Wick Renormalized Parabolic Stochastic Quantization Equations on Rough Metric Measure Spaces
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911735617683456 |
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| author | Chen, Hongyi Yifan Yang |
| author_facet | Chen, Hongyi Yifan Yang |
| contents | On metric measure spaces with sub-Gaussian heat kernel behavior in small time, we obtain a sufficient condition to solve Wick renormalized stochastic quantization equations with polynomial interaction. Given the power of the nonlinearity, the local solution condition depends on the Hausdorff dimension $d_h$, the walk dimension $d_w$, and the maximal spatial Hölder regularity of the heat kernel $Θ$. A slightly more restrictive condition based on the same parameters is required for a global solution. For all global solutions, we construct an invariant measure for the Markov process defined by the solution. Our results apply to many rough spaces such as Barlow--Kigami type fractals as well as their Cartesian products and open up the possibility of making rigorous various structures in quantum field theory and statistical mechanics in non-integer dimensions. In the process, we build entirely from the short-time heat semigroup the necessary analytic framework that accommodates the issues which come with allowing rough local geometry. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_05442 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Wick Renormalized Parabolic Stochastic Quantization Equations on Rough Metric Measure Spaces Chen, Hongyi Yifan Yang Probability Mathematical Physics Analysis of PDEs 60H17, 28A80, 31C25, 46E35, 81T08 On metric measure spaces with sub-Gaussian heat kernel behavior in small time, we obtain a sufficient condition to solve Wick renormalized stochastic quantization equations with polynomial interaction. Given the power of the nonlinearity, the local solution condition depends on the Hausdorff dimension $d_h$, the walk dimension $d_w$, and the maximal spatial Hölder regularity of the heat kernel $Θ$. A slightly more restrictive condition based on the same parameters is required for a global solution. For all global solutions, we construct an invariant measure for the Markov process defined by the solution. Our results apply to many rough spaces such as Barlow--Kigami type fractals as well as their Cartesian products and open up the possibility of making rigorous various structures in quantum field theory and statistical mechanics in non-integer dimensions. In the process, we build entirely from the short-time heat semigroup the necessary analytic framework that accommodates the issues which come with allowing rough local geometry. |
| title | Wick Renormalized Parabolic Stochastic Quantization Equations on Rough Metric Measure Spaces |
| topic | Probability Mathematical Physics Analysis of PDEs 60H17, 28A80, 31C25, 46E35, 81T08 |
| url | https://arxiv.org/abs/2605.05442 |