论文标题
抛物线型抛物线型安德森模型的空间平稳性，Ergodicity和CLT具有Delta初始条件的尺寸$ d \ geq 1 $
Spatial stationarity, ergodicity and CLT for parabolic Anderson model with delta initial condition in dimension $d\geq 1$
论文作者
论文摘要
Suppose that $\{u(t\,, x)\}_{t >0, x \in\mathbb{R}^d}$ is the solution to a $d$-dimensional parabolic Anderson model with delta initial condition and driven by a Gaussian noise that is white in time and has a spatially homogeneous covariance given by a nonnegative-definite measure $f$ which satisfies达兰的状况。令$ \ boldsymbol {p} _t（x）：=（2πt）^{ - d/2} \ exp \ { - \ | x \ |^2/（2t）\} $表示标准的高斯热kernel在$ \ mathbb {r}^d $上。我们证明，对于所有$ t> 0 $，过程$ u（t）：= \ {u（t \ ,, x）/\ boldsymbol {p} _t _t（x）：x \ in \ in \ mathbb {r}^d \} $是使用Feynman-Kac的固定条件，并且是feynman-kac的其他条件，并且是其他条件的，并且$ \ hat {f} \ {0 \} = 0 $，其中$ \ hat {f} $是$ f $的傅立叶变换。此外，使用Malliavin-Stein方法，我们根据$ f $的定量分析研究了$ u（t）$的各种中心限制定理。特别是，当$ f $由riesz kernel给出时，即$ f（\ m athrm {d} x）= \ | x \ | |^{ - β} \ m m mathrm {d} x $，我们获得了$ u（t）$ u（t）$ u（t）$的多相过渡，来自$ u（t）$β\ in（t） $β\ in（1 \ ,, d \ wedge 2）$。
Suppose that $\{u(t\,, x)\}_{t >0, x \in\mathbb{R}^d}$ is the solution to a $d$-dimensional parabolic Anderson model with delta initial condition and driven by a Gaussian noise that is white in time and has a spatially homogeneous covariance given by a nonnegative-definite measure $f$ which satisfies Dalang's condition. Let $\boldsymbol{p}_t(x):=(2πt)^{-d/2}\exp\{-\|x\|^2/(2t)\}$ denote the standard Gaussian heat kernel on $\mathbb{R}^d$. We prove that for all $t>0$, the process $U(t):=\{u(t\,, x)/\boldsymbol{p}_t(x): x\in \mathbb{R}^d\}$ is stationary using Feynman-Kac's formula, and is ergodic under the additional condition $\hat{f}\{0\}=0$, where $\hat{f}$ is the Fourier transform of $f$. Moreover, using Malliavin-Stein method, we investigate various central limit theorems for $U(t)$ based on the quantitative analysis of $f$. In particular, when $f$ is given by Riesz kernel, i.e., $f(\mathrm{d} x) = \|x\|^{-β}\mathrm{d} x$, we obtain a multiple phase transition for the CLT for $U(t)$ from $β\in(0\,,1)$ to $β=1$ to $β\in(1\,,d\wedge 2)$.
