math.PR — Probability
Stochastic Connections on Surface Bundles
We construct local stochastic connection dynamics on arbitrary principal bundles over closed oriented Riemannian surfaces. For each smooth reference connection, the bare additive spatial heat approximations of the Yang–Mills–DeTurck equation converge to a maximal local strong Markov family in a separable negative Hölder space. A finite Gaussian enhancement defines the singular products, and cancellation of the total logarithmic mass identifies the limiting equation. Rough comparison gauges and predictable rotations of the Wiener noise identify the local orbit laws for different references. The finite quadratic tensor anomaly vanishes after the two-dimensional Yang–Mills contraction, with the finite normalisation fixed by the spatial heat approximation. Continuing through bounded representatives gives a reference-independent maximal local strong Markov family on the full rough orbit space. Gaussian line primitives have every Hölder exponent below one on bounded regular curve families and define non-Abelian Young parallel transport. On a dense, gauge-invariant Borel full-heat trace domain, polygonal holonomies separate rough gauge orbits, admit Borel representative maps, and recover the bundle class. The orbit dynamics has continuous holonomy paths on this domain, and the regularised holonomies converge uniformly on bounded regular curve families and compact surviving positive-time intervals. We also prove regularisation-independent convergence of first-order Gaussian currents after time integration, including a harmonic example of infinite variation and zero quadratic variation. At fixed regularisation scale, the field-dependent covariant heat approximation has classical solutions and an explicit Itô correction in Coulomb coordinates. In every Abelian sector, the quotient diffusion is global, reference independent, and reversible with an explicitly identified unique invariant law.