Skip to main content
Log in

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Chen, X.: Parabolic Anderson model with rough or critical Gaussian noise. Ann. Inst. Henri Poincaré Probab. Stat. 55(2), 941–976 (2019)

    Article  MathSciNet  Google Scholar 

  2. Chen, X., Deya, A., Ouyang, C., Tindel, S.: Moment estimates for some renormalized parabolic Anderson models (Submitted) (2020). arXiv:2003.14367

  3. Deya, A.: On a modelled rough heat equation. Probab. Theory Relat. Fields 166, 1–65 (2016)

    Article  MathSciNet  Google Scholar 

  4. Deya, A.: Construction and Shorohod representation of a fractional \(K\)-rough path. Electron. J. Probab. 22, 1–40 (2017)

    Article  MathSciNet  Google Scholar 

  5. Hairer, M.: A theory of regularity structures. Invent. Math. 198(2), 269–504 (2014)

    Article  MathSciNet  Google Scholar 

  6. Hairer, M., Labbé, C.: Multiplicative stochastic heat equations on the whole space. J. Eur. Math. Soc. 20(4), 1005–1054 (2018)

    Article  MathSciNet  Google Scholar 

  7. Hu, Y., Huang, J., Nualart, D., Tindel, S.: Stochastic heat equations with general multiplicative Gaussian noises: Hölder continuity and intermittency. Electron. J. Probab. 20(55), 1–50 (2015)

    MathSciNet  MATH  Google Scholar 

  8. Hu, Y., Nualart, D.: Stochastic heat equation driven by fractional noise and local time. Probab. Theory Relat. Fields 143(1–2), 285–328 (2009)

    Article  MathSciNet  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Aurélien Deya.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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

$$\begin{aligned} {\mathcal {F}}{p}(\lambda ,\xi )=\left( \frac{|\xi |^2}{2}+\imath \lambda \right) ^{-1}. \end{aligned}$$
(6.1)

Therefore, the integral under consideration can be bounded as

$$\begin{aligned} \mathcal {J}_{\rho ,H_0,\mathbf {H}} \le \mathcal {J}_{\infty } +\mathcal {J}_{0}, \end{aligned}$$
(6.2)

where we consider a compact region \(\mathcal {D}_\mathfrak {s}\) of \(\mathbb {R}^{d+1}\) defined by

$$\begin{aligned} \mathcal {D}_\mathfrak {s}:=\{(\lambda ,\xi )\in \mathbb {R}^{d+1}: \, \lambda ^2+\xi _1^4+\cdots +\xi _d^4\le 1\}, \end{aligned}$$
(6.3)

and where the quantities \(\mathcal {J}_\infty , \mathcal {J}_0\) are respectively defined by

$$\begin{aligned} \mathcal {J}_{\infty }:= & {} \int _{\mathbb {R}^{d+1}\backslash \mathcal {D}_\mathfrak {s}} \frac{ d\lambda d\xi }{(\lambda ^2+\xi _1^4 +\cdots +\xi _d^4)^{1/2}}|\mathcal F{\rho }(\lambda ,\xi )|^2 \mathcal {N}_{H_0,\mathbf {H}}(\lambda ,\xi ) \nonumber \\ \mathcal {J}_{0}:= & {} \int _{\mathcal {D}_\mathfrak {s}} \frac{ d\lambda d\xi }{(\lambda ^2+\xi _1^4 +\cdots +\xi _d^4)^{1/2}}|{\mathcal {F}}{\rho }(\lambda ,\xi )|^2 \mathcal {N}_{H_0,\mathbf {H}}(\lambda ,\xi ). \end{aligned}$$
(6.4)

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

$$\begin{aligned} \Lambda _0:=\left\{ (\lambda ,\xi _1,\ldots ,\xi _d): \, \lambda ^2\ge \frac{1}{d+1}\right\} \quad \text {and} \quad \Lambda _i:=\left\{ (\lambda ,\xi _1,\ldots ,\xi _d): \, \xi _i^4\ge \frac{1}{d+1}\right\} . \end{aligned}$$

According to this decomposition we write

$$\begin{aligned} \mathcal {J}_{\infty }\le \sum _{i=0}^d\mathcal {J}_{\infty ,i}, \end{aligned}$$
(6.5)

where the terms \(\mathcal {J}_{\infty ,i}\) can be written as

$$\begin{aligned} \mathcal {J}_{\infty ,i}:=\int _{\Lambda _i} \frac{ d\lambda d\xi }{(\lambda ^2+\xi _1^4 +\cdots +\xi _d^4)^{1/2}}|{\mathcal {F}}{\rho }(\lambda ,\xi )|^2 \mathcal {N}_{H_0,\mathbf {H}}(\lambda ,\xi ). \end{aligned}$$
(6.6)

