Renormalisation in the presence of variance blowup

2024/01/19 by Hairer, Martin · 3 citations
#60H10 #60H15 #60L20 #60L30 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2401.10868

Abstract

We show that if one drives the KPZ equation by the derivative of a space-time white noise smoothened out at scale ε ≪ 1 and multiplied by ε^3/4 then, as ε → 0, solutions converge to the Cole-Hopf solutions to the KPZ equation driven by space-time white noise. In the same vein, we also show that if one drives an SDE by fractional Brownian motion with Hurst parameter H < 1/4, smoothened out at scale ε ≪ 1 and multiplied by ε^1/4-H then, as ε → 0, solutions converge to an SDE driven by white noise. The mechanism giving rise to both results is the same, but the proof techniques differ substantially.

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• Transportation cost inequalities for singular SPDEs
• Probabilistic well-posedness of dispersive PDEs beyond variance blowup I: Benjamin-Bona-Mahony equation
• Renormalisation of singular SPDEs with Correlated Coefficients

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