Stratonovich

The stochastic differential equation and the second moment of the random force are insufficient to determine which calculus is to be preferred. The two calculus correspond to different physical models [11,12]. It is beyond the scope of the present article to describe the difference in details. We only note that the Ito calculus consider r t) to be a function of the edge of the interval while the Stratonovich calculus takes an average value. Hence, in the Ito calculus using a discrete representation rf t) becomes r] tn) i]n — y n — A i) -I- j At. Developing the determinant of the Jacobian -... 

The function a(x, t) appearing in the FPE is called the drift coefficient, which, due to Stratonovich s definition of stochastic integral, has the form [2]... 

R. L. Stratonovich, Topics of the Theory of Random Noise, Gordon and Breach, New York, 1963. 

MSN. 136. 1. Prigogine and 1. Antoniou, From microscopic irreversibility to macroscopic ireversibility. Discussion remarks and comments to R. E. Stratonovich, Z. Phys. Chem. 170, 219-221 (1992). 

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. 

In both the Ito and Stratonovich formulations, the randomness in a set of SDEs is generated by an auxiliary set of statistically independent Wiener processes [12,16]. The solution of an SDE is defined by a hmiting process (which is different in different interpretations) that yields a unique solution to any stochastic initial value problem for each possible reahzation of this underlying set of Wiener processes. A Wiener process W t) is a Gaussian Markov diffusion process for which the change in value W t) — W(t ) between any two times t and t has a mean and variance... 

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 solution of SDE (2.238) was originally defined by Stratonovich [30] as the limit At 0 of a sequence of discrete Markov processes, for which... 

The drift velocity and diffusivity for a Stratonovich SDE may be obtained by using Eq. (2.239) to calculate the first and second moments of AX to an accuracy of At). To calculate the drift velocity, we evaluate the average of the RHS of Eq. (2.244) for AX . To obtain the required accuracy of At), we must Taylor expand the midpoint value of that appears in Eq. (2.244) to first order in AX about its value at the initial position X , giving the approximation... 

Note that for a Stratonovich SDE, unlike an Ito SDE, the drift velocity generally differs from the drift coefficient that appears in the SDE. 

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. 

To construct a sequence of ODEs whose solutions converge to that of a corresponding Stratonovich SDE driven by a known set of Weiner processes W f),..., WM f), the random functions /m(f) may be taken as the time derivatives f ( ) = dWm (f)fdt of a sequence of differentiable functions Wm t) that approach the specified Weiner processes Wm t) in the limit e 0, so that... 

The essential properties of a Stratonovich SDE, which may be derived from either of the two limiting processes presented above, are... 

Fokker-Planck Equation. A Stratonovich SDE obeys a Fokker-Planck equation of the form given in Eq. (2.222) with the drift velocity V (X) given in Eq. (2.243), and the diffusivity given in Eq. (2.229). The resulting diffusion equation may be written in terms of the drift coefficient... 

Note the differences between the form of the diffusion term that appears in this Stratonovich form of the diffusion equation and those that appear in the Ito (or forward Kolmogorov) form of Eq. (2.222) and in the physical diffusion equation of Eq. (2.78). 

Stratonovich SDEs, unlike Ito SDEs, may thus be manipulated using the familiar calculus of differentiable functions, rather than the Ito calculus. This property of a Stratonovich SDE may be shown to follow from the Ito transformation rule for the equivalent Ito SDE. It also follows immediately from the definition of the Stratonovich SDE as the white-noise limit of an ordinary differential equation, since the coefficients in the underlying ODE may be legitimately manipulated by the usual rules of calculus. 

A set of Stratonovich SDEs for a constrained mechanical system may be formulated either as a set off SDEs for the soft coordinates or as a corresponding set of 3N SDEs for the Cartesian bead positions. The Stratonovich SDEs for the generalized coordinates are of the form given in Eq. (2.238), with a drift coefficient... 

Note that, in either system of coordinates, the value of the Stratonovich drift coefficient required to produce a given drift velocity depends on derivatives of... 

The Stratonovich SDEs for either generalized or Cartesian