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Abstract We define a state space and a Markov process associated to the stochastic quantisation equation of Yang–Mills–Higgs (YMH) theories. The state space$$\mathcal{S}$$ $S$is a nonlinear metric space of distributions, elements of which can be used as initial conditions for the (deterministic and stochastic) YMH flow with good continuity properties. Using gauge covariance of the deterministic YMH flow, we extend gauge equivalence ∼ to$$\mathcal{S}$$ $S$and thus define a quotient space of “gauge orbits”$$\mathfrak {O}$$ $O$. We use the theory of regularity structures to prove local in time solutions to the renormalised stochastic YMH flow. Moreover, by leveraging symmetry arguments in the small noise limit, we show that there is a unique choice of renormalisation counterterms such that these solutions are gauge covariant in law. This allows us to define a canonical Markov process on$$\mathfrak {O}$$ $O$(up to a potential finite time blow-up) associated to the stochastic YMH flow. 

In this Review, we review some recent rigorous results on large N problems in quantum field theory, stochastic quantization, and singular stochastic partial differential equations (SPDEs) and their mean field limit problems. In particular, we discuss the O( N) linear sigma model on a two- and three-dimensional torus. The stochastic quantization procedure leads to a coupled system of N interacting Φ 4 equations. In d = 2, we show uniformity in N bounds for the dynamics and convergence to a mean-field singular SPDE. For large enough mass or small enough coupling, the invariant measures [i.e., the O( N) linear sigma model] converge to the massive Gaussian free field, the unique invariant measure of the mean-field dynamics, in a Wasserstein distance. We also obtain tightness for certain O( N) invariant observables as random fields in suitable Besov spaces as N → ∞, along with exact descriptions of the limiting correlations. In d = 3, the estimates become more involved since the equation is more singular. We discuss in this case how to prove convergence to the massive Gaussian free field. The proofs of these results build on the recent progress of singular SPDE theory and combine many new techniques, such as uniformity in N estimates and dynamical mean field theory. These are based on joint papers with Scott Smith, Rongchan Zhu, and Xiangchan Zhu. 

Abstract We define a natural state space and Markov process associated to the stochastic Yang–Mills heat flow in two dimensions. To accomplish this we first introduce a space of distributional connections for which holonomies along sufficiently regular curves (Wilson loop observables) and the action of an associated group of gauge transformations are both well-defined and satisfy good continuity properties. The desired state space is obtained as the corresponding space of orbits under this group action and is shown to be a Polish space when equipped with a natural Hausdorff metric. To construct the Markov process we show that the stochastic Yang–Mills heat flow takes values in our space of connections and use the “DeTurck trick” of introducing a time dependent gauge transformation to show invariance, in law, of the solution under gauge transformations. Our main tool for solving for the Yang–Mills heat flow is the theory of regularity structures and along the way we also develop a “basis-free” framework for applying the theory of regularity structures in the context of vector-valued noise – this provides a conceptual framework for interpreting several previous constructions and we expect this framework to be of independent interest. 

Abstract In this paper we introduce the stochastic Ricci flow (SRF) in two spatial dimensions. The flow is symmetric with respect to a measure induced by Liouville conformal field theory. Using the theory of Dirichlet forms, we construct a weak solution to the associated equation of the area measure on a flat torus, in the full “$L^1$ regime” $$\sigma < \sigma _{L^1}=2 \sqrt \pi $$ where $$\sigma $$ is the noise strength. We also describe the main necessary modifications needed for the SRF on general compact surfaces and list some open questions.