The Gaussian structure of the singular stochastic Burgers equation
Forum of Mathematics, Sigma, Tome 10 (2022)
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We consider the stochastically forced Burgers equation with an emphasis on spatially rough driving noise. We show that the law of the process at a fixed time t, conditioned on no explosions, is absolutely continuous with respect to the stochastic heat equation obtained by removing the nonlinearity from the equation. This establishes a form of ellipticity in this infinite-dimensional setting. The results follow from a recasting of the Girsanov Theorem to handle less spatially regular solutions while only proving absolute continuity at a fixed time and not on path-space. The results are proven by decomposing the solution into the sum of auxiliary processes, which are then shown to be absolutely continuous in law to a stochastic heat equation. The number of levels in this decomposition diverges to infinite as we move to the stochastically forced Burgers equation associated to the KPZ equation, which we conjecture is just beyond the validity of our results (and certainly the current proof). The analysis provides insights into the structure of the solution as we approach the regularity of KPZ. A number of techniques from singular SPDEs are employed, as we are beyond the regime of classical solutions for much of the paper.
@article{10_1017_fms_2022_64,
author = {Jonathan C. Mattingly and Marco Romito and Langxuan Su},
title = {The {Gaussian} structure of the singular stochastic {Burgers} equation},
journal = {Forum of Mathematics, Sigma},
publisher = {mathdoc},
volume = {10},
year = {2022},
doi = {10.1017/fms.2022.64},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2022.64/}
}
TY - JOUR AU - Jonathan C. Mattingly AU - Marco Romito AU - Langxuan Su TI - The Gaussian structure of the singular stochastic Burgers equation JO - Forum of Mathematics, Sigma PY - 2022 VL - 10 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.1017/fms.2022.64/ DO - 10.1017/fms.2022.64 LA - en ID - 10_1017_fms_2022_64 ER -
%0 Journal Article %A Jonathan C. Mattingly %A Marco Romito %A Langxuan Su %T The Gaussian structure of the singular stochastic Burgers equation %J Forum of Mathematics, Sigma %D 2022 %V 10 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.1017/fms.2022.64/ %R 10.1017/fms.2022.64 %G en %F 10_1017_fms_2022_64
Jonathan C. Mattingly; Marco Romito; Langxuan Su. The Gaussian structure of the singular stochastic Burgers equation. Forum of Mathematics, Sigma, Tome 10 (2022). doi: 10.1017/fms.2022.64
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