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Stochastic Lagrangians for Statistical Dynamics

The concept of stochastic Lagrangian and its use in statistical dynamics is illustrated theoretically, and with some examples. Dynamical variables undergoing stochastic differential equations are stochastic processes themselves, and their realization probability functional within a given time interval arises from the interplay between the deterministic parts of dynamics and noise statistics. The stochastic Lagrangian is a tool to formulate realization probabilities via functional integrals, once the statistics of noises involved in the stochastic dynamical equations is known. In principle, it allows to highlight the invariance properties of the statistical dynamics of the system. In this work, after a review of the stochastic Lagrangian formalism, some applications of it to physically relevant cases are illustrated.


Introduction
After Isaac Newton's big work about "the motion of bodies" [1], it was understood that physical systems are generally governed by equations of motion, expressing the variability with time of the state of the system in terms of the state of what acts on the system. When one writes F = m a, the time variability d p dt of "the state of motion" of the pointlike particle p (namely, its momentum) is put in relationship with "the force" F as and this force is a function of the state of what acts on the system. As an example, think about Kepler's problem, in which the point of mass m at position x undergoes the gravitational force exerted by the presence of another body of mass M at position x , according to the law the state of what acts on the pointlike particle is described indeed by the force , containing the value of the gravitational mass M of the second body and its relative position with respect to the point particle of momentum p, i.e. x − x. By the way, notice that in the force the state of the system itself appears as x, rendering (1) and (2) proper differential equations for the state of the system ψ = ( x, p). As Newton's equation F = m a is re-written considering also the relationship between momentum and velocity, one writes here it is stressed that the "force" is a function of the system state, of time t and of the "state of the environment" acting on the system, indicated as X env . A system as (3) is properly a system of ordinary differential equations (ODE), in which the variability of the state ψ, of the portion of universe we are intersted in, is put in relationship with how this portion of universe interacts with the environment. Similar systems of ODEs may be written as: In such a formulation, the system is described by a state ψ ∈ V, where V is a certain mathematical ambient through which the state ψ moves as time flows; V is referred to as the phase space of the system.
Calculus teaches us that when ODEs as (4) are equipped with some initial condition ψ (t 0 ) = ψ 0 , the problem admits a unique solution ψ (t) = Ψ ϕ (t, ψ 0 ) for all t ≥ t 0 . To be honest, this happens only if the Cauchy Problem (5) has the expression ϕ (ψ, t, X env ) that has particularly favorable conditions with respect to ψ, which we assume to happen "always" (however, see [2] for a not-that-problematic counterexample). From here on, the statement means that there exists a (suitably regular) map Ψ ϕ : R × V → V depending on the dynamics ϕ, associating the state ψ (t) to the initial condition ψ (t 0 ) in an injective way. This map Ψ ϕ is what one calls evolution. As a note, let us introduce here the system velocity space W, or dynamical flow space, so that ϕ (ψ, t, X env ) ∈ W and ϕ : V × R × V env → W (here V env is the phase space of the environment acting on the system).
Uniqueness of the solution of initial value problems (5) nourishes the Deterministic Paradigm (DP), according to which once a system's initial conditions are given, its future history will be completely determined, as long as what acts on it is known for all the future times. This is strongly accepted all through the Classical Physics, and after all it is true also in Quantum Mechanics [3], just considering the ODE (4) to be Schrödinger equation i d dt |ψ = H |ψ , according to which the motion of the quantum state |ψ is a perfectly deterministic trajectory through the quantum state Hilbert space H S .
The intelligent criticism to the DP (that actually turns out to be a generalization of the DP itself) must be based on the observation that, in order for (5) to have a unique solution, the quantities X env and ψ 0 appearing there, and the full mathematical construction of ϕ (ψ, t, X env ), must be known perfectly, i.e. with no uncertainty.
For instance, as the initial condition ψ 0 is not known perfectly, instead one knows just ψ 0 to belong to some subset A 0 ⊆ V of the phase space V, as one must admit the state of the system at time t ≥ t 0 to be any state Ψ (t, ψ (t 0 )) for any ψ (t 0 ) ∈ A 0 , i.e.: being A 0 a finite size set 1 , and so the set Ψ ϕ (t, A 0 ). Put in a simpler way, with a coarse grained knowledge of the initial condition as ψ (t 0 ) ∈ A 0 instead of the sharp ψ (t 0 ) = ψ 0 , one has to get content with a coarse grained knowledge of the state ψ (t) at later times. The uncertainty in the initial conditions will render uncertain the evolution of the system, setting a natural limit to the DP due to our finite precision and to how fast a finite size initial condition set A 0 may be deformed by the dynamics in (5): the whole querelle between the DP and chaos theory comes precisely from the capacity of non-linear dynamics to deform the initial condition set A 0 and diffuse it all over extended and complicated regions of the phase space, rendering the evolution unpredictable to an arbitrarily high precision.
