The differential structure of spaces of rough paths

1简要描述的背景下的研究，包括潜在的impactRough路径是数学对象，近年来已经获得了显着的关注，由于各种各样的应用，他们享受在纯数学和应用领域。粗糙路径空间是非线性的，这使得人们很难理解它的微分结构。虽然关于如何理解非线性空间上的可微性的理论已经发展了很多，但粗糙路空间的微分结构还没有得到充分的解释。光滑函数遍及数学，光滑的定义有很大的不同。在真实的分析中，Ck空间是光滑函数的空间，其中光滑性被定义为函数的k阶导数的连续性。一个重要的领域，光滑函数是至关重要的相关性是研究微分方程。在那里，我们可以说微分方程有属于Ck的解，对于某个k。Sobolev空间是函数空间，在某种意义上，它是Ck空间的推广。虽然Sobolev空间是与Ck空间一样多的光滑函数的空间，但函数属于Sobolev空间的要求是它具有一定程度的光滑性，但光滑性在弱意义上理解。Sobolev空间帮助我们研究微分方程的解，它允许我们找到某些偏微分方程在没有强解的情况下的弱解，即属于Ck空间的解。理解粗糙路的微分结构将使我们能够将Sobolev空间推广到粗糙路空间上。反过来，这将有助于我们更好地理解空间中找到解决方案的某些类别的微分方程，其中粗糙路径发挥了至关重要的作用：随机微分方程（SDEs）.2目的和objectivesThe本研究项目的目标是建立一个帐户的Sobolev空间的空间粗糙路径。这样一个帐户有可能使我们更好地解决SDES，这反过来又有广泛的纯数学和实际应用。3研究方法的新奇由于粗糙路径空间的微分结构的一般性描述尚未发展，因此正在发展的理论是原创的。本项目福尔斯属于EPSRC数学研究领域，其中数学分析和非线性系统是一些主题或研究领域

1 Brief description of the context of the research including potential impactRough paths are mathematical objects that have garnered significant attention in recent years thanks to the variety of applications they enjoy both in pure mathematics and applied fields. The space of rough paths is non-linear, which makes it difficult to understand its differential structure. While many theories about how to understand differentiability on non-linear spaces have been developed, a full account of what the differential structure of rough paths spaces has not been developed yet. Smooth functions pervade mathematics, and the definitions of smoothness vary significantly. In real analysis, Ck spaces are spaces of smooth functions, where smoothness is defined as the continuity of the k-th derivative of the function. One important area where smooth functions are of crucial relevance is the study of differential equations. There, we can say that differential equations have solutions which belong to Ck for a certain k. Sobolev spaces are spaces of functions that, in a sense, generalise Ck spaces. While Sobolev spaces are spaces of smooth functions as much as Ck spaces, the requirement for a function to belong to a Sobolev space is that it has a certain degree of smoothness, but smoothness is understood in the weak sense. Sobolev spaces help us in the study of solutions to differential equations by allowing us to find weak solutions to certain partial differential equations in cases where there is no no strong solution, that is a solution that belongs to a Ck space.Understanding the differential structure of rough paths will thus enable us to characterise Sobolev spaces on the space of rough paths. In turn, this will help us better understand the spaces in which to find solutions to certain classes of differential equations where rough paths play a crucial role: stochastic differential equations (SDEs).2 Aims and objectivesThe goal of this research project is to build an account of Sobolev spaces on the spaces of rough paths. Such an account has the potential to allow us to better characterise solutions to SDEs, which in turn have a broad range of pure mathematics and practical applications. 3 Novelty of the research methodologyThe theories being developed are original since only a general account of the differential structure of spaces of rough paths has not been developed yetThis project falls within the EPSRC Mathematics research area, where Mathematical Analysis and Non-Linear Systems are some of the themes or research areas