Rough evolution equations
Massimiliano Gubinelli, Samy Tindel
TL;DR
This paper extends Lyons' rough paths theory to provide a pathwise framework for certain nonlinear infinite-dimensional evolution equations driven by irregular noise, with applications to specific 1D SPDEs.
Contribution
It generalizes rough paths theory to infinite-dimensional settings, enabling pathwise analysis of complex SPDEs driven by singular Gaussian noise.
Findings
Established a pathwise interpretation for nonlinear infinite-dimensional evolution equations.
Applied the theory to analyze specific 1D SPDEs with singular space covariance.
Demonstrated the framework's effectiveness in handling irregular noise in evolution equations.
Abstract
We generalize Lyons' rough paths theory in order to give a pathwise meaning to some nonlinear infinite-dimensional evolution equation associated to an analytic semigroup and driven by an irregular noise. As an illustration, we discuss a class of linear and nonlinear 1d SPDEs driven by a space--time Gaussian noise with singular space covariance and Brownian time dependence.
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