AbstractWe consider the integro‐differential equation for the classical trajectory of an oscillator coupled to another one. On the quantum level the elimination of the coordinateAof the “unvisible” oscillator leads to an effective path integral (X, Ξ, μ) for the associated imaginary time stochastic processtϵ, (‐∞,∞) →x(t). We prove reflection positivity of the measuredμ ≈F·dξ, wheredξ governes the free oscillatorxandFis the counterpart of Feynman's influence functional. Finally, realizing the Hamiltonian semigroup exp(‐tH),t≧ 0, in the physical Hilbert space ℋ︁ =L2(X, Γ, μ), where Γ ⊆ Ξ+,