Abstract
We study a full discretization scheme for the stochastic linear heat equation
when is a very rough space-time fractional noise.
The discretization procedure is divised into three steps: regularization of the noise through a mollifying-type approach; discretization of the (smoothened) noise as a finite sum of Gaussian variables over rectangles in ; discretization of the heat operator on the (non-compact) domain , along the principles of Galerkin finite elements method.
We establish the convergence of the resulting approximation to , which, in such a specific rough framework, can only hold in a space of distributions. We also provide some partial simulations of the algorithm.
Acknowledgments
We are grateful to two anonymous reviewers for their careful reading and relevant suggestions, as well as for drawing our attention to the two references [2, 29].
Citation
Aurélien Deya. Renaud Marty. "A full discretization of the rough fractional linear heat equation." Electron. J. Probab. 27 1 - 41, 2022. https://doi.org/10.1214/22-EJP839
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