The differential structure of spaces of rough paths

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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