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KPZ equation with fractional derivatives of white noise

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Abstract

In this paper, we consider the KPZ equation driven by space-time white noise replaced with its fractional derivatives of order \(\gamma >0\) in spatial variable. A well-posedness theory for the KPZ equation is established by Hairer (Invent Math 198:269–504, 2014) as an application of the theory of regularity structures. Our aim is to see to what extent his theory works if noises become rougher. We can expect that his theory works if and only if \(\gamma <1/2\). However, we show that the renormalization like “\((\partial _x h)^2-\infty \)” is well-posed only if \(\gamma <1/4\).

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Acknowledgments

This work was supported by JSPS KAKENHI, Grant-in-Aid for JSPS Fellows, 16J03010. The author would like to thank Professor T. Funaki for leading him to the problem discussed in the present paper and asking Professor M. Hairer about this problem, who kindly pointed out that \(\gamma =\frac{1}{4}\) is the border. The author also thanks anonymous referees for their helpful remarks.

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  1. The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan

    Masato Hoshino

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  1. Masato Hoshino
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Correspondence to Masato Hoshino.

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Hoshino, M. KPZ equation with fractional derivatives of white noise. Stoch PDE: Anal Comp 4, 827–890 (2016). https://doi.org/10.1007/s40072-016-0078-x

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  • DOI: https://doi.org/10.1007/s40072-016-0078-x

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