通过随机量子化构造 ${\mathbb R}^3$ 上的非高斯且旋转不变的 $\Phi^4$-测度及相关流
Construction of a non-Gaussian and rotation-invariant $Φ^4$-measure and associated flow on ${\mathbb R}^3$ through stochastic quantization
AI总结：
本文结合半群与随机偏微分方程方法，构造了量子场论 $\phi^4_3$-模型对应的非高斯、旋转不变且反射正的概率测度及其随机量子化流，并利用加权 Besov 空间理论证明了相关过程的紧性与平稳测度性质。
摘要：
本文提出了一种与量子场论的 $\phi^4_3$-模型相关的非高斯、旋转不变且反射正的概率测度 $\mu$ 的新构造方法。我们的构造结合了半群方法，以及随机偏微分方程 (SPDEs) 的方法，用于寻找与 $\phi^4_3$-模型相关的自然随机量子化的解和平稳测度。我们的出发点是我们打算构造的测度 $\mu$ 的一个合适近似 $\mu_{M,N}$。$\mu_{M,N}$ 由依赖于 $M$ 的空间截断函数 $\rho_M: {\mathbb R}^3\rightarrow {\mathbb R}$ 和依赖于 $N$ 的动量截断函数 $\psi_N: \widehat{\mathbb R}^3 \cong {\mathbb R}^3 \rightarrow {\mathbb R}$ 参数化，它们作用于相互作用项（非线性项和抵消项）。相应的随机量子化方程族产生了解 $(X_t^{M,N}, t\geq 0)$，其以 $\mu_{M,N}$ 作为不变概率测度。通过将概率论和泛函分析方法相结合，用于处理具有负指标加权 Besov 空间（具有旋转不变权重）上的奇异随机微分方程，我们证明了连续过程族 $(X_t^{M,N},t \geq 0)_{M,N}$ 的紧性。当 $M$ 和 $N$ 均发散至 $+\infty$ 时，依分布收敛意义上的极限点存在。极限过程 $(X_t; t\geq 0)$ 在合适的 Besov 空间的交集上是连续的，且 $\mu_{M,N}$ 的任何极限点 $\mu$ 都是 $X$ 的平稳测度。我们证明了 $\mu$ 是一个旋转不变的非高斯概率测度，并提供了关于其支撑的结果。本文还证明了 $\mu$ 满足属于欧几里得量子场公理族的另一个重要性质，即反射正性。
英文摘要：
A new construction of non-Gaussian, rotation-invariant and reflection positive probability measures $μ$ associated with the $φ^4_3$-model of quantum field theory is presented. Our construction uses a combination of semigroup methods, and methods of stochastic partial differential equations (SPDEs) for finding solutions and stationary measures of the natural stochastic quantization associated with the $φ^4_3$-model. Our starting point is a suitable approximation $μ_{M,N}$ of the measure $μ$ we intend to construct. $μ_{M,N}$ is parametrized by an $M$-dependent space cut-off function $ρ_M: {\mathbb R}^3\rightarrow {\mathbb R}$ and an $N$-dependent momentum cut-off function $ψ_N: \widehat{\mathbb R}^3 \cong {\mathbb R}^3 \rightarrow {\mathbb R}$, that act on the interaction term (nonlinear term and counterterms). The corresponding family of stochastic quantization equations yields solutions $(X_t^{M,N}, t\geq 0)$ that have $μ_{M,N}$ as an invariant probability measure. By a combination of probabilistic and functional analytic methods for singular stochastic differential equations on negative-indices weighted Besov spaces (with rotation invariant weights) we prove the tightness of the family of continuous processes $(X_t^{M,N},t \geq 0)_{M,N}$. Limit points in the sense of convergence in law exist, when both $M$ and $N$ diverge to $+\infty$. The limit processes $(X_t; t\geq 0)$ are continuous on the intersection of suitable Besov spaces and any limit point $μ$ of the $μ_{M,N}$ is a stationary measure of $X$. $μ$ is shown to be a rotation-invariant and non-Gaussian probability measure and we provide results on its support. It is also proven that $μ$ satisfies a further important property belonging to the family of axioms for Euclidean quantum fields, it is namely reflection positive.
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