/2) and ? ∈ ?∞,∞
−2+?
(?
2
) for some κ > 0 . This
covers the case of ? = (−∆)
?/2
? for ? < 1 and ? a spatial white noise on ?
2
.
Depending on the relation between ? and ? , our setting is subcritical, critical or
supercritical in the language of Hairer's regularity structures [Hai14]. Based on purely
analytical tools from convex integration and without the need of any probabilistic arguments
including renormalization, we prove existence of infinitely many analytically weak solutions
in ????
?
(0, ∞; ?∞,1
−1/2
) ∩ ??([0, ∞); ?∞,1
−1/2−?
) ∩ ??([0, ∞); ?∞,1
−3/2−?
) for all ? ∈ [1, ∞)
and ? > 0. We are able to prescribe an initial as well as a terminal condition at a finite time
? > 0 , and to construct steady state, i.e. time independent, solutions. In all cases, the
solutions are non-Gaussian, but we may as well prescribe Gaussianity at some given times.
Moreover, a coming down from infinity with respect to the perturbation and the initial
condition holds. Finally, we show that the our solutions generate statistically stationary
solutions as limits of ergodic averages, and we obtain existence of infinitely many nonGaussian time dependent ergodic stationary solutions. We also extend our results to a more
general class of singular SPDEs.
