On the Necessary and Sufficient Conditions to Solve A Heat Equation with General Additive Gaussian Noise
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In this note, we consider stochastic heat equation with general additive Gaussian noise. Our aim is to derive some necessary and sufficient conditions on the Gaussian noise in order to solve the corresponding heat equation. We investigate this problem invoking two different methods, respectively, based on variance computations and on path-wise considerations in Besov spaces. We are going to see that, as anticipated, both approaches lead to the same necessary and sufficient condition on the noise. In addition, the path-wise approach brings out regularity results for the solution.
Key wordsKey words Stochastic heat equation general Gaussian noise L2 solution sufficient and necessary condition Wong-Zakai approximation pathwise solution Hölder continuity Besov space
2010 MR Subject Classification60G15 60H07 60H15
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