论文标题
小球概率和随机热方程的支持定理
Small ball probabilities and a support theorem for the stochastic heat equation
论文作者
论文摘要
我们考虑以下随机部分微分方程在$ t \ geq 0，x \ in [0，j]，j \ geq 1 $中，我们考虑$ [0，j] $作为确定端点的圆圈：\ begin {qore {equation*} \ partial_t { {\ MathBf U}（t，X） + {\ MathBf G}（t，x，x，x，\ MathBf U） + {\ MathBfσ}}（t，t，x，x，X，\ Mathbf u}）\ dot {\ dot {\ Mathbf W}是2-参数$ d $ - 维矢量的价值白噪声，$ {\ mathbfσ} $是从$ {\ Mathbb r} _+\ times {\ Mathbb r} \ times {\ Mathbb r} \ times {\ Mathbb r} Lipschitz以$ \ Mathbf u $。我们假设$σ$是均匀的椭圆形，并且$ \ mathbf g $均匀界定。假设$ {\ mathbf u}（0，x）\ equiv \ mathbf 0 $，我们证明了解决方案$ \ mathbf u $的小球概率。当$ {\ mathbf u}（0，x）$不一定为零时，我们还证明了解决方案的支持定理。
We consider the following stochastic partial differential equation on $t \geq 0, x\in[0,J], J \geq 1$ where we consider $[0,J]$ to be the circle with end points identified: \begin{equation*} \partial_t{\mathbf u}(t,x) =\frac{1}{2}\,\partial_x^2 {\mathbf u}(t,x) + {\mathbf g}(t,x,\mathbf u) + {\mathbf σ}(t,x, {\mathbf u})\dot {\mathbf W}(t,x) , \end{equation*} and $\dot {\mathbf W }(t,x)$ is 2-parameter $d$-dimensional vector valued white noise and ${\mathbf σ}$ is function from ${\mathbb R}_+\times {\mathbb R} \times {\mathbb R}^d \rightarrow {\mathbb R}^d$ to space of symmetric $d\times d$ matrices which is Lipschitz in $\mathbf u$. We assume that $σ$ is uniformly elliptic and that $\mathbf g$ is uniformly bounded. Assuming that ${\mathbf u}(0,x) \equiv \mathbf 0$, we prove small-ball probabilities for the solution $\mathbf u$. We also prove a support theorem for solutions, when ${\mathbf u}(0,x)$ is not necessarily zero.
