On the long term behaviour of stochastic heat equations

许多现象都是在随机输入的影响下演变的。例如，DNA链的运动、太阳的内部结构以及粒子系统中的许多现象都涉及随机性。为了模拟这些情况，人们通常可以使用随机偏微分方程组(SPDEs)，它粗略地描述了受某种随机性影响的运动。与偏微分方程组理论相比，离散偏微分方程组的理论还不够发达。幸运的是，这种情况正在迅速改变；在过去的几十年里，许多研究人员一直在研究SPDEs，并取得了一些重大进展。本课题的基础已经奠定，人们可以选择三种不同的方法：希尔伯特空间方法、令测度方法和L_p空间方法。这一建议的主要目的是研究一大类随机微分方程的长期行为。更准确地说，我们建议研究一种称为“间歇性”的现象，即在很长一段时间内，SPDE的解都有高峰值。在空间被假定为离散格子的特殊情况下，间歇性的研究非常活跃，到目前为止，只有少数连续的空间方程被证明存在这种现象。这项提议旨在表明，更多类别的SPDE表现出这些行为。这将阐明许多涉及随机性的现象。

Many phenomena evolve under the influence of random inputs. For instance, the motion of a strand of DNA, the internal structure of the sun and many phenomena in particle systems all involve randomness. To model these situations, one can often use stochastic partial differential equations(SPDEs) which roughly speaking, describe motions which are under the influence of some kind of randomness. Compared with the theory of partial differential equations, the theory of SPDEs is not well developed. Fortunately, this situation is changing quite rapidly; a lot of researchers have been studying SPDEs during the past decades and some major advances have been made. The foundations of the subject have been settled and one has the option of three different approaches; the Hilbert space approach, the martingale measure approach and the L_p space approach.The main aim of this proposal is to study the long term behaviour of a wide class of stochastic differential equations. More precisely, we propose to study a phenomenon called "intermittency" whereby for large times, the solutions of the SPDE have high peaks. Intermittency is actively studied in the special case where space is assumed to be a discrete lattice and so far only a small number of continuous space equations have been proved to exhibit this phenomenon. This proposal aims to show that a much wider class of SPDEs exhibit these kinds of behaviour. This will shed light on a lot of phenomena which involve randomness.