Namely, for any configuration of the noise, ξ ξ , and an initial condition, x ( t ′ ) = x ′ ∈ ∈ X , SDE defines a unique solution/trajectory, x ( t ) ∈ ∈ X . Even for noise configurations that are non-differentiable with respect to time, t , the solution is differentiable with respect to the initial condition, x ′ . In other words, SDE defines the family of the noise-configuration-dependent diffeomorphisms of the phase space to itself, M t t ′ : X → → X . This object can be understood as a collection and/or definition of all the noise-configuration-dependent trajectories, x ( t ) = M t t ′ ( x ′ ) . The diffeomorphisms induce actions or pullbacks, M t ′ t ∗ ∗ : Ω Ω ( X ) → → Ω Ω ( X ) . Unlike, say, trajectories in X , pullbacks are linear objects even for nonlinear X . Linear objects can be averaged and averaging M t ′ t ∗ ∗ over the noise configurations, ξ ξ , results in the finite-time SEO which is unique and corresponds to the Stratonovich–Weyl interpretation of the BRST procedure approach to SDEs, M ^ ^ t t ′ = ⟨ ⟨ M t ′ t ∗ ∗ ⟩ ⟩ noise = e − − ( t − − t ′ ) H ^ ^ 1 / 2 .