一类奇异随机泛函偏微分方程的性质研究

A large number of models were found that could be described by partial differential equations with random parameters, such as the .coefficients or the forcing term. As a result, the study of SDE in infinite dimensional space has begun to attract a lot of attention of many researchers. In this field, the existence and uniqueness of the solutions and the properties of the solutions to a class of stochastic partial differential equations driven by irregular noise (the space time white noise) remain open for many years..Recently, the development of this problem has made a lot progress. One approach is Martin Hairer's regularity structure theory, the other is the method of paracontrolled distribution introduced by Gubinelli, Imkeller and Perkowski. The project applicant wishes to carry out the study of the stochastic partial differential equations driven by irregular noise (space time white noise) by using these two approaches..On the other hand, recently stochastic partial differential equations (SPDE) with delays has been paid a lot of attention. There is a large number of literature on the mathematical theory and on applications of stochastic functional (or delay) differential equations. When one wants to model some evolution phenomena arising in physics, biology, engineering, etc., some hereditary characteristics such as after-effect, time-lag, time-delay can appear in the variables. Typical examples can be found in the researches of materials with termal memory, biochemical reactions, population models, etc. The applicant wants to study the properties of stochastic functional (delay) equations in infinite dimensions including existence and uniqueness of the solutions, ergodicity and related properties.

无穷维空间上的随机微分方程，是一个重要的研究课题, 是目前国际上概率论领域的一个热门方向。在随机微分方程领域中,一类带有奇异噪声(时空白噪声)的随机微分方程解的存在唯一性以及解的性质一直是公开问题.最近国际上关于这个问题有了重大进展, 主要的方法是由Martin Hairer提出的regularity structure理论和Gubinelli 等人提出的paracontrolled distribution的方法。项目申请人希望用这两种方法来进一步研究带有奇异的噪声的随机微分方程解的存在唯一性以及解的性质。另一方面，随机泛函（时滞）微分方程作为一种重要的数学模型，可以视为既考虑了随机微分方程模型问题又考虑了时滞因素的影响，被广泛地应用到控制论、神经网络、生物学、金融学等众多领域。申请人准备研究无穷维随机泛函（时滞）微分方程解的各种性质,包括解的存在唯一性,遍历性等。

无穷维空间上的随机微分方程，是一个重要的研究课题, 是目前国际上概率论领域的一个热门方向。在随机微分方程领域中,一类带有奇异噪声(时空白噪声)的随机微分方程解的存在唯一性以及解的性质一直是公开问题.最近国际上关于这个问题有了重大进展, 主要的方法是由Martin Hairer提出的regularity structure理论和Gubinelli 等人提出的paracontrolled distribution的方法。项目申请人用这两种方法得到了带有奇异的噪声的随机微分方程解的存在唯一性以及解的性质。另一方面，随机泛函（时滞）微分方程作为一种重要的数学模型，可以视为既考虑了随机微分方程模型问题又考虑了时滞因素的影响，被广泛地应用到控制论、神经网络、生物学、金融学等众多领域。申请人得到了局部单调条件下无穷维随机泛函（时滞）微分方程解的存在唯一性。