On Sobolev rough paths. (English) Zbl 1496.60119
The authors consider spaces of rough paths with Sobolev regularity and associated controlled rough differential equations. First they define the space of Sobolev rough paths using a fractional Sobolev norm. Typical estimates used for rough paths work for \(p\)-variation or related seminorms, but not for the fractional Sobolev norm presenting the main obstacle. In a suitable topology, the authors introduce the controlled rough paths of Sobolev type requiring a remainder term in the mixed Hölder-variation space. Then they show that solutions to the controlled rough differential equations driven by Sobolev rough paths also possess Sobolev regularity. Finally the authors prove local Lipschitz continuity of the Itô-Lyons map on the space of Sobolev rough paths with arbitrary low regularity with respect to the initial value, vector field and the driving signal.
Reviewer: Maria Gordina (Storrs)
MSC:
| 60L20 | Rough paths |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
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