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The recent interest in structure preserving stochastic Lagrangian andHamiltonian systems raises questions regarding how such models are to beunderstood and the principles through which they are to be derived. Byconsidering a mathematically sound extension of the Hamilton-Pontryaginprinciple, we derive a stochastic analogue of the Euler-Lagrange equations,driven by independent semimartingales. Using this as a starting point, we canapply symmetry reduction carefully to derive non-canonical stochasticLagrangian / Hamiltonian systems, including the stochastic Euler-Poincar\'e /Lie-Poisson equations, studied extensively in the literature. Furthermore, wedevelop a framework to include dissipation that balances thestructure-preserving noise in such a way that the overall stochastic dynamicspreserves the Gibbs measure on the symplectic manifold, where the dynamicseffectively take place. In particular, this leads to a new derivation ofdouble-bracket dissipation by considering Lie group invariant stochasticdissipative dynamics, taking place on the cotangent bundle of the group.

Semimartingale driven mechanics and reduction by symmetry for stochastic and dissipative dynamical systems