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

$$\begin{aligned} \mathcal {J}_{\infty ,0}\lesssim \bigg (\int _{\lambda ^2\ge \frac{1}{d+1}} \frac{ d\lambda }{|\lambda |^{2H_0+2}}\bigg ) \prod _{i=1}^d \bigg \{ \int _{|\xi _i|\le 1} \frac{d\xi _i}{|\xi _i|^{2H_i-1}}+\int _{|\xi _i|\ge 1} \frac{d\xi _i}{|\xi _i|^{2H_i+1}} \bigg \} \ < \ \infty , \end{aligned}$$
(6.7)

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

$$\begin{aligned} \mathcal {J}_0\lesssim \int _{\mathcal {D}_\mathfrak {s}\cap \mathbb {R}_+^{d+1}} \frac{ d\lambda d\xi }{(\lambda ^2+\xi _1^4 +\cdots +\xi _d^4)^{1/2}}\mathcal {N}_{H_0,\mathbf {H}}(\lambda ,\xi ) . \end{aligned}$$
(6.8)

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

$$\begin{aligned} \mathcal {J}_0&\lesssim \int _{\mathcal {B}(0,1)\cap \mathbb {R}_+^{d+1}} \frac{d\lambda d\tilde{\xi }}{(\lambda ^2+\tilde{\xi }_1^2 +\cdots +\tilde{\xi }_d^2)^{1/2}} \frac{1}{|\lambda |^{2H_0-1}}\bigg (\prod _{i=1}^d \frac{1}{|\tilde{\xi }_i|^{1/2}}\bigg ) \mathcal {N}_{H_0,\mathbf {H}}\big (\lambda ,\tilde{\xi }_1^{1/2},\ldots ,\tilde{\xi }_d^{1/2}\big )\nonumber \\&\lesssim \int _{\mathcal {B}(0,1)\cap \mathbb {R}_+^{d+1}} \frac{d\lambda d\tilde{\xi }}{(\lambda ^2+\tilde{\xi }_1^2 +\cdots +\tilde{\xi }_d^2)^{1/2}} \frac{1}{|\lambda |^{2H_0-1}} \prod _{i=1}^d \frac{1}{|\tilde{\xi }_i|^{H_i}}\lesssim \int _0^1 \frac{dr}{r^{2H_0+H-d}} , \end{aligned}$$
(6.9)

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

$$\begin{aligned}&\int _{|\lambda |+|\xi |^2\ge 2^{-2n}} |{\mathcal {F}}{\rho }(\lambda ,\xi )|^2 {\mathcal {F}}{p}(\lambda ,\xi ) \mathcal {N}_{H_0,\mathbf {H}}(\lambda ,\xi )\, d\lambda d\xi \nonumber \\&\quad =\int _{2^{-2n}\le |\lambda |+|\xi |^2\le 1} {\mathcal {F}}{p}(\lambda ,\xi ) \mathcal {N}_{H_0,\mathbf {H}}(\lambda ,\xi )\, d\lambda d\xi \nonumber \\&\quad +\int _{2^{-2n}\le |\lambda |+|\xi |^2\le 1} \big \{|{\mathcal {F}}{\rho }(\lambda ,\xi )|^2-1\big \} {\mathcal {F}}{p}(\lambda ,\xi ) \mathcal {N}_{H_0,\mathbf {H}}(\lambda ,\xi )\, d\lambda d\xi +O(1). \end{aligned}$$
(6.10)

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,

$$\begin{aligned}&\int _{2^{-2n}\le |\lambda |+|\xi |^2\le 1} {\mathcal {F}}{p}(\lambda ,\xi ) \mathcal {N}_{H_0,\mathbf {H}}(\lambda ,\xi )\, d\lambda d\xi =\int _{2^{-2n}\le |\lambda |+|\xi |^2\le 1} \frac{d\lambda d\xi }{\frac{|\xi |^2}{2}+\imath \lambda } \frac{1}{|\lambda |^{2H_0-1}} \prod _{i=1}^d \frac{1}{|\xi _i|^{2H_i-1}}\\&\quad =C_{H_0,\mathbf {H}} \int _0^\infty dr \int _{2^{-2n}\le |\lambda |+r^2\le 1}\frac{d\lambda }{\frac{r^2}{2}+\imath \lambda } \frac{r^{2d-2H-1}}{|\lambda |^{2H_0-1}}\\&\quad =C_{H_0,\mathbf {H}} \int _0^\infty dr \int _0^\infty d\lambda \, \mathbf{1}_{2^{-2n}\le \lambda +r^2\le 1}\bigg [\frac{1}{\frac{r^2}{2}+\imath \lambda } +\frac{1}{\frac{r^2}{2}-\imath \lambda } \bigg ] \frac{r^{2d-2H-1}}{|\lambda |^{2H_0-1}}\\&\quad =C_{H_0,\mathbf {H}} \int _0^\infty dr \int _0^\infty d\lambda \, \mathbf{1}_{2^{-2n}\le \lambda +r^2\le 1}\bigg (\frac{r^2}{\frac{r^4}{4}+ \lambda ^2}\bigg ) \frac{r^{2d-2H-1}}{|\lambda |^{2H_0-1}}\\&\quad =C_{H_0,\mathbf {H}} \int _0^\infty dr \int _0^\infty d\tilde{\lambda }\, \mathbf{1}_{2^{-2n}\le \tilde{\lambda }^2+r^2\le 1}\frac{\tilde{\lambda }}{\frac{r^4}{4}+ \tilde{\lambda }^4} \frac{r^{2d-2H+1}}{|\tilde{\lambda }|^{4H_0-2}}\\&\quad =C_{H_0,\mathbf {H}} \bigg (\int _0^\infty d\rho \frac{\mathbf{1}_{2^{-2n}\le \rho ^2\le 1}}{\rho ^{2(2H_0+H)-2d-1}}\bigg ) \bigg (\int _0^{\frac{\pi }{2}} \, \frac{d\theta }{\frac{\cos ^4 \theta }{4}+ \sin ^4 \theta } \frac{(\cos \theta )^{2d-2H+1}}{(\sin \theta )^{4H_0-3}}\bigg ) \end{aligned}$$

