Finite $p$-variation of solutions to (stochastic) evolution equations and applications in numerical analysis
A talk in the BI.discrete Workshop series by
Raphael Kruse from Halle (Saale)
| Abstract: | In numerical analysis of (stochastic) evolution equations one often depends on estimates of the temporal regularity of the exact solution to derive the optimal order of convergence for discretization methods. For instance, the order of convergence of the backward Euler method typically agrees with the exponent $\gamma \in (0,1]$ of Hölder continuity of the exact solution. In this talk we discuss how to measure the temporal regularity of the exact solution in terms of the $p$-variation semi-norm. As it turns out, this notion of regularity is weaker than that of Hölder continuity. In particular, if the solution is $\gamma$-Hölder continuous then it is also of finite $p$-variation for any $p \in [\frac{1}{\gamma}, \infty)$. In addition, it allows to treat (stochastic) evolution equations with non-smooth initial values or whose regularity is measured in terms of fractional Sobolev spaces $W^{\sigma,q}(0,T;H)$, $\sigma \in (\frac{1}{q},1)$, in a unified setting. At the same time, the $p$-variation semi-norm still allows to derive the optimal order of convergence of numerical methods for the temporal discretization of stochastic evolution equations. This is joint work with Johanna Weinberger and Rico Weiske (both MLU Halle-Wittenberg). Attendance is only possible after registration with the organizers and with 3G-certificate. Within the CRC this talk is associated to the project(s): A7, B7 |