Pathwise solutions to stochastic evolution equations

Monday, October 22, 2012 - 3:15pm - 4:05pm
Keller 3-180
María Garrido-Atienza (University of Sevilla)
In this talk, we combine tools from classical fractional calculus and the Rough Path Theory to study the existence and uniqueness of mild solutions to evolutions equations driven by a Hölder continuous function with Hölder exponent in the interval $(1/3,1/2)$. The stochastic integral is given by a generalization of the well-known Young integral and can be defined independently of the initial condition. In order to formulate an operator equation solving the problem we need a second equation for the so called area in the space of tensors, and the key ingredient to get this is to construct a tensor depending on the noise path and also on the semigroup. We prove in a first step the existence of a unique Hölder continuous solution of the system of equations, consisting of the path and the area components, if the nonlinear term and the initial condition are sufficiently smooth. In a second part, we also prove similar results when considering more general initial states, by modifying accordingly the phase space.

The abstract theory is applicable to evolution equations driven by a fractional Brownian motion $B^H$ with Hurst parameter $H in (1/3,1/2]$. One important result is that the pathwise definition of the stochastic integral allows to prove that the solution process generates a random dynamical system.

This is a join work with Kening Lu (BYU, Provo) and Björn Schmalfuss (Universität Jena).
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