Equivariant degree approach to random periodic solutions in second-order stochastic systems with symmetry

We study second-order stochastic systems with finite symmetry through an associated deterministic mean-field equation. This reduction provides a symmetry-preserving setting in which equivariant degree methods can be applied to the periodic problem. After reformulating the mean-field equation as a compact perturbation of the identity, we define a mean-field equivariant degree and introduce a degree-jump invariant for detecting bifurcation from the trivial branch. A nonzero degree jump yields nontrivial $2\pi$-periodic solutions, while its Burnside-ring components provide information on possible symmetry types of the bifurcating branches. We then formulate conditional lifting principles connecting mean-field periodic solutions with random periodic solutions of the original stochastic system through measurable complete random trajectories with prescribed mean motion. A sufficient framework based on a compact invariant random set and exponential contraction is also established. Examples with dihedral symmetry illustrate the computation of the invariant and the detection of maximal orbit types.