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arXiv 1812.04422math.PRmath-phmath.MP
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椭圆随机量子化

Elliptic stochastic quantization

Sergio Albeverio, Francesco C. De Vecchi, Massimiliano Gubinelli

备注：
51 pages, an appendix added
发表信息：
Ann. Probab. Volume 48, Number 4 (2020), 1693-1741
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摘要：

证明了一类椭圆型SPDE在$\mathbb{R}^2$中解的零点律的一个显式公式。这个公式是Parisi和Sourlas（1979）在物理学文献中发现的降维的最简单的例子，它将$d + 2$维中的椭圆SPDE定律与$d$维中的吉布斯测度联系起来。这种现象类似于$\mathbb{R}^{d + 1}$维抛物型随机微分方程与其$\mathbb{R}^d$维不变测度之间的关系。因此，椭圆随机偏微分方程的降维可以被认为是Nelson（1966）和Parisi and Wu（1981）意义下的一种椭圆随机量化过程。我们的证明使用在一个基本的方式表示的法律的SPDE作为一个超对称的量子场论。Klein等人（1984）已经建立了超对称理论的降维。我们修复了他们的证明中的一个微妙的差距，并通过提供椭圆SPDE和超对称模型之间的联系来完成降维图像。即使在我们的$d = 0$上下文中的参数是非平凡的和非超对称的，基本的证明似乎只有在高斯的情况下。

英文摘要：

We prove an explicit formula for the law in zero of the solution of a class of elliptic SPDE in $\mathbb{R}^2$. This formula is the simplest instance of dimensional reduction, discovered in the physics literature by Parisi and Sourlas (1979), which links the law of an elliptic SPDE in $d + 2$ dimension with a Gibbs measure in $d$ dimensions. This phenomenon is similar to the relation between an $\mathbb{R}^{d + 1}$ dimensional parabolic SPDE and its $\mathbb{R}^d$ dimensional invariant measure. As such, dimensional reduction of elliptic SPDEs can be considered a sort of elliptic stochastic quantisation procedure in the sense of Nelson (1966) and Parisi and Wu (1981). Our proof uses in a fundamental way the representation of the law of the SPDE as a supersymmetric quantum field theory. Dimensional reduction for the supersymmetric theory was already established by Klein et al. (1984). We fix a subtle gap in their proof and also complete the dimensional reduction picture by providing the link between the elliptic SPDE and the supersymmetric model. Even in our $d = 0$ context the arguments are non-trivial and a non-supersymmetric, elementary proof seems only to be available in the Gaussian case.

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