On the long term behaviour of stochastic heat equations

点击翻译按钮获取中文摘要

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.