In this work we will examine a different "limit of the DP": namely, we will deal with what happens when some elements of the mathematical expression of ϕ are known only to some statistical extent, i.e. when this mathematical function of ψ, t and X env has terms whose exact values is unknown, and of which one can only state they appear according to some given probability distribution. As a simple example, consider ψ to be a real variable, and the dynamics ϕ to read where K is a coefficient, and the time-dependent term γ (t) is an addendum that is "extracted from some real set Ξ" at each t > t 0 , according to the probability distribution function P t . A situation as that written in (8) represents the case in which the dynamics ϕ is obtained, at each time, by a first "completely known" term Kψ 2 plus some term γ about which one only knows that it can be a value within the set Ξ: as nothing sharper can be stated on γ, the value γ (t) comes randomly within Ξ at each different time, with a probability P t (γ * ) δ to fall in any interval γ * − δ 2 , γ * + δ 2 . Terms as the γ (t) in (8) are referred to as stochastic terms or, more simply, noises. In general, "noise terms" may appear in a variety of ways in the dynamics ϕ, due to different "physical" reasons: typically, as one distinguishes the dynamical variables ψ assigning the state of the system from "everything else" encoded in X env , noise terms will sensibly describe the degree of uncertainty about X env , which is the other possible source of uncertainty in (5), next to the initial conditions. For instance, in (8) one might imagine that ψ self-evolves with Kψ 2 , and undergoes the action of the "random kicks" γ.
Considering that noise terms typically come from what one refers to as "en-1 Stating anything about the "size" of a set in V has not sense until a proper definition of "size", or better "measure", is defined on V, which hasn't been done, and won't be done, here. Of course, if V is a metric space, the built-in distance d : V × V → R + may be used to define the size of A ⊆ V, as (A) = max (d (x, y)) / x, y ∈ A, and this is also true if a measure is defined on V, as when it is treated as the sample space of some probability. In the latter case, if one is able to define a probability density on V, a physically useful measure of A could be the Shannon entropy associated to this probability relative to all the points in A: again, this is beyond the scope of the present exposition.
vironment", the general form of (8) may read: In the presence of noise terms, the equation (4) is named stochastic dynamical equation (SDE). The stochastic version of (5) will read: The most relevant fact passing from (5) to (10) is that the uniqueness of the solution is lost, even in the presence of perfect knowledge of the initial condition ψ (t 0 ) = ψ 0 . Indeed, depending on what point in Ξ is picked at each time t to play the role of γ (t), one has different possible curves hence a different "histories" ϕ (ψ, t, X env (γ (t))) of the dynamics of the system. Since the noises γ may describe any continuous trajectory for t ∈ [t i , t f ], as illustrated in the cartoon of Figure 1, also the corresponding dynamics ϕ (ψ, t, X env (γ (t))) will have any continuous shape as a curve in W, see Figure  2. This explains in a pictorial way the fail of uniqueness of the solution to the stochastic Cauchy problem (10): the randomness transits from noises γ to dynamics ϕ, and from dynamics to the solution of the Cauchy problem ψ (t), see Figure 3 for the final step of the pictorial explanation. Note that, even if all of the three solutions to (10) start at the same initial value, they develop very differently and the uniqueness invoked by DP is lost.
The random nature of each term γ (t) in (9) and (10) renders it useful to consider the sample space Σ (t i , t f ) of all the possible continuous trajectories γ (t) in Ξ with t ∈ [t i , t f ], i.e. the sample space of all the possible realizations of the stochastic process γ : ; a probability measure defined on it, should be possibly based upon the functional counterpart of what an ordinary probability distribution function is for finite dimensional sample spaces, the square bracket left to γ means that P depends on the infinite number of values γ (t) (one per each t ∈ [t i , t f ]), while the round bracket right after t f indicates that P depends also on the two real variables t i and t f . The P [γ; t i , t f ) is referred to as realization probability functional (RPF) of γ, and its normalization condition reads: Figure 1: Three possible realizations of the noise γ (t) between t i and t f : as γ (t) can be any possible continuous curve in Ξ, here we have drawn two rather regular curves γ 1 (t) and γ 3 (t), and a less regular, sharp-cornered one γ 2 (t).
In this (11) the symbol [dγ] indicates the functional integral on Σ (t i , t f ) of the stochastic terms, namely the continuous product: (see [4] for a thorough explanation of this). About the RPF P [γ; t i , t f ), it must be stressed that this will be expressable as the continuous product of all the P t (γ) dt (see (9) and (10)), for t ∈ [t i , t f ], only for time-δ-correlated noises, see § 2.2: assigning the functional P [γ; t i , t f ) is then more powerful than giving all the time-local PDFs P t (γ), because P [γ; t i , t f ) contains any type of correlation among noises, that the collection of the PDFs does not, and indeed the latter is equivalent to the RPF only for the δ-correlated stochastic variables. In fully mathematical terms, one could write: In what follows, the program is to start with some particular form of the SDE in (10), in which one will understand how to pass from the ensemble statistics of noises γ (t) to that of the system realizations ψ (t), considering the interplay between the deterministic and the stochastic features of ϕ: where A [ψ; t i , t f ) is the RPF of the stochastic process ψ : [t i , t f ] → V, while the script ϕ → means that this pass is done thanks to the form of ϕ. For the types of SDE we are going to work with, the scheme to perform the transition (12) includes the introduction of the concept of stochastic Lagrangian (SL), that will be introduced in § 2. Then, in § 3 the machinery will be applied to some particular case of physical relevance. Figure 2: The cartoon of the three realizations of the process ϕ (ψ, t, X env (γ (t))) corresponding to the realizations γ 1 (t), γ 2 (t) and γ 3 (t) cited in Figure 1.

