Abstract
We construct a piecewise linear approximation for the dynamical \(\Phi_3^4\) model on \(\mathbb{T}^3\). The approximation is based on the theory of regularity structures developed by Hairer (2014). They proved that renormalization in a dynamical \(\Phi_3^4\) model is necessary for defining the nonlinear term. In contrast to Hairer (2014), we apply piecewise linear approximations to space-time white noise, and prove that the solutions of the approximating equations converge to the solution of the dynamical \(\Phi_3^4\) model. In this case, the renormalization corresponds to multiplying the solution by a t-dependent function, and adding it to the approximating equation.
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This work was supported by National Natural Science Foundation of China (Grant Nos. 11671035 and 11771037).
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Zhu, R., Zhu, X. Piecewise linear approximation for the dynamical \(\Phi_3^4\) model. Sci. China Math. 63, 381–410 (2020). https://doi.org/10.1007/s11425-017-9269-1
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DOI: https://doi.org/10.1007/s11425-017-9269-1