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Mathematics  2014 

Large deviations for white-noise driven, nonlinear stochastic PDEs in two and three dimensions

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Abstract:

We study the stochastic Allen-Cahn equation driven by a noise term with intensity $\sqrt{\varepsilon}$ and correlation length $\delta$ in two and three spatial dimensions. We study diagonal limits $\delta, \varepsilon \to 0$ and describe fully the large deviation behaviour depending on the relationship between $\delta$ and $\varepsilon$. The recently developed theory of regularity structures allows to fully analyse the behaviour of solutions for vanishing correlation length $\delta$ and fixed noise intensity $\varepsilon$. One key fact is that in order to get non-trivial limits as $\delta \to 0$, it is necessary to introduce diverging counterterms. The theory of regularity structures allows to rigorously analyse this renormalisation procedure for a number of interesting equations. Our main result is a large deviation principle for these renormalised solutions. One interesting feature of this result is that the diverging renormalisation constants disappear at the level of the large deviations rate function. We apply this result to derive a sharp condition on $\delta, \varepsilon$ that guarantees a large deviation principle for diagonal schemes $\varepsilon, \delta \to 0$ for the equation without renormalisation.

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