Functional Formalism for Statistical Dynamics
The mathematical quantity ψ, that describes the state of the system, is in general given by a certain number of components, indicated as ψ I , so locally V will be described as some R n (or C n ) space, as I = 1, ..., n. Remarkably, an infinite dimensional state may well be necessary, e.g. in a classical continuum theory, in which the system is described by the local properties of the continuum, so that the index I of ψ will rather be some continuous position x in the three dimensional space. The functional formalism is described here for a general finite dimensional system, while in [5,6,7,8] an infinite dimensional system, namely magneto-hydrodynamics, is considered.
Let's go back to the SDE in (10), and consider a particular class of functions ϕ (ψ, t, X env (γ (t))) of the noises γ. Let us assume that the noises enter the dynamical equation (DE) of ψ both in an additive form, as (8), and in a multiplicative way. In particular, as we assume the phase space V to be ndimensional, let us consider two stochastic n-vectors, one of components f I and the other of components g J , whose statistics is assigned through their RPF P [f, g; t i , t f ), and appearing in the SDE of ψ as: In (13) one has the co-existence of deterministic expressions, depending on ψ with that of the noise terms f and g, of which one supposes to know the statistics via P [f, g; t i , t f ). Equation (13) might appear rather particular and simple, as a form of SDE: it is clearly more complicated than the simplest case (8), but one may well think more involuted expressions, as for instance ..., being V = R for simplicity. The point is that: • the "simple" form (13) is the one allowing for the formalism of stochastic Lagrangian, that is of interest here; • however, as well explained in [5,6,7,8], and in [9], important realistic systems as space plasmas may be described by (13).
Our work will then concentrate on the stochastic Cauchy problem: As the paper [10] by Phythian is the first paper introducing this formalism, we will refer to (13) as Langevin-Phythian Equation (LPE) (as far as the Author is aware of, the formalism of the stochastic Lagrangian 2 has been introduced in [10] and in [11]). Before going ahead, notice that in the SDE (13), and in the system (14), noises seem to appear "directly" without the mediation of the environmental variables X env in (10): this is only an appearance, as the reader may get convinced of going through [5]. In that case, e.g., noise terms are identified with the terms − ∂ × ζ · J , J ρ and − ∂p ρ , being ζ the plasma conductivity tensor, J the electric current, ρ the plasma mass density and p the plasma pressure: as the state of the system ψ = V, B included, there, the plasma bulk velocity and the magnetic induction vector, the quantities ζ, J, ρ and p could be understood as "environmental variables" forcing ψ. Following this suggestion, one forms noises with quantities determined by the microscopic nature of the continuum, the microscopic stochastically treated degrees of freedom (μSTDoF, see also [12] for this concept) of which are the "environment" for the otherwise isolated fluid variable system.
The dynamics governed by conditions (14) will produce a multi-history evolution for ψ: as pictorially indicated in Figure 3, the system history between t i and t f is "statistically distributed" along many elements of that is the version of (12) adapted to the dynamics (14). The theoretical program we want to pursue here is to calculate the map just mimicked in (15), i.e. to obtain a (closed as possible) expression of the RPF of ψ from that of the RPF of the noises f and g, and the deterministic parts Λ and Γ of (14) (precisely, the competition between chance and necessity about which Haken speaks in his book [13] about "Synergetics").
Once the RPF A [ψ; t i , t f ) is given, the following program may be realized: 1. evaluate any statistical quantity F , for any functional F [ψ], on the ensemble Σ (t i , t f ) of trajectories through V admitted for the dynamics (14); 2. calculate the transition probability P ψi →ψ f (t i , t f ) for the system from an initial condition ψ i = ψ (t i ) and a final one Both any F and the transition probability where it is intended that F [ψ] depends on the whole trajectory ψ (t) for t ∈ [t i , t f ]: actually, the prescription (16) may work also for a time-local function F (ψ), that can always be expressed as an integration with the presence of some δ t −t , providedt is the instant of interest to calculate the function.
, while the initial and final configurations ψ (t i ) and ψ (t f ) are not integrated on, and kept fixed to the values of interest ψ i and ψ f respectively. One may write This may be also written as

