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Rough path theory, differential equations and stochastic analysis

Final Report Summary - RPT (Rough path theory, differential equations and stochastic analysis)

We study stochastic (genuine and partial) differential equations and various topics of stochastic analysis with the aid of Lyons' rough path theory.

1) There is deep link, due to Malliavin, between the theory of hypoelliptic second order differential operators and certain smoothness properties of diffusion processes, constructed via stochastic differential equations. It was understood that the Markovian structure is dispensable and that Hoermander type results are a robust feature of differential equations driven by non-degenerate Gaussian processes. The full resolutions of this matter turned out to be a truly international undertaking, and also involved T. Lyons (Oxford) and M. Hairer (Warwick).

2) We return to the work of P.L. Lions and P. Souganidis (1998-2003) on a path-wise theory of fully non-linear stochastic partial differential equations in viscosity sense. A “rough” path-wise theory for such equations was proposed, a point of view which offers a number of benefits. For instance, concrete numerical (splitting) schemes for classes of non-linear stochastic partial differential equations are easily deduced from rough path stability properties. But even for linear equations, the results are non-trivial and led us to a resolution of a long-standing open problem of filtering theory. From a broader perspective, the use of rough paths for other classes of non-linear stochastic partial differential equations - notably those subject to space-time white noise – has received an enormous boost by Hairer's work on KPZ and his subsequent development of regularity structures.

3) Rough path theory has been related to a variety of weak- and strong numerical schemes (cubature, multi-level Monte Carlo ...). We furthermore decived a satisfactory theory of rough path with jumps, mimicking the general stochastic integration theory know from Ito calculus.