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Directed polymers and rough paths

Abstract : Stochastic Partial Differential Equations are an essential tool for the analysis of scaling limits of a diverse array of microscopic models coming from other fields such as physics and chemistry.This type of equations correspond to classical partial differential equations to which one has added a random forcing which is typically very irregular ; the most basic example is perhaps the Stochastic Heat Equation, one of whose versions is studied in this thesis. The roughness of the potential turns the analysis of solutions to these probles a lot more difficult than the classic case. In fact, there are cases where solutions can be understood only in the sense of distributions, i.e. as generalised functions. There are some critical cases, such the Kardar-Parisi-Zhang (KPZ)equation where, even though the solutions can be shown to be continuous (even Hölder continuous) they are not regular enough so that some non-linear terms appearing in this equation are well defined. In the last 20 years certain techniques have been developed for the analysis of these equations, among which there is the theory of Rough Paths by T. Lyons (1998), their branched version introduced by M. Gubinelli (2010) and more recently the theory of Regularity Structures of M. Hairer (2014) and for which he was awarded the Fields Medal in 2014. All these techniques have as main idea that of renormalisation, coming from physics. In particular, Wick renormalisation plays an essential role in Regularity Structures. In this work we develop Wick products and polynomials from a Hopf-algebraic point of view, inspired by G.-C. Rota's Umbral Calculus. We also explore the general theory of Rough Paths and in particular in their branched version, where we show some new results in the direction of incorporating an analogue of Wick renormalisation as found in Hairer's Regularity Structures. Finally, the semi-discrete multi-layer polymer model, introduced by I. Corwin and A. Hammond (2014) is studied. We show the convergence of its partition function towards a stochastic process known as (the solution to) "the multi-layer Stochastic Heat Equation" introduced by N. O'Connell and J. Warren (2011) some years earlier. We remark that at the time of writing of this work there were no results allowing to interpret this last process as the solution to a singular SPDE as is the case, for example, for the KPZ equation. This was one of the main sources of inspiration of this work.
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Nikolas Tapia. Directed polymers and rough paths. Probability [math.PR]. Sorbonne Université; Universidad de Chile, 2018. English. ⟨NNT : 2018SORUS363⟩. ⟨tel-02865419⟩

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