论文标题
带有彩色噪声的随机部分微分方程的解决方案的紧凑型支持属性
The compact support property for solutions to the stochastic partial differential equations with colored noise
论文作者
论文摘要
我们研究了以下随机部分差分方程的解决方案的紧凑型支持属性：$ \ partial_t u = a^{ij} u_ {x^ix^j}（x^ix^j}（t，t，x）+b^{i} i} u_ {x^i} （t，x）\ in（0，\ infty）\ times {\ bf {r}}^d，$$，其中$ \ dot {f} $是一种在空间上是均匀的高斯噪声，它是白色的，在太空中是白色的，$ h（t，x，x，x，x，x，x，x，x，x，x，x，x，x，x，x，x，u）$ k^{ - 1}} | u（us） k（1+ | u |）$ for $λ\ in（0,1）$和$ k \ geq 1 $。我们表明，如果初始数据$ u_0 \ geq 0 $具有紧凑的支持，那么，在加强达兰在$ \ dot {f} $上的条件下（保证了弱解决方案的存在和Hölder的连续性），所有非弱解决方案的弱解决方案$ u（t，\ cdot）$均具有$ t的功能。 [Probab。理论相关。 Fields，93（3）：325--358，1992]和Krylov [Probab。理论相关。 Fields，108（4）：543--557，1997]，其中仅显示由$（0，\ infty）\ times \ bf {r} $驱动的一维SPDES的紧凑型支持属性。
We study the compact support property for solutions of the following stochastic partial differential equations: $$\partial_t u = a^{ij}u_{x^ix^j}(t,x)+b^{i}u_{x^i}(t,x)+cu+h(t,x,u(t,x))\dot{F}(t,x),\quad (t,x)\in (0,\infty)\times{\bf{R}}^d,$$ where $\dot{F}$ is a spatially homogeneous Gaussian noise that is white in time and colored in space, and $h(t, x, u)$ satisfies $K^{-1}|u|^λ\leq h(t, x, u)\leq K(1+|u|)$ for $λ\in(0,1)$ and $K\geq 1$. We show that if the initial data $u_0\geq 0$ has a compact support, then, under the reinforced Dalang's condition on $\dot{F}$ (which guarantees the existence and the Hölder continuity of a weak solution), all nonnegative weak solutions $u(t, \cdot)$ have the compact support for all $t>0$ with probability 1. Our results extend the works by Mueller-Perkins [Probab. Theory Relat. Fields, 93(3):325--358, 1992] and Krylov [Probab. Theory Relat. Fields, 108(4):543--557, 1997], in which they show the compact support property only for the one-dimensional SPDEs driven by space-time white noise on $(0, \infty)\times \bf{R}$.
