Abstract
We consider the linear stochastic heat and wave equations with generalized Gaussian noise that is white in time and spatially correlated. Under the assumption that the homogeneous spatial correlation f satisfies some mild conditions, we show that the solutions to the linear stochastic partial differential equations (SPDEs) exhibit tall peaks in macroscopic scales, which means they are macroscopically multi-fractal. We compute the macroscopic Hausdorff dimension of the peaks for Gaussian random fields with vanishing correlation and then apply this result to the solution of the linear SPDEs. We also study the spatio-temporal multi-fractality of the linear SPDEs.
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Notes
Dirac measure at 0.
For all reals a and b, \(a\vee b\) denotes \(\max \{a,b\}\). Similarly, \(a \wedge b := \min \{a,b\}\).
For a set \(E\in \mathbb {R}^d\), we define \(\inf (A) := x_0\) where \(x_0\) satisfies \(\min _{x\in A} \Vert x\Vert = x_0\). We also define \(\inf (\varnothing ) := \infty \).
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Acknowledgements
The author would like to thank Professor Kunwoo Kim for his helpful comments and continued support. The author also would like to thank the anonymous referee for advice improving the presentation of the paper.
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Research supported by the NRF (National Research Foundation of Korea) grants 2019R1A5A1028324 and 2020R1A2C4002077.
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Yi, J. Macroscopic Multi-fractality of Gaussian Random Fields and Linear Stochastic Partial Differential Equations with Colored Noise. J Theor Probab 36, 926–947 (2023). https://doi.org/10.1007/s10959-022-01198-6
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DOI: https://doi.org/10.1007/s10959-022-01198-6
Keywords
- Stochastic heat equations
- Stochastic wave equations
- Multi-fractal
- Macroscopic Hausdorff dimension
- Gaussian random field
- colored noise