论文标题
Wiener空间中的亚批判性高斯乘法混乱：构造，时刻和音量衰减
Subcritical Gaussian Multiplicative Chaos in the Wiener Space: Construction, Moments and Volume Decay
论文作者
论文摘要
我们构建和研究了Kahane的高斯乘法混乱理论的无限维度类似物的特性{k85}。也就是说，如果$ h_t（ω）$是一个随机字段定义的W.R.T.时空白噪声$ \ dot b $和集成的W.R.T. $ d \ geq 3 $中的布朗路径，我们考虑重新归一化的，加权的W.R.T. Wiener量$ \ MATHBB P_0 $。我们在{\ IT弱障碍（次临界）}制度中构建了几乎确定的限制$μ_γ$，并将其称为Wiener空间中的{\ IT亚临界GMC}。我们表明，$μ_γ\ big \ {ω：\ lim_ {t \ to \ infty} \ frac {h_t（ω）} { $γ$ - {\ it厚路径}，因此，归一化版本是单数W.R.T.维纳措施。我们以唯一的限制来表征$μ_γ$ W.R.T. shamov \ cite {s14}和随机{\ it rooted} $ \ mathbb q_ {μ_γ}（d \ dot bdΩ）=μ_γ（d \ dome b）p（d \ dot b）p（d \ d \ dot b）$。 然后，我们确定该度量的分形特性左右$γ$ -THICK路径：$ -C_2 \ leq \ liminf_ {r \ to 0} r^2 \ log \ log \ log \wideHatμ_γ（\ | | | |ω\ | <| <| <r）\ leq \ leq \ limsup_ \wideHatμ_γ（\ |ω-η\ | <r）\ leq -c_1 $ w.r.t加权规范$ \ | \ cdot \ | $。这里$ C_1> 0 $和$ C_2 <\ infty $是整个弱疾病制度中{\ it明确}的统一鞋面（分别为suppersise）hölder指数。此外，随着疾病接近零，它们会收敛到维纳措施的缩放指数。最后，我们建立了负面质量$μ_γ$的负质量的负面和$ l^p $（$ p> 1 $）的瞬间。
We construct and study properties of an infinite dimensional analog of Kahane's theory of Gaussian multiplicative chaos \cite{K85}. Namely, if $H_T(ω)$ is a random field defined w.r.t. space-time white noise $\dot B$ and integrated w.r.t. Brownian paths in $d\geq 3$, we consider the renormalized exponential, weighted w.r.t. the Wiener measure $\mathbb P_0$. We construct the almost sure limit $μ_γ$ in the {\it entire weak disorder (subcritical)} regime and call it {\it subcritical GMC} on the Wiener space. We show that $$ μ_γ\Big\{ω: \lim_{T\to\infty} \frac{H_T(ω)}{T(ϕ\starϕ)(0)} \ne γ\Big\}=0 \qquad \mbox{almost surely,} $$ meaning, $μ_γ$ is supported only on $γ$-{\it thick paths}, and consequently, the normalized version is singular w.r.t. the Wiener measure. We characterize uniquely the limit $μ_γ$ w.r.t. the mollification scheme $ϕ$ in the sense of Shamov \cite{S14} and the random {\it rooted} measure $\mathbb Q_{μ_γ}(d\dot B dω)= μ_γ(dω,\dot B)P(d\dot B)$. We then determine the fractal properties of the measure around $γ$-thick paths: $-C_2 \leq \liminf_{r\to 0} r^2 \log \widehatμ_γ(\|ω\| < r) \leq \limsup_{r\to 0}\sup_ηr^2 \log \widehatμ_γ(\|ω-η\| < r) \leq -C_1$ w.r.t a weighted norm $\|\cdot\|$. Here $C_1>0$ and $C_2<\infty$ are the uniform upper (resp. pointwise lower) Hölder exponents which are {\it explicit} in the entire weak disorder regime. Moreover, they converge to the scaling exponent of the Wiener measure as the disorder approaches zero. Finally, we establish negative and $L^p$ ($p>1$) moments for the total mass of $μ_γ$ in the weak disorder regime.
