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In the realm of complex systems, dynamics is often modeled in terms of anon-linear, stochastic, ordinary differential equation (SDE) with either anadditive or a multiplicative Gaussian white noise. In addition to awell-established collection of results proving existence and uniqueness of thesolutions, it is of particular relevance the explicit computation ofexpectation values and correlation functions, since they encode the keyphysical information of the system under investigation. A pragmaticallyefficient way to dig out these quantities consists of the Martin-Siggia-Rose(MSR) formalism which establishes a correspondence between a large class ofSDEs and suitably constructed field theories formulated by means of a pathintegral approach. Despite the effectiveness of this duality, there is nocorresponding, mathematically rigorous proof of such correspondence. We addressthis issue using techniques proper of the algebraic approach to quantum fieldtheories which is known to provide a valuable framework to discuss rigorouslythe path integral formulation of field theories as well as the solution theoryboth of ordinary and of partial, stochastic differential equations. Inparticular, working in this framework, we establish rigorously, albeit at thelevel of perturbation theory, a correspondence between correlation functionsand expectation values computed either in the SDE or in the MSR formalism.

An algebraic correspondence between stochastic differential equations and the Martin-Siggia-Rose formalism