Stochastic Lagrangian
The program stated in Points 1 and 2 before needs the knowledge of A [ψ; t i , t f ): in this work the RPF of ψ is represented through a time-local function L dψ dt , ψ , so that Clearly, the term N 0 (t i , t f ) is a normalization factor, so that the condition holds. The time-local function L dψ dt , ψ is referred to as stochastic Lagrangian (SL).
Since the relationship between this L dψ dt , ψ and the RPF A of the stochastic process is the same that one has between the classical Lagrangian function and the quantum amplitude in Quantum Mechanics [14], also here one may expect the "symmetries" of L to turn into invariances of the system statistics. One has to add that, for real ψ, as transition probabilities P ψi →ψ f (t i , t f ) must be real positive numbers, the quantity L dψ dt , ψ must be an imaginary number, as it happens indeed in real cases [5,7].
About the stochastic Lagrangian L dψ dt , ψ one has to stress that it is a function intrinsically different from what one calls "Lagrangian" in Analytical Mechanics: indeed, this Lagrangian L dψ dt , ψ , contains the maximum timederivative appearing in the ODE of the system (13), that is an intrinsically first order ODE, while the Lagrangian of Analytical Mechanics L dq dt , q does not contain the maximum order time-derivative of the evolution equation of the system, as Euler-Lagrange equations d dt ∂L ∂q − ∂L ∂q = 0 are second order ODEs (here the dot means time-derivative, asq where the case of a point particle undergoing classical and stochastic forces is treated, the difference between L and the would-be-Lagrangian L of the point prticle without noise is apparent: indeed, where the traditional Lagrangian would be L ˙ x, x , a function of position and velocity, the stochastic Lagrangian is some more complicated function L ˙ x,˙ p, x, p , depending on position, velocity, momentum and momentum derivative. Last difference we need to stress between the Analytical Mechanics Lagrangian L and the stochastic Lagrangian L is that, while L contains "all the Physics" of the system only for conservative systems, L encodes the whole Physics of any stochastic system in the form (13), regardless it is conservative (Hamiltonian, see § 3.2.1), or dissipative (metriplectic, see §

