Persistence of Algebraic Symplectic Orthogonality in Dissipative Stochastic Linear Oscillators

We analyze a linear Hamiltonian oscillator in R2 subject to isotropic linear dissipation, ad- ditive Gaussian noise, and sparse impulse perturbations modeled as jumps. We prove three rigorous statements under explicit moment assumptions: (1) uniform boundedness of the sec- ond moment (equivalently mean-square boundedness of quadratic energy); (2) a precise “or- thogonality” identity separating the Hamiltonian (skew) drift from the dissipative (symmetric) drift, clarifying that the flow is not symplectic but admits an algebraic generator decomposition with commuting parts in the isotropic case; and (3) existence (and, under mild non-degeneracy, uniqueness) of a non-Dirac invariant probability measure for the resulting Ornstein–Uhlenbeck– Lévy process. These results provide a mathematically auditable baseline for separating struc- tural rotation from stochastic entropy injection.