带奇异系数的随机(偏)微分方程

Stochastic(partial)differential equations with singular coefficients

熟知,在Gauss噪声的扰动下,微分方程的性质可以得到本质的改善.比如:系数仅为Hlder连续的常微分方程不具备适定性,但是在Brown运动的驱动下,只要系数具有某种局部可积性(此时系数仅为几乎处处定义的)就可保证方程的适定性;随机微分方程可以将粗糙的函数磨光为光滑的函数.本文简要介绍关于奇异系数随机微分方程解的存在唯一性研究的基本思想,提供关于随机偏微分方程、泛函随机微分方程以及带跳随机微分方程等模型研究的前沿文献,并着重展示在可积性条件下关于随机微分方程所取得的最新研究进展.

It is well known that a nice noise may actually improve the properties of differential equations.For instance, an ODE is ill-posed if the coefficient is merely Ho1der continuous, but with a Gaussian noise theequation is well-posed if the drift is only locally integrable with power larger than dimension and the noise maymake singular distribution smooth. Recent progress on integrability conditions for the non-explosion andregularity estimates of the invariant probability measures for stochastic differential equations driven byBrownian motion is surveyed. References on the study for more general models are also introduced.