Ito Formula for RPDEs and boundedness of solutions
A talk in the Bielefeld Stochastic Afternoon series by
Antoine Hocquet from TU Berlin
| Abstract: | An important question is whether Stochastic Partial differential equations can be treated path by path. In the case of an SDE driven by a Fractional Brownian motion B, it is well-known that any reference to a probability space can be avoided, provided one has a suitable definition for the iterated integrals of B against itself (see e.g. Lyons '98, Gubinelli '04).
Generalizations of such results in the case of SPDEs is however not at all straightforward, even in the "easier case" of SPDEs driven by a finite dimensional signal. Recently, much effort has been done in that direction (see e.g. Gubinelli-Imkeller-Perkowski '12, Hairer '14, see also Friz and coauthors). Although these approaches are quite successful, they rely mainly on the existence of a mild formulation (i.e. using the heat semigroup). Some treatments are also possible using flow transformations (the SPDE is then equivalent to a deterministic equation with random coefficients), but this limits the range of possible RPDEs to deal with.
In order to overcome these limitations, we will introduce a formulation using test functions, which is somewhat more faithful to the classical treatment of partial differential equations.
We will show that for a relatively large class of equations, this formulation allows us to have an "Ito Formula" (i.e. a chain rule), which is crucial in this context. We will provide an application to the maximum principle. |