论文标题
二-D随机热方程的卷曲中的布朗粒子
Brownian particle in the curl of 2-d stochastic heat equations
论文作者
论文摘要
我们研究了在动态随机环境中演变的棕色粒子的长时间行为。最近，[G。 Cannizzaro，L。Haunschmid-Sibitz，F。Toninelli，Preprint Arxiv：2106.06264]被证明是敏锐的$ \ sqrt {log} $ - 在2-D-D-D-D-D-D-D-D-D-D-D-D-D-D-D-field Field Field（Gff）$ plute（Gff）$ plude（GFF）$ \ fline-flownline $ \ fline $ \ fline $ \ fline $ fline $ curl中的棕色粒子的超级分散界限。我们考虑了马尔可夫人和高斯动态环境的一个参数家族，相对于$ \usevenlineΩ$的定律是可逆的。通过调整他们的方法，我们表明，如果$ s \ ge1 $，$ s = 1 $对应于标准随机热方程，则粒子保持$ \ sqrt {log} $ - 超级扩散，而如果$ s <1 $，则对应于分数热方程，则粒子变得差异。实际上，对于$ s <1 $，我们表明这是[t的特殊情况。 Komorowski，S。Olla，J。Func。肛门，2003]，通过扇形条件产生不变性原理。我们的主要结果与Alder-Wainwright的缩放论点一致（请参阅[B. Alder，T。Wainright，Phys。Rev。Lett。1967]）。 Tóth，B.Valkó，J。Stat。 Phys。，2012]，以预测$ \ log $ - 校正以扩散性。我们还提供了显示$ log^a $ -super扩散行为的示例，以$ a \ in（0,1/2] $。
We study the long time behaviour of a Brownian particle evolving in a dynamic random environment. Recently, [G. Cannizzaro, L. Haunschmid-Sibitz, F. Toninelli, preprint arXiv:2106.06264] proved sharp $\sqrt{log}$-super diffusive bounds for a Brownian particle in the curl of (a regularisation of) the 2-d Gaussian Free Field (GFF) $\underlineω$. We consider a one parameter family of Markovian and Gaussian dynamic environments which are reversible with respect to the law of $\underlineω$. Adapting their method, we show that if $s\ge1$, with $s=1$ corresponding to the standard stochastic heat equation, then the particle stays $\sqrt{log}$-super diffusive, whereas if $s<1$, corresponding to a fractional heat equation, then the particle becomes diffusive. In fact, for $s<1$, we show that this is a particular case of [T. Komorowski, S. Olla, J. Func. Anal., 2003], which yields an invariance principle through a Sector Condition result. Our main results agree with the Alder-Wainwright scaling argument (see [B. Alder, T. Wainright, Phys. Rev. Lett. 1967]) used originally in [B. Tóth, B. Valkó, J. Stat. Phys., 2012] to predict the $\log$-corrections to diffusivity. We also provide examples which display $log^a$-super diffusive behaviour for $a\in(0,1/2]$.
