Stochastic Calculus

About the non-Markovian Stochastic structures

Since the pioneer work of Langevin, stochastic differential equations have become a powerful tool for the study of systems where fluctuations play a fundamental role. The basic idea of this approach consists in adding explicitly random elements in the proper system evolution, and then to characterize the statistical properties of the non-equilibrium dynamics by averaging the evolution over a set of noise realizations. For physical systems provided with a thermodynamical equilibrium state, fluctuation and dissipation appear in a linked way as
demanded by the fluctuation dissipation Kubo-theorem. Except for this situation, fluctuations and dissipation can be considered as independent elements, whose characteristics depend on each particular physical situation. Thus, the fluctuations in general may be non-Gaussian and the dissipative dynamics introduces arbitrary correlation effects or memory contributions.

Memory effects can be rigorously derived by using projector operator techniques like the one from Zwanzig and Mori. This method applies for linear sub-systems embedded in a bigger one. Nevertheless, in general it is not possible to use this procedure, and the memory contributions follows from a phenomenological description. In fact, in many natural and physical situations, the memory effects arise as a consequence of an intrinsic delay
mechanism, which implies that the dissipative evolution depends on the state of the system in a shifted previous time. Remarkable examples of this situation arise in physics, biology, physiology, etc. This particular signature in the dissipative dynamics motivated the study of differential delay equations and delay Langevin equations. An exact analytic treatment of these equations is in general extremely difficult. Nevertheless, some progress was achieved in the characterization of linear stochastic evolutions driven by Gaussian fluctuations. One of the goals of the present project is to go, in this analysis, beyond the Gaussian fluctuations. As was previously mentioned, non-Gaussian fluctuation appears in a natural way in many situations of interest. Thus, the characterization of linear delay Langevin equations in the presence of any kind of fluctuations is of great value. We remark that independently of the type of fluctuations, a linear delay Langevin equation is inherently a non-Markovian process.

The study of non-Markovian Langevin equations have received a lot of attention. From a rigorous point of view, these equations can only be completely characterized after knowing the full Kolmogorov hierarchy,
i.e., any n-joint probability, or equivalently any n-time correlation. All this information is encoded in the characteristic functional of the process. In fact, this object allows to get the n-characteristic function of the process, from which any n-joint probability follows from an inverse Fourier transform, and any n-moment or cumulant follow from an n-derivative operation.

In a set of works we have presented a procedure to get the characteristic functional of processes defined by
linear stochastic Langevin equations with local and non-local dissipation. Then, this procedure can be also applied in the context of delay Langevin equations. Functional techniques have also been introduced by another authors for studying disordered systems and multiplicative stochastic equations.

In the present project we will apply our functional technique to study the transient and stationary properties of linear delay Langevin equations driven by arbitrary noises defined through their characteristic functional. The basic idea consists in obtaining an explicit expression for the realizations of the delay stochastic process in terms of the dissipative delay Green function, and then to get the characteristic functional of the stochastic process, in terms of that of the driving noise. 

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