and so, recalling that \(2H_0+H=d+1\), we end up with

$$\begin{aligned} \int _{2^{-2n}\le |\lambda |+|\xi |^2\le 1} {\mathcal {F}}{p}(\lambda ,\xi ) \mathcal {N}_{H_0,\mathbf {H}}(\lambda ,\xi )\, d\lambda d\xi =C_{H_0,\mathbf {H}} \bigg (\int _{2^{-n}}^1 \frac{d\rho }{\rho } \bigg )=C_{H_0,\mathbf {H}} \cdot n. \end{aligned}$$
(6.11)

On the other hand, thanks to Assumption \((\rho )\)-(i)-(ii), we have

$$\begin{aligned}&\int _{2^{-2n}\le |\lambda |+|\xi |^2\le 1} \big | |{\mathcal {F}}{\rho }(\lambda ,\xi )|^2-1\big | \big | {\mathcal {F}}{p}(\lambda ,\xi )\big | \mathcal {N}_{H_0,\mathbf {H}}(\lambda ,\xi )\, d\lambda d\xi \\&\quad =\int _{2^{-2n}\le |\lambda |+|\xi |^2\le 1} \big | |{\mathcal {F}}{\rho }(\lambda ,\xi )|^2-|{\mathcal {F}}{\rho }(0,0)|^2\big | \big | {\mathcal {F}}{p}(\lambda ,\xi )\big | \mathcal {N}_{H_0,\mathbf {H}}(\lambda ,\xi )\, d\lambda d\xi \\&\quad \lesssim \int _{0\le |\lambda |+|\xi |^2\le 1} \big \{ |\lambda |+|\xi |\big \} \big | {\mathcal {F}}{p}(\lambda ,\xi )\big | \mathcal {N}_{H_0,\mathbf {H}}(\lambda ,\xi )\, d\lambda d\xi \\&\quad \lesssim \int _0^\infty dr \int _0^\infty d\lambda \, \mathbf{1}_{0\le \lambda +r^2\le 1}\, \big \{ \lambda +r\big \} \frac{r^{2d-2H-1}}{r^2+\lambda } \frac{1}{\lambda ^{2H_0-1}}\\&\quad \lesssim \int _0^\infty dr \int _0^\infty d\lambda \, \mathbf{1}_{0\le \lambda ^2+r^2\le 1}\, \lambda \big \{ \lambda ^2+r\big \} \frac{r^{2d-2H-1}}{r^2+\lambda ^2} \frac{1}{\lambda ^{4H_0-2}}\\&\quad \lesssim \int _{0\le \rho ^2\le 1} d\rho \, \rho ^3 \frac{\rho ^{2d-2H-1}}{\rho ^2} \frac{1}{\rho ^{4H_0-2}} \lesssim \int _{0\le \rho ^2\le 1} \frac{d\rho }{\rho ^{2(2H_0+H)-2d-2}} \lesssim 1 , \end{aligned}$$

where the last inequality is immediately derived from the assumption \(2H_0+H=d+1\). Thus,

$$\begin{aligned} \sup _{n\ge 1} \bigg |\int _{2^{-2n}\le |\lambda |+|\xi |^2\le 1} \big \{|{\mathcal {F}}{\rho }(\lambda ,\xi )|^2-1\big \} {\mathcal {F}}{p}(\lambda ,\xi ) \mathcal {N}_{H_0,\mathbf {H}}(\lambda ,\xi )\, d\lambda d\xi \bigg | \ < \ \infty .\nonumber \\ \end{aligned}$$
(6.12)

Finally, injecting (6.11) and (6.12) into (6.10), we deduce the desired decomposition (3.17).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40072-020-00186-3

Keywords

Mathematics Subject Classification

Navigation