A probabilistic approach to the Yang-Mills heat equation.
Let \(E\) be a metric vector bundle over a compact Riemannian manifold \(M\), and let \(R^\nabla\) denote the curvature 2-form of any metric connection \(\nabla\) on \(E\). Such connection \(v\) is said to be ``Yang-Mills'' when \(\int_M| R^\nabla|^2 d\text{\,vol}\) is critical, and when \((d^\nabla)^* R^\nabla\equiv 0\). Denote by \(X_t\) a Brownian motion on \(M\), started from \(x\in M\), by \(\|^t_0\) ist associated stochastic Levi-Civita parallel transport, and fix \(u\in T_x M\). Set then \(X^a_t:= \exp_{X_t}(a\sqrt{r}\,\|^t_0 u)\), and let \(U^a_{r,s}\) denote the parallel transport in \(E\) along \(X^a_s\) with respect to \(\nabla(T-s)\), satisfying \(U^a_{r,r}= \text{id}\), where \((\nabla(t);\,0\leq t\leq T)\) denotes a smooth curve in the space of metric connections on \(E\). The basic object studied in this article is the semimartingale \(N_{r,s}:= (U^0_{r,s})^{-1} (\nabla_a U^a_{r,s})|_{a=0}\), \(r\leq s\leq T\). It is first proved that \(N_{r,s}\) is a local martingale if and only if \(\partial_t \nabla(t)=-{1\over 2}(d^{\nabla(t)})^* R^{\nabla(t)}\). Next, for some \(\beta\in ]0,1[\), consider the energy \(\phi_\beta(s):= {1\over 2}\mathbb{E}[\| N_{\beta s,s}\|^2]\), which is shown to be nondecreasing. It is proved that \(\lim_{r \downarrow 0}\,\mathbb{E}[\| N_{r,T}\|^2]/\log(T/r)\) exists, and that if \(\phi_\beta(s)\) is small enough for small \(s\), then \(\phi_\beta\) is locally bounded. This allows in turn to establish a criterion of non-explosion for the Yang-Mills heat flow, which is fulfilled if \(\dim M\leq 3\).
scientific article; zbMATH DE number 1985989