论文标题
具有超线性扩散系数和空间均匀的有色噪声的随机部分微分方程的规律性理论
A regularity theory for stochastic partial differential equations with a super-linear diffusion coefficient and a spatially homogeneous colored noise
论文作者
论文摘要
对于具有彩色噪声$ f $的随机PDE，获得了强大解决方案的存在，唯一性和规律性，其超线性扩散系数：$$ du =（a^{ij {ij} u_ {x^ix^ix^ix^j}+b^iu_ （t，x）\ in（0，\ infty）\ times \ times \ mathbb {r}^d，$$其中$λ\ geq 0 $，系数取决于$（ω，t，x）$。 The strategy of handling nonlinearity of the diffusion coefficient is to find a sharp estimation for a general Lipschitz case, and apply it to the super-linear case.此外，对估算值的调查提供了$λ$的范围，这是独特的解决性的足够条件，其中范围取决于$ f $的空间协方差和空间尺寸$ d $。
Existence, uniqueness, and regularity of a strong solution are obtained for stochastic PDEs with a colored noise $F$ and its super-linear diffusion coefficient: $$ du=(a^{ij}u_{x^ix^j}+b^iu_{x^i}+cu)dt+ξ|u|^{1+λ}dF, \quad (t,x)\in(0,\infty)\times\mathbb{R}^d, $$ where $λ\geq 0$ and the coefficients depend on $(ω,t,x)$. The strategy of handling nonlinearity of the diffusion coefficient is to find a sharp estimation for a general Lipschitz case, and apply it to the super-linear case. Moreover, investigation for the estimate provides a range of $λ$, a sufficient condition for the unique solvability, where the range depends on the spatial covariance of $F$ and the spatial dimension $d$.
