论文标题
由彩色噪声驱动的分数随机热方程的小球概率
Small Ball Probabilities for the Fractional Stochastic Heat Equation Driven by a Colored Noise
论文作者
论文摘要
我们考虑$ d $ d $ d $ d $ \ mathbb {t}^d：= \ left [ - \ frac {1} {2}，\ frac {1} {1} {2} {2} \ orirt] u（t，\ textbf {x}）= - （ - δ）^{α/2} u（t，t，\ textbf {x}）+σ（t，t，\ textbf {x}，u）\ dot {f}（f}（f}（t，t，t，t，t，\ textbf {x}） \ Mathbb {t}^d，t \ in \ Mathbb {r} _+，\]其中$α\ in（1,2] $和$ \ dot {f}（t，t，\ textbf {x}）$是一般的高斯噪声，是$ with in As the us and ys $。 $ u（0，\ textbf {x}）\ equiv 0 $时，解决方案$ u $的小球概率。
We consider the fractional stochastic heat equation on the $d$-dimensional torus $\mathbb{T}^d:=\left[-\frac{1}{2},\frac{1}{2}\right]^d$, $d\geq 1$, with periodic boundary conditions: \[ \partial_t u(t,\textbf{x})= -(-Δ)^{α/2}u(t,\textbf{x})+σ(t,\textbf{x},u)\dot{F}(t,\textbf{x})\quad \textbf{x}\in \mathbb{T}^d,t\in\mathbb{R}_+ ,\] where $α\in(1,2]$ and $\dot{F}(t,\textbf{x})$ 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 $u(0,\textbf{x})\equiv 0$.
