论文标题
带有彩色噪声的随机热方程的小球概率
Small ball probabilities for the stochastic heat equation with colored noise
论文作者
论文摘要
我们考虑一维圆环$ \ mathbb {t}的随机热方程：= \ left [-1,1 \右] $具有周期性边界条件：$ \ partial_t u（t，t，x）= \ partial^partial^partial^2_x u（t，x）+σ（t，x） \ Mathbb {t}，t \ in \ mathbb {r} _+，$$，其中$ \ dot {f}（t，x）$是一种通用的高斯噪声，时间是白色的，在太空中是彩色的。假设$σ$是$ u $且均匀边界的Lipschitz，我们估计解决方案$ u $时的小球概率（0，x）\ equiv 0 $。
We consider the stochastic heat equation on the 1-dimensional torus $\mathbb{T}:=\left[-1,1\right]$ with periodic boundary conditions: $$ \partial_t u(t,x)=\partial^2_x u(t,x)+σ(t,x,u)\dot{F}(t,x),\quad x\in \mathbb{T},t\in\mathbb{R}_+, $$ where $\dot{F}(t,x)$ is a generalized Gaussian noise, which is white in time and colored in space. Assuming that $σ$ is Lipschitz in $u$ and uniformly bounded, we estimate small ball probabilities for the solution $u$ when $u(0,x)\equiv 0$.
