Interacting Particle Systems and Stochastic PDEs

热方程是一个数学方程，用来描述热通过纯导电物质的流动。这个方程和它的许多变体已经被研究了大约200年，数学理论现在已经很发达了。在过去的几十年里，数学家和物理学家一直对更现实的随机热方程感兴趣，该方程考虑了杂质随机散布在介质中的影响。这个方程的某些变换也被用来描述许多其他物理现象，如湍流流体的运动、雪的沉积和细菌的生长。尽管我们对随机热方程的理解已经取得了许多进展，但仍有更多的东西有待发现。这项提议的目标之一是描述介质中一大类杂质的这个方程的特征。该提案还探讨了逼近该方程的方法，以便人们可以在计算机上模拟该方程，这反过来将给我们对其行为的进一步直观。我们还将探索这个方程与其他具有随机性的系统之间的联系。

The heat equation is a mathematical equation used to describe the flow of heat through a purely conducting substance. This equation and many variations of it have been studied for around two hundred years, and the mathematical theory is now well developed. Over the last few decades mathematicians and physicists have been interested in the more realistic stochastic heat equation which considers the effect of having impurities randomly scattered in the medium. Certain transformations of this equation are also used to describe many other physical phenomena, for example motion of a turbulent fluid, deposition of snow and bacterial growth.Although many advances have been made in our understanding of the stochastic heat equation, much more remains to be discovered. One of the goals of this proposal is to describe features of this equation for a wide class of impurities in the medium. The proposal also looks into ways of approximating this equation so that one can simulate the equation on a computer, which in turn will give us further intuition on its behaviour. We shall also explore connections of this equation to other systems exhibiting randomness.