Brownian particle in the curl of 2-D stochastic heat equations

DOI10.1007/S10955-023-03224-1arXiv2211.02194OpenAlexW4391287592WikidataQ129080982 ScholiaQ129080982MaRDI QIDQ6189805

Author name not available (Why is that?)

Publication date: 5 February 2024

Published in: (Search for Journal in Brave)

Abstract: 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 $sqrtlog$-super diffusive bounds for a Brownian particle in the curl of (a regularisation of) the 2-d Gaussian Free Field (GFF) $underlineomega$. We consider a one parameter family of Markovian and Gaussian dynamic environments which are reversible with respect to the law of $underlineomega$. Adapting their method, we show that if $sge1$, with $s=1$ corresponding to the standard stochastic heat equation, then the particle stays $sqrtlog$-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. T'oth, B. Valk'o, J. Stat. Phys., 2012] to predict the $log$-corrections to diffusivity. We also provide examples which display $lo{g}^a$-super diffusive behaviour for $ain(0,1/2]$.

Full work available at URL: https://arxiv.org/abs/2211.02194

zbMATH Keywords

No records found.

Mathematics Subject Classification ID

No records found.

This page was built for publication: Brownian particle in the curl of 2-D stochastic heat equations

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q6189805)