Interacting particle systems in random environment

当潜在的单粒子随机漫步在空间中是均匀的时，随机相互作用系统的许多方面都得到了很好的理解。例如，基于Z^d上的简单（对称或不对称）随机漫步的不相容过程和零范围过程的大规模极限（稳态波动和水动力极限）已经得到了很多研究，并且现在已经很好地理解了。当潜在的空间结构中存在一些先验的随机性时，就很少有人知道了，比如随机跳跃率或依赖于随机环境的随机局部漂移。在这个项目中，我们计划研究无散度随机漂移场中的相互作用粒子系统。这本质上意味着图边是随机定向的，这样平均局部通量为零，然而系统远非可逆。在这类随机环境中，随机游走是扩散的还是超扩散的，这取决于随机漂移场相关的某个条件是否成立。我们计划研究这种二分法如何反映在基于这些随机漫步的相互作用粒子系统的大尺度行为上。极端平稳测度仍然是产品结构。这一事实使研究流体动力极限和稳态波动成为现实的希望。然而，大规模的限制可能与通常的限制非常不同。我们将研究这类模型的水动力极限和稳态涨落场。

Many aspects of stochastic interacting systems are well understood when the underlying single-particle random walks are homogeneous in space. E.g., large scale limits (steady state fluctuations and hydrodynamic limits) of exclusion and zero-range processes based on simple (symmetric or asymmetric) random walks on Z^d have been much studied and are well understood by now. Much less is known when there is some randomness a priori encoded in the underlying spatial structure, like random jump rates or random local drift depending on the random environment.In this project we plan to investigate interacting particle systems in divergence-free random drift fields. This means essentially that the graph edges are randomly oriented in such a way that the average local fluxes are nil, however the systems are far from being reversible. It is known that the random walk in this type of random environment is either diffusive or superdiffusive depending on whether a certain condition on the correlations of the random drift field are valid or not. We plan to investigate how this dichotomy is reflected on the large scale behaviour of the interacting particle systems based on these random walks. Extremal stationary measures will still be of product structure. This fact makes it realistically hopeful that hydrodynamic limits and steady state fluctuations can be investigated. However, the large scale limits can be very different from the usual ones. We will investigate the hydrodynamic limit and the steady state fluctuation fields of these kind of models.