论文标题
在$ \ Mathbb r^d $中使用空间色噪声的随机分数热方程的梯度分析
Analysis of the gradient for the stochastic fractional heat equation with spatially-colored noise in $\mathbb R^d$
论文作者
论文摘要
考虑随机部分微分方程$$ \ frac {\ partial} {\ partial t} u_t（\ mathbf {x}）= - （ - δ）^{\fracα{2}} u_t（\ mathbf {x}） +b \ left（u_t（\ MathBf {x}）\右） +σ\ left（u_t（\ m mathbf {x}））\ right）\ dot f（t，\ mathbf {x}），\ \ \ \ \ \ \ \ \ \ \ \ t \ t \ ge0，\ ge0，\ mathbf { 其中$ - （ - δ）^{\fracα{2}} $表示具有$ $ $α/2 \ in（1/2,1] $的功率$α/2 \的分数laplacian，而驱动噪声$ \ dot f $是一个时代和空间同质范围内的互联性行为，是一个中心的高斯领域。渐变$ u_t（\ Mathbf {x}） - u_t（\ MathBf {x} - \ Varepsilon \ Mathbf E）在任何固定时间$ t> 0 $，$ \ varepsilon \ varepsilon \ downarrow 0 $，在$ \ mathbf e $中，是$ \ there $ \ m m iate $ \ mathbb r^d $。对数和$ Q $ - $ Q $ - 跨太空中解决方案的行为。
Consider the stochastic partial differential equation $$ \frac{\partial }{\partial t}u_t(\mathbf{x})= -(-Δ)^{\fracα{2}}u_t(\mathbf{x}) +b\left(u_t(\mathbf{x})\right)+σ\left(u_t(\mathbf{x})\right) \dot F(t, \mathbf{x}), \ \ \ t\ge0, \mathbf{x}\in \mathbb R^d, $$ where $-(-Δ)^{\fracα{2}}$ denotes the fractional Laplacian with the power $α/2\in (1/2,1]$, and the driving noise $\dot F$ is a centered Gaussian field which is white in time and with a spatial homogeneous covariance given by the Riesz kernel. We study the detailed behavior of the approximation spatial gradient $u_t(\mathbf{x})-u_t(\mathbf{x}-\varepsilon \mathbf e)$ at any fixed time $t>0$, as $\varepsilon\downarrow 0$, where $\mathbf e$ is the unit vector in $\mathbb R^d$. As applications, we deduce the law of iterated logarithm and the behavior of the $q$-variations of the solution in space.
