Stratonovich interpretation

The present analysis follows the approach taken by aU three of these authors, in which SDEs are constructed by choosing the drift and diffusivity coefficients so as to yield a desired diffusion equation. Peters [13] has pioneered an alternative approach, in which expressions for the drift and diffusivity are derived from a direct, but rather subtle, analysis of the underlying inertial equations of motion, in which (for rigid systems) he integrates the instantaneous equations of motion over time intervals much greater than the autocorrelation time of the particle velocities. Peters has expressed his results both as standard Ito SDEs and in a nonstandard interpretation that he describes heuristically as a mixture of Stratonovich and Ito interpretations. Peters mixed Ito—Stratonovich interpretation is equivalent to the kinetic interpretation discussed here. Here, we recover several of Peters results, but do not imitate his method. 

The Stratonovich interpretation of a generic set of L SDEs driven by M Wiener processes will be indicated in what follows by the notation... 

The notation o (the Stratonovich circle ) is used to distinguish this from the corresponding set of Ito SDEs. The Stratonovich interpretation of such a set of equations may be defined by either of two hmiting procedures, which have been shown to yield equivalent hmits, and are discussed separately below ... 

The Stratonovich interpretation of Eq. (2.238) may also be obtained [31,32] from the white-noise limit of a sequence of stochastic ordinary differential equations (ODEs) of the form... 

Here, 8(f — f ) represents a sharply peaked but finite and differentiable autocorrelation function with a small but nonzero autocorrelation time, which is assumed to be an even function of t — t. The Stratonovich interpretation is obtained in the limit of vanishing autocorrelation time. 

The interpretation of the Langevin equation presents conceptual difficulties that are not present in the Ito and Stratonovich interpretation. These difficulties are the result of the fact that the probability distribution for the random force rip(f) cannot be fully specihed a priori when the diffusivity and friction tensors are functions of the system coordinates. The resulting dependence of the statistical properties of the random forces on the system s trajectories is not present in the Ito and Stratonovich interpretations, in which the randomness is generated by standard Wiener processes Wm(f) whose complete probability distribution is known a priori. 

Neither the Ito nor the Stratonovich interpretation of an SDE leads naturally to a term of this form. The Ito interpretation yields a diffusion equation of the form given in Eq. (2.222), in which the diffusivity instead appears inside two derivatives, while the Stratonovich interpretation yields Eq. (2.255), in which is decomposed into two factors of B, one of which appears inside both derivatives and the other between them. 

Stratonovich interpretation of the Langevin equation, it is the use of a midstep value of C (X) that causes the unwanted bias in the random forces. 

We now show that the same drift velocity is obtained from a Stratonovich interpretation of the Langevin equation with unprojected and projected random forces that have the same soft components but different hard components. We consider an unprojected random force ri (t) and a corresponding projected random force Tj (f) that are related by a generalization of Eq. (2.300), in which... 

It can be proved that this choice leads to (4.8). This shows that our naive use of the transformation of variables amounted to opting for the Stratonovich interpretation. 

Exercise. The damping of a Brownian particle is described in terms of the energy by E = -2yE. Construct a Langevin force which gives the correct fluctuations in the Stratonovich interpretation. Find another Langevin force that is correct in the Ito interpretation. 

As long as L(t) is not singular one is free to apply any transformation of y using the familiar rules of calculus in particular one may transform (4.5) into the quasilinear equation (4.3). In the latter the limit rc — 0 gives no problem and the result is (4.4). Transforming back to the original variables gives (4.8), as we found before. Hence, whenever the delta function stands for a sharp, but not infinitely sharp peak the Stratonovich interpretation is appropriate. The Ito interpretation cannot even be formulated unless tc is strictly zero. 

A crucial feature of the multiplicative noise case is that the noise term in (4.97) has a nonzero mean value. Using Novikov s theorem [324] for Gaussian noise in the Stratonovich interpretation, we find that... 

Historically, the two major interpretations are from Ito and Stratonovich. In both these formulations, (x) = 5(x) l However, (x) = A(x) in Ito interpretation while (x) = A(x)-jd B(x) in the Stratonovich interpretation. From the practical perspective, Ito interpretation allows one to simulate the SDE using the usual forward Euler scheme. However, special differentiation and integration rules are required for analytical calculations. On the other hand, Stratonovich interpretation allows using the regular rules of calculus but has to be simulated using implicit schemes. We emphasize that the FPE does not suffer from such ambiguity of interpretation SDEs corresponding to different interpretations of the same FPE lead to the same physical results [3, 7]. 

In cases where the divergence of the mobility vanishes, dUiajp/ fjp = 0, the Ito and Stratonovich interpretations coincide. 

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