Abstract

We are interested in studied an exact functional formalism to deal with linear Langevin equations with arbitrary memory kernels and driven by an arbitrary noise structure characterized through its characteristic functional. No other hypothesis is assumed over the noise, neither we use the fluctuation dissipation theorem. We have find that the characteristic functional of the linear process can be expressed in terms of noise's functional and the Green function of the deterministic (memory-like) dissipative dynamics. This yields a procedure for calculating the full Kolmogorov hierarchy of the non-Markov process. As examples we have characterized through the 1-time probability a noise-induced interplay between the dissipative dynamics and the structure of different noises. Conditions that lead to non-Gaussian statistics and distributions with long tails have been analyzed. The introduction of arbitrary fluctuations in fractional Langevin-like equations is also studied.

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INTRODUCTION
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Noise is a basic ingredient of many types of model in physics, mathematics, economy, as well as in engineering. While in each area of research the fluctuations have very different origins, in many cases the evolution equation governing the system of interest can be approximated by a suitable stochastic differential equation.

In general the driving forces may be any source of fluctuations and the system can be characterized by a given potential. The most popular of those stochastic differential equations are those driven by white Gaussian fluctuations, where the problem can immediately be reduced to the well known Fokker-Planck dynamics. If the fluctuations are not Gaussian we are faced with a problem that is hard to solve, but among the different types of fluctuations the so call dichotomic noise is a good candidate to study, because in general for any potential some
conclusions can be drawn. If the fluctuations are neither Gaussian or dichotomic noise, in general it is not possible to solve the problem for an arbitrary potential.

In many situations the technical complications of the model can be reduced by studying the system in a linear approximation, i.e., around the fixed points of the dissipative dynamics. This leads to the study of linear stochastic differential equations with arbitrary noises. Besides the simplicity of this kind of equations they have been the subject of extensive theoretical investigation, and they also provide non-trivial models for the study of many different mechanisms of relaxation in non-equilibrium physics, biology and other research areas.

From the above considerations, it is clear that the usefulness of a linear Langevin equation arises from the possibility of working with different kinds of noise structures. Therefore, one is faced with the characterization, in general, of non-Markov processes. These processes can only be completely characterized through the whole Kolmogorov hierarchy of the stochastic process.

By using our functional technique, we have been able to characterize arbitrary linear Langevin equations with local dissipation, finding therefore a procedure to calculate the whole Kolmogorov hierarchy of the process and any n-time moment. In this project we are going to studied our functional technique to tackle the more general situation when the system is extended and the dissipative term is non-local in time and the noise is also arbitrary, i.e., a stochastic field governed by a linear Langevin equation with arbitrary memory and driven by any noise structure. The fluctuation term (i.e., the external noise) is always characterized by its associated functional, and the "dissipation" by an arbitrary memory kernel.

This type of generalized Langevin equation arises quite naturally (considering Gaussian fluctuations) in the context of the Zwanzig-Mori projector operator technique; in this case the fluctuation-dissipation theorem is required, which imposes restriction on the memory kernel. Therefore the dissipative memory must be consistent
with the structure of the correlation of the Gaussian fluctuations. Nevertheless, if the system is far away from equilibrium, the fluctuation-dissipation theorem does not apply and in general the Gaussian assumption is not a good candidate to describe the fluctuations of the system. Therefore, in a general situation, both the kernel and the noise properties must be considered independent elements whose interplay will determine the full stochastic dynamics of the process. We will be interested in characterizing this noise-induced interplay by assuming different kinds of noises and memory kernels, both in the transient as in the long-time regime.

We are interested in analyzing the case of stable noises. Then the interplay between the kernel and the noise
properties is studied in different physical systems as turbulent fluids and granular matter. The necessary properties that guarantee a stationary distribution with a long-tail are analized. We have introduced the Abel noise that induces a long-tail asymptotic behavior. In addition the characterization of a fractional Langevin equation with arbitrary noise is also studied. The interplay between the fractional property of the differential operator and Levy noise is analyzed.

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