Invariance principle for the KPZ equation arising in stochastic flows of kernels
Abstract
We consider a generalized model of random walk in dynamical random environment, and we show that the multiplicativenoise stochastic heat equation (SHE) describes the fluctuations of the quenched density at a certain precise location in the tail. The distribution of transition kernels is fixed rather than changing under the diffusive rescaling of spacetime, i.e., there is no critical tuning of the model parameters when scaling to the stochastic PDE limit. The proof is done by pushing the methods developed in [arxiv 2304.14279, arXiv 2311.09151] to their maximum, substantially weakening the assumptions and obtaining fairly sharp conditions under which one expects to see the SHE arise in a wide variety of random walk models in random media. In particular we are able to get rid of conditions such as nearestneighbor interaction as well as spatial independence of quenched transition kernels. Moreover, we observe an entire hierarchy of moderate deviation exponents at which the SHE can be found, confirming a physics prediction of J. Hass.
 Publication:

arXiv eprints
 Pub Date:
 January 2024
 DOI:
 10.48550/arXiv.2401.06073
 arXiv:
 arXiv:2401.06073
 Bibcode:
 2024arXiv240106073P
 Keywords:

 Mathematics  Probability;
 Mathematical Physics
 EPrint:
 76 pages