论文标题
稳定噪声的随机热方程的解决方案的紧凑型支持属性
The compact support property for solutions to stochastic heat equations with stable noise
论文作者
论文摘要
我们考虑对随机部分差分方程\ [\ partial_t y（t，x）=Δy（t，x） + y（t，x）^γ\ dot {l}（t，x），\ in $（t，t，x）\ in \ mathBb {r} r} p { 0 $和$ \ dot {l} $是（1,2）$的索引$α\的单面稳定噪声。我们证明，如果$ d = 1 $ $γ\ in（2-α，1）$ for $ d = 1 $，并且如果$γ\ in [1/α，1）$ in [2,2/（α-1））\ cap \ cap \ mathbb {n n} $。这与高斯噪声有关方程的解决方案的结果补充。 我们还建立了一个随机积分公式，以用于溶液的密度和相关的矩界，该公式在定义溶液的所有维度中。
We consider weak non-negative solutions to the stochastic partial differential equation \[ \partial_t Y(t,x) = ΔY(t,x) + Y(t,x)^γ\dot{L}(t,x), \] for $(t,x) \in \mathbb{R}_+ \times \mathbb{R}^d$, where $γ> 0$ and $\dot{L}$ is a one-sided stable noise of index $α\in (1,2)$. We prove that solutions with compactly supported initial data have compact support for all times if $γ\in (2-α, 1)$ for $d=1$, and if $γ\in [1/α,1)$ in dimensions $d \in [2,2/(α-1)) \cap \mathbb{N}$. This complements known results on solutions to the equation with Gaussian noise. We also establish a stochastic integral formula for the density of a solution and associated moment bounds which hold in all dimensions for which solutions are defined.