3.2.2).
It is also of use to give the definition of stochastic actionŞ [ψ, t i , t f ) as the time-integral of the Lagrangian L: so that A = N 0 exp (−iŞ). The caveat to make no confusion between thisŞ and the action of Analytical Mechanics is the same one as that of not confusing the stochastic L and the mechanical Lagrangian L discussed just now. The construction of the SL is described in [10], and inspired by the previous literature cited therein. In particular, thanks to the mathematical nature of the LPE (13), it is possible to introduce the kernel A [ψ; t i , t f ) starting from the definition of an ensemble statistical average F , based on noise statistics F def = F f,g , as noises represent the only element giving to (14) a stochastic character. The definition of the average F f,g is obviously our program is to obtain an expression of A so that The first step taken in [10] to obtain A satisfying (22), is to consider n auxiliary variables χ I=1,...,n that in a sense represent the Fourier momenta conjugated with the additive noises f I=1,...,n in the functional space Σ (t i , t f ). Then, an auxiliary kernel, depending on ψ and χ is defined as: is the term in which the noise statistics ends up being encoded, as

Once the auxiliary kernel
The calculation indicated in (24) is nothing but obvious, its feasibility is definitely not for grant. Indeed, summing over all the possible histories χ (t) / t ∈ [t i , t f ] is still matter of being able to do a functional integration, that we know is not an easy task in general. In § 2.2 we will see this integration is rather tractable if noises f and g are Gaussian fluctuations δ-correlated in time, as in (25).
The great effort is then re-casting the RPF defined as (24) and (23) in the Lagrangian form (19).

δ-correlated Gaussian noise
The example in which the calculations indicated in (23) and (24) are completely, and rather easily, feasible, is that in which f (t) and g (t) are time-δ-correlated noises with t-local Gaussian PDF. Let us assume that the two noises f and g have t-local probability density functions As noises taken at different times are independent of each other, the whole RPF is a continuous product of the time-local PDFs, i.e.: (in the expression (26) the emergence of the integration t f ti dt... in the argument of the exponential, as well as the power dt 2 in the normalization factor, come from the continuous product, see [4]). About the coefficients λ and µ in (25) and (26), one has to note that their relationship with the standard deviation σ f and σ g of noises reads: (all the quantities in (27) are R n,n matrices). A further assumption necessary for (26) to represent the correct RPF of those noises, is that f and g are uncorrelated with each other, i.e.: The RPF (26) gives rise to the average over noises: Without loss of generality one may put (23) is calculated as follows: (in these scripts, repeated nearby indices mean to be summed over). This (29) is usefully re-written separating the functional depending only on ψ from the one depending also on χ: We are now in the position of calculating A [ψ, χ; t i , t f ) by using this expression in (23), as done in § A.2, with the important (and not generality-losing) assumption (28). Moreover, a suitable symmetric may be defined, that is easily shown to be positive definite where the symbol τ is the determinant of the matrix of τ . Moreover, a vector β in V is defined as: so that all in all one remains with:

A.2 is interesting because, as the definition of J in (30) is considered
it is possible to define a total SL This is used as: The SL of the problem (14) with f and g time-δ-correlated Gaussian noises, so Ldt , is the function (36). In what follows, relevant examples of system stirred by time-δ-correlated Gaussian noises will be described, making reference to the SL (36).

Examples
In this Section some mechanical examples are reported, to show few, physically relevant cases in which the construction of A and L is performed completely. These are: the case of a pointlike particle of classical mechanics, that of Hamiltonian and metriplectic systems, and that of a general Leibniz dynamics. Only in the very simple case of the particle undergoing a deterministic viscous force, plus noise, the functional measure [dψ] appearing in (16) and (17)  From the examples mentioned here, the reader should learn at least a general idea about the meaning and use of SLs; however, a more complex and practical application of this formulation may be found in papers [5,6,15,7,8,9,4], where the same formulation described here is applied to space physics and geophysical examples.

