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We study ergodic properties of stochastic geometric wave equations on aparticular model with the target being the 2D sphere while considering thespace variable-independent solutions only. This simplification leads to adegenerate stochastic equation in the tangent bundle of the 2D sphere. Studyingthis equation, we prove existence and non-uniqueness of invariant probabilitymeasures for the original problem and we obtain also results on attractivitytowards an invariant measure. We also present a suitable numerical scheme forapproximating the solutions subject to a sphere constraint.

Ergodicity for a stochastic geodesic equation in the tangent bundle of the 2D sphere