Itô perspective on variance renormalisation
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2026
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| _version_ | 1866914414694760448 |
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| author | Dareiotis, Konstantinos Gerencsér, Máté |
| author_facet | Dareiotis, Konstantinos Gerencsér, Máté |
| contents | We show that the Itô solutions of the nonlinear stochastic heat equation $$ \partial_t u^\varepsilon- Δu^\varepsilon =\varepsilon^{3/4} g (u^\varepsilon) \nabla ξ_\varepsilon, $$ where $ ξ_\varepsilon$ denotes the mollification in space at scale $\varepsilon>0$ of a space-time white noise $ξ$, converge in law, as $\varepsilon\to 0$, to the solution of the stochastic heat equation with right-hand side $cg'g(u)ξ$ with a constant $c>0$. Since the noise $\nablaξ$ is supercritical, the small prefactor is not unexpected to obtain a limit, but the exponent $3/4$ is not predicted by naive scaling arguments. The case $g(u)=u$, modulo a Cole-Hopf transform, corresponds to the result of [Hai25] for the KPZ equation. Our argument is relatively short and relies solely on stochastic analytic techniques. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_22272 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Itô perspective on variance renormalisation Dareiotis, Konstantinos Gerencsér, Máté Probability 60H15 We show that the Itô solutions of the nonlinear stochastic heat equation $$ \partial_t u^\varepsilon- Δu^\varepsilon =\varepsilon^{3/4} g (u^\varepsilon) \nabla ξ_\varepsilon, $$ where $ ξ_\varepsilon$ denotes the mollification in space at scale $\varepsilon>0$ of a space-time white noise $ξ$, converge in law, as $\varepsilon\to 0$, to the solution of the stochastic heat equation with right-hand side $cg'g(u)ξ$ with a constant $c>0$. Since the noise $\nablaξ$ is supercritical, the small prefactor is not unexpected to obtain a limit, but the exponent $3/4$ is not predicted by naive scaling arguments. The case $g(u)=u$, modulo a Cole-Hopf transform, corresponds to the result of [Hai25] for the KPZ equation. Our argument is relatively short and relies solely on stochastic analytic techniques. |
| title | Itô perspective on variance renormalisation |
| topic | Probability 60H15 |
| url | https://arxiv.org/abs/2603.22272 |