Technische Universitat Berlin

Recently, Deya, Gubinelli, Hofmanova and Tindel ('16) (also Bailleul-Gubinelli '15) have provided a general approach in order to obtain a priori estimates for rough partial differential equations of the form
(*)    du = Au dt + Bu dX
where X is a two-step rough path, A is a second order operator (elliptic), while B is first order. We will pursue the line of this work by presenting an L^p theory "à la Krylov" for generalized versions of (*). We will give an application of this theory by proving boundedness of solutions for a certain class

Duality methods can be very powerful tools for the analysis of stochastic

processes. However, there seems to be no general theory available

yet. In this talk, I will discuss and aim to clarify various notions

of duality, give some recent rather striking examples (applied to

stochastic PDEs, interacting particle systems and combinatorial stochastic

processes)

and try to give some systematic insight into the type of questions

that can in principle be tackled. Finally, I will try to provide you

with some intuition for this fascinating technique.

In the past the theory of rough paths has proven to be an elegant tool for deriving support theorems and large deviation principles. In this talk I will explain how this approach can be used in the analysis of stochastic flows generated by Kunita SDE's. As driving processes I will consider general Banach space valued Wiener processes