Point Particle with noise
Consider a point particle described by its position x and its momentum p, with the relationship p = m d x dt . Consider also Newton's DE where the force vector F ( x, p, t) may be given by "any" force law. In general, the convenient thing is to consider an F given by the sum of a pure deterministic, smooth addendum Φ ( x, p, t), plus a noise terms ξ (t), the statistics of which is going to be specified in a moment: The equations of motion read: In equations (38), one has six components of the state vector while the independent additive noises are only three ones, the ξ components. In order to be cautiously compliant with the scheme of [10], let us introduce an auxiliary noise term η (t), so that the ODEs (38) become the complete equivalence between these (40) with the ODEs (38) will be weakly restored, in the sense that the noises η will be supposed to have their PDF δ-like peaked in zero identically, i.e. for all times t.
The first thing to do in order to apply what described in § 2.2 to the system (40), one has to construct the Lagrangian (36). As no multiplicative noise exists in the example (40), one can adapt what obtained before to systems with pure additive noise just putting everywhere: This reduces the SL to: Also, the expressions of β and τ must be modified according to (41): the definition of β in (33) reads: while that of τ in (31) now becomes: The tensor τ is trivially inverted so the expression of L λ,ψ, ψ reads: The set of necessary functions to write the SL in (44) is completed with the T (V)-vector Λ collecting the deterministic part of the dynamics, that from (40) reads: From (45) one has: The SL reads: is constructed as follows: Considering (27), one may conclude: being σ η k and σ ξ h the standard deviations or the respective noise components. When (46) is used in (37), one obtains: In order to reproduce the system (38) one has to consider an identically zero noise η, which is realized in (47) "simply" considering the δ-family 4 so that we are interested in: being N 0 (t i , t f ) the suitable, necessary normalization constant, generally different from the N 0 (t i , t f ) in (47). Note that in (49) the symbol δ d x dt − p m is a functional δ, being effective at any time t ∈ [t i , t f ], so that the prescription d x dt = p m will be enforced as the integration over d 3 x , or over d 3 p , takes place. The example of Φ one might consider reads: This force is the sum of a gradient force − ∂V ( x), as in classical gravitation, elasticity or electrostatics, a Lorenz force q m p × B ( x) mimicking the effect of with f being a function in the suitable space rendering each dxϕ (x, α) f (x) and dxϕ 0 (x) f (x) finite. a magnetic field B on a particle of electric chage q, and a dissipative, viscous friction force − ζ m p. Considering the calculations in § B.1, one may write and this (51) is inserted into (49), obtaining: 2m (t f −ti) , so we may write: (in (52) the components Φ h are those forming the vector Φ in (50)) Some comments are needed about this result: the exponential will, again, converge to a Dirac δ functional of the argumentṗ h − Φ h ( x, p, t) as σ ξ h tends to zero, i.e. as the noise vector ξ is distributed according to a more and more peaked Gaussian, provided a suitable term O is admitted in the normalization factor M 0 (ζ, m; t i , t f ). In other words: in the "classical limit" σ ξ h → 0, in which one has the certainty of having zero ξ, the RPF à la Phythian of the stochastic trajectories admitted by (38) converges to the certainty that the trajectory will obey the "silent 5 ODEs" d x dt = p m and d p dt = Φ. Before going to more examples of stochastic systems, we would like to repeat, through the formalism introduced here, the calculations reported in § 6.1 of the book [14], and make an explicit calculation of the functional measure [dψ] in the case of the point particle undergoing additive noise. We will do this for the simplified case in which Φ is a pure viscous force, i.e. Φ = − ζ m p, so the ODEs of the system read: 5 "Silent" here means "without noise", classical, fully deterministic.
The kernel in use reads: The term 2ζ m λ (i)ṗi p i in the integral can be simplified: so one re-writes: This kernel is the central quantity allowing for the discussion of the statistical dynamics of the system (53), as it represents the statisical weight of the realization chosen in the integrand The current use of the functional A [ x, p; t i , t f ) needs the determination of the constantÑ 0 (ζ; t i , t f ), that is prescribed in order to obtain: so that its value is determined as: dt Lp(˙ p, p) .
(56) As in the kernel in (54), and hence in (56), no dependence on x appears in the integrand, the integration can be considered, simply turning (56) into: The way out to find a closed expression for is to make use of time-discretization as done by Feynman and Hibbs in their book [14], in which they subdivide the interval [t i , t f ] into -long pieces The full integration [d p] may be represented as while the integration in the exponential in (57) is discretized as: (60) Defining in a closed form the functional measure [d p] precisely means finding sensible expressions for the factors W p (k) in (59).
Feynman and Hibbs suggest the way to interpret the quantityṗ i (k) as: while the quantity p i (k) is better approximated as The replacements (61) and (62) are peformed, and calculations are done in § B.1. The integral in (60) reads: In anticipation of calculating the limit → 0 invoked in (59), the expressions calculated in (97) are reported neglecting o ( ), so that one may re-write: giving rise to the following approximation for the termŞ p in (63): Due to (65), one has the following normalization: All the integrations R dp i (t k ) are "the same" apart from the integration over p i (t i ), because it contains also the factor e ζ m λ (i) p 2 i (ti) inherited from (54). So, it is useful to separate the integration over the initial momentum p i (t i ) from all the other ones in (66): (68) The quantity J 0 reads: The integral I k instead reads: It also the case to calculate I 0 as: All in all, one may write: where one has given the definition: Once the quantityÑ −1 0 (ζ; t i , t f ) has been calculated as done in § B.2, and represented as in (70), one may write: where the definition V p (t) all in all, (72) implies also: The relationship (73) may be of use when ensemble calculations are to be per- Going back to (71), if one wants to keep finiteÑ 0 (ζ; t i , t f ), the factor N k=0 W p (k) must be chosen accordingly. Provided one defines as suggested in § B.3, the factorÑ 0 remains finite. This amounts to defining the functional measure:

Leibniz systems
Leibniz systems are dynamical systems whose ODEs may be expressed in tensor terms as where the quantity F is a function from V to R indicated as Leibniz-generator of the motion, while T ∈ T (V) ⊗ T (V) is a tensor, referred to as Leibniz tensor [16]. Suppose to deal with a stochastic dynamics, the deterministic part of which is (76), i.e. some version of (14) with Provided the SDE reads and f and g are Gaussian time-δ-correlated noises with the statistics described in § 2.2, one must work with the functionals (37). The calculations needed are shown in § C, and one has: .
(78) The more tractable version of (78) is the one with additive noises only: to obtain this case in the smoothest way possible one may resort to (41) and to the relationship (27). Then, one will write: In the following two § §, the Hamiltonian and the metriplectic systems are treated, as particularly relevant cases of Leibniz systems.

Hamiltonian Systems
A Hamiltonian system is a DS as in (76), with T IJ being a Jocobi tensor [17]; moreover, the dyamics generator F is what they call Hamiltonian of the system, namely its energy: In the relationships (78) and (79), this means in practical that T IJ = −T JI , and that T IJ S = 0; besides this, T IJ will be rather indicated as J IJ , i.e. the Jacobi tensor. All in all, one may write the functional formalism for the statistical dynamics of a general Hamiltonian system, stirred by both additive f and multiplicative g Gaussian time-δ-correlated noises, as follows: .

Metriplectic Systems
Metriplectic systems are Leibniz systems in which the tensor T is the sum of a Jacobi tensor J and a semimetric tensor G. Moreover, the generator F takes the form of free energy function, namely the sum of a Hamiltonian H and an entropy S, weighted by some coefficient α: F = H + αS. The two tensors J and G must have the particular following relationships with the gradients of H and S: As explained thoroughly in [12] and references therein, a metriplectic system is basically the algebrization of an energetically closed, otherwise Hamiltonian system, to which some dissipation is added, due to degrees of freedom whose entropy is represented by S. All in all, one has energy conservationḢ = 0 and entropy growthṠ ≥ 0, thanks to the ODEs: The stochastic version of the metriplectic system is simply written as: Of course, if the noises f and g have completely general statistics, then all the formalism in § 2.1 is applicable, but we are interested to the simpler particular case of time-δ-correlated Gaussian noises, so that what reported in the general part of this § 3.2 is sufficient. Particular reference is made to the relationship as the compatibility conditions (82) are considered, the quantity − i 2 ∂ ∂ψ I T IJ (ψ) ∂F (ψ) ∂ψ J will be calculated as: When the additive noise f and the multiplicative one g are both present, the kernel of the metriplectic stochastic system reads: The kernel for the metriplectic system with purely additive, Gaussian, time-δ-correlated noise f will instead read:

Conclusions
The whole science about "stochastic processes" appears to advance a criticism to the DP described in § 1, which is, instead, an upgrade of it. Indeed, nondeterministic finite dimensional systems, to which one resorts because of the only statistical knowledge of some elements of the initial value problem, are promoted to deterministic infinite-dimensional systems, as the mathematical quantities used to described them are not their state variables ψ but, rather, statistical distributions of ψ.
In statistical dynamics the central tool for such a theory are master equations [18], i.e. the evolution equations for probability distributions [13]. An alternative way to describe the same problem, appearing even more powerful to the Author of this paper, is that of functional formalism [10], in which the goal is to calculate the probability that a certain particular historyψ (t) takes place between t i and t f , what we refer to as RPF in this work. The program of functional formalism in Statistical Dynamics is, then, to start from the knowledge of the noise history probability P [γ; t i , t f ) and of the equations of motion of ψ, and find the history probability A [ψ; t i , t f ) of the state variables ψ of the system, namely studying the map (12).
Under the particular condition that the SDE of the system is in the form of the ODE in (10), i.e. the Langevin-Phythian equatioṅ ψ = Λ (ψ) + g T · Γ (ψ) + f, (f and g being noises of assigned RPF P [f, g; t i , t f )), it is possible to obtain A [ψ; t i , t f ) in the form of imaginary exponential of a time-local function L ψ (t) , ψ (t) , t , referred to as the stochastic Lagrangian: As the whole statistical dynamics of such a system is constructed by using A [ψ; t i , t f ), to some extent one could say it is encoded in L ψ , ψ, t , precisely as it happens for Quantum Mechanics when Feynman's path integral approach is adopted [14]. Once all the ingredients in (88) are found, one will use them to calculate "anything" of the ensemble statistics of the system, e.g. averages, transition probabilities or correlations: This promises to be a very powerful tool. Two big difficulties exist in obtaining things in (88), and using them as in (89): first of all, as underlined in § 2.1, obtaining the RPF of ψ getting rid of the auxiliary variables χ is in general a rather hard task, that appears to be "simple" only if f and g are time-δ-correlated Gaussian noises, see § 2.2; this impeds to obtain A [ψ; t i , t f ) from the more immediately defined A [χ, ψ; t i , t f ) (this difficulty is partly circumvented by the recipes in [11], where the kernel A [χ, ψ; t i , t f ) is used to calculate suskeptibilities and other statistical observables). Moreover, in all the statistical quantities as those in (89), there is the necessity of writing in a closed form the functional measure [dψ], that is the celebrated big difficulty of any branch of Physics making use of path integrals! Here, we have made the explicit calculation of the functional measure As we have shown in § § 17 and 25, the functional formalism of the stochastic Lagrangian appears to be applicable to a very wide range of physical problems, from Newton's mechanics plus noise, to Leibniz systems, included Hamiltonian or metriplectic systems, see § § 26 and 27. Fluid dynamical and plasma physics examples are treated in the references quoted here, and for sure much wider fields will be covered in the future.
Last but not least, I feel indebted with Polish and Romanian people, who are using the beautiful letters "Ł" and "Ş" respectively, that I could use for the stochastic Lagrangian L and stochastic actionŞ: mathematics is also a way to acknowledge the wonderful attitude of mankind to create symbols mimicking our voices.
A Calculation for § 2.2 First of all, let us calculate the expression for C [χ, Γ; t i , t f ): Without loss of generality one may put so that: and continue the calculation as: It is of use to give the definition: and to continue as: Pivoting on the relationship i.e.

(95)
It is time to conclude our calculation rendering the expression β · τ −1 · β T more explicit. Considering (31) and (33), once one has defined α IJ (µ, ψ) After all, one has:Ş B.2 Calculating J 0 and I k In order to make the calculations needed in (68), we start by computing J 0 : Let us, then, calculate the expression of I k as defined in (67) and then compute the product N −1 k=1 I k : B.3 KeepingÑ 0 (ζ; t i , t f ) finite In order to keepÑ 0 (ζ; t i , t f ) finite in (72), the reasoning is the following one: