The Parisi–Sourlas method of BRST procedure approach to Langevin SDEs have also been adapted to classical mechanics, stochastic generalization of classical mechanics, higher order Langevin SDEs, and, more recently, to SDEs of arbitrary form. While there exist standard techniques that allow to consider models with colored noises, higher-dimensional "base spaces" described by partial SDEs etc., the key elements of STS can be discussed using the following basic class of SDEs, x ˙ ˙ ( t ) = F ( x ( t ) ) + ( 2 Θ Θ ) 1 / 2 e a ( x ( t ) ) ξ ξ a ( t ) , where x ∈ ∈ X is a point in the phase space assumed for simplicity a closed topological manifold, F ( x ) ∈ ∈ T X x is a sufficiently smooth vector field, called flow vector field, from the tangent space of X , and e a ∈ ∈ T X , a = 1 , … … , dim ⁡ ⁡ X is a set of sufficiently smooth vector fields that specify how the system is coupled to the noise, which is called additive / multiplicative depending on whether e a 's are independent/dependent on the position on X .