Abstract
We construct a K-rough path [along the terminology of Deya (Probab Theory Relat Fields 166:1–65, 2016)] above either a space-time or a spatial fractional Brownian motion, in any space dimension d. This allows us to provide an interpretation and a unique solution for the corresponding parabolic Anderson model, understood in the renormalized sense. We also consider the case of a spatial fractional noise.
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Acknowledgements
We are grateful to an anonymous reviewer for his/her careful reading of the paper and his/her comments about it. In particular, we would like to thank him/her for drawing our attention to the other possible approach to the renormalization issue evoked in Remark 3.7.
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C. Ouyang is supported in part by Simons Grant #355480. S. Tindel is supported by the NSF Grant DMS-1952966.
Appendix
Appendix
1.1 Proof of Lemma 3.2
We only focus on the treatment of \(\mathcal {J}_{\rho ,H_0,\mathbf {H}}\) [defined in (3.8)] when \(2H_0+H<d+1\). It should however be clear to the reader that the subsequent arguments could also be used to prove the finiteness of the integral in (3.7) when \(2H_0+H=d+1\).
According to the definition (2.3) of the heat kernel p and recalling that \(\mathcal {F}\) stands for the space-time Fourier transform, it is readily checked that for \((\lambda ,\xi )\in \mathbb {R}^{d+1}\) we have
Therefore, the integral under consideration can be bounded as
where we consider a compact region \(\mathcal {D}_\mathfrak {s}\) of \(\mathbb {R}^{d+1}\) defined by
and where the quantities \(\mathcal {J}_\infty , \mathcal {J}_0\) are respectively defined by
We now proceed to the evaluation of those two terms.
In order to estimate \(\mathcal {J}_\infty \), note that \((\mathbb {R}^{d+1}\backslash \mathcal {D}_\mathfrak {s}) \subset \cup _{i=0}^d \Lambda _i\), where the regions \(\Lambda _i\) are defined by
According to this decomposition we write
where the terms \(\mathcal {J}_{\infty ,i}\) can be written as
Let us now show how to bound \(\mathcal {J}_{\infty ,0}\) above. To this aim we invoke our bound (3.2) in two different ways. Namely we take \(\tau _0=1\), and \(\tau _i=0\) if \(|\xi _i|\le 1\), while \(\tau _i=1\) if \(|\xi _i|\ge 1\). Together with the trivial inequality \(\lambda ^2+\sum _{i=1}^d\xi _i^4\ge \lambda ^2\), the term \(\mathcal {J}_{\infty ,0}\) given in (6.6) can be bounded as follows
where the last inequality is immediate. The terms \(\mathcal {J}_{\infty ,i}\) for \(i=1,\ldots ,d\) in (6.6) are handled similarly, and we omit the details for the sake of conciseness. Taking into account the upper bound (6.5), we end up with the relation \(\mathcal {J}_\infty <\infty \).
We now turn to a bound on \(\mathcal {J}_0\) defined by (6.4), for which we invoke (3.2) with \(\tau _i=0,\) for all \( i=0,\ldots ,d\). We get
To see that the latter integral is indeed finite, let us set \(\tilde{\xi }_i:=\xi _i^2\), so that \((\lambda ,\xi _1,\ldots ,\xi _d)\in \mathcal {D}_\mathfrak {s}\cap \mathbb {R}_+^{d+1}\) if and only if \((\lambda ,\tilde{\xi }_1,\ldots ,\tilde{\xi }_d)\in \mathcal {B}(0,1) \cap \mathbb {R}_+^{d+1}\), where \(\mathcal {B}(0,1)\) stands for the standard Euclidean unit ball. This yields
where we have used spherical coordinates to derive the last inequality. The finiteness of \(\mathcal {J}_0\) now follows from the assumption \(2H_0+H<d+1\).
Summarizing our computations, we have seen that \(\mathcal {J}_0<\infty \) and \(\mathcal {J}_\infty <\infty \). Recalling relation (6.2), this proves our claim \(\mathcal {J}_{\rho ,H_0,\mathbf {H}}<\infty \).
1.2 Proof of Proposition 3.8
Let us decompose the integral under consideration as
Using a series of elementary changes of variable, we get, for some constant \(C_{H_0,\mathbf {H}}\) that may change from line to line,
and so, recalling that \(2H_0+H=d+1\), we end up with
On the other hand, thanks to Assumption \((\rho )\)-(i)-(ii), we have
where the last inequality is immediately derived from the assumption \(2H_0+H=d+1\). Thus,
Finally, injecting (6.11) and (6.12) into (6.10), we deduce the desired decomposition (3.17).
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Chen, X., Deya, A., Ouyang, C. et al. A K-rough path above the space-time fractional Brownian motion. Stoch PDE: Anal Comp 9, 819–866 (2021). https://doi.org/10.1007/s40072-020-00186-3
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DOI: https://doi.org/10.1007/s40072-020-00186-3
Keywords
- Parabolic Anderson model
- Regularity structures
- Stratonovich equation
- Space-time fractional Brownian motion