RG transformations are conditional expectations with respect to a scale filtration, so the RG flow is a martingale exploration process, a viewpoint previously developed by Bauerschmidt and Bodineau and by the author and Bauer.
Wilson-It\^o diffusions
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We introduce Wilson-It\^o diffusions, a class of random fields on $\mathbb{R}^d$ that change continuously along a scale parameter via a Markovian dynamics with local coefficients. Described via forward-backward stochastic differential equations, their observables naturally form a pre-factorization algebra \`a la Costello-Gwilliam. We argue that this is a new non-perturbative quantization method applicable also to gauge theories and independent of a path-integral formulation. Whenever a path-integral is available, this approach reproduces the setting of Wilson-Polchinski flow equations.
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The Renormalization Group as a Stochastic Exploration Process
RG transformations are conditional expectations with respect to a scale filtration, so the RG flow is a martingale exploration process, a viewpoint previously developed by Bauerschmidt and Bodineau and by the author and Bauer.