Existence, uniqueness, and universality of global dynamics for the fractional hyperbolic ${\mathrm{\Phi }}_3^4$-model

We study the fractional ${\mathrm{\Phi }}_3^4$-measure (with order $\alpha >1$) and the dynamical problem of its canonical stochastic quantization: the three-dimensional stochastic damped fractional nonlinear wave equation with a cubic nonlinearity, also called the fractional hyperbolic ${\mathrm{\Phi }}_3^4$-model. We first construct the fractional ${\mathrm{\Phi }}_3^4$-measure via the variational approach by Barashkov-Gubinelli (2020). When $\alpha \le \frac{9}{8}$, this fractional ${\mathrm{\Phi }}_3^4$-measure turns out to be singular with respect to the base Gaussian measure. We then prove almost sure global well-posedness of the fractional hyperbolic ${\mathrm{\Phi }}_3^4$-model and invariance of the fractional ${\mathrm{\Phi }}_3^4$-measure for all $\alpha >1$ by further developing the globalization framework due to Oh-Okamoto-Tolomeo (2024) on the hyperbolic ${\mathrm{\Phi }}_3^3$-model. Furthermore, when $\alpha >\frac{9}{8}$ , we prove weak universality of the fractional hyperbolic ${\mathrm{\Phi }}_3^4$-model by utilizing the convergence of Gibbs measures.