论文标题
随机热方程的适合性，分布漂移和偏斜随机热方程
Well-posedness of stochastic heat equation with distributional drift and skew stochastic heat equation
论文作者
论文摘要
我们研究随机反应 - - 接口方程$ \ partial_tu_t（x）= \ frac12 \ partial^2_ {xx} u_t（x） $\mathcal{B}^β_{q,\infty}({\mathbb R})$, $D\subset{\mathbb R}$ and $\dot W$ is a space-time white noise on ${\mathbb R}_+\times D$.每当$β-1/q \ ge-1 $，$β> -1 $和$ q \ in [1，\ infty] $中，我们将解决方案的概念引入了该方程式，并获得强大解决方案的存在和独特性。此类包括$ b $是度量的方程式，尤其是$ b =Δ_0$，与偏斜的随机热方程相对应。对于$β -1/q> -3/2 $，我们获得了弱解决方案的存在。我们的结果将Bass and Chen（2001）的工作扩展到随机部分微分方程的框架，并将Gyöngy和Pardoux（1993）的结果推广到分布漂移。为了建立这些结果，我们通过基于Lê〜（2020）中引入的随机缝纫引理的新策略来利用白噪声的正则化效应。
We study stochastic reaction--diffusion equation $$ \partial_tu_t(x)=\frac12 \partial^2_{xx}u_t(x)+b(u_t(x))+\dot{W}_{t}(x), \quad t>0,\, x\in D $$ where $b$ is a generalized function in the Besov space $\mathcal{B}^β_{q,\infty}({\mathbb R})$, $D\subset{\mathbb R}$ and $\dot W$ is a space-time white noise on ${\mathbb R}_+\times D$. We introduce a notion of a solution to this equation and obtain existence and uniqueness of a strong solution whenever $β-1/q\ge-1$, $β>-1$ and $q\in[1,\infty]$. This class includes equations with $b$ being measures, in particular, $b=δ_0$ which corresponds to the skewed stochastic heat equation. For $β-1/q > -3/2$, we obtain existence of a weak solution. Our results extend the work of Bass and Chen (2001) to the framework of stochastic partial differential equations and generalizes the results of Gyöngy and Pardoux (1993) to distributional drifts. To establish these results, we exploit the regularization effect of the white noise through a new strategy based on the stochastic sewing lemma introduced in Lê~(2020).
