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

February 21, 2023

In the realm of complex systems, dynamics is often modeled in terms of a
non-linear, stochastic, ordinary differential equation (SDE) with either an
additive or a multiplicative Gaussian white noise. In addition to a
well-established collection of results proving existence and uniqueness of the
solutions, it is of particular relevance the explicit computation of
expectation values and correlation functions, since they encode the key
physical information of the system under investigation. A pragmatically
efficient way to dig out these quantities consists of the Martin-Siggia-Rose
(MSR) formalism which establishes a correspondence between a large class of
SDEs and suitably constructed field theories formulated by means of a path
integral approach. Despite the effectiveness of this duality, there is no
corresponding, mathematically rigorous proof of such correspondence. We address
this issue using techniques proper of the algebraic approach to quantum field
theories which is known to provide a valuable framework to discuss rigorously
the path integral formulation of field theories as well as the solution theory
both of ordinary and of partial, stochastic differential equations. In
particular, working in this framework, we establish rigorously, albeit at the
level of perturbation theory, a correspondence between correlation functions
and expectation values computed either in the SDE or in the MSR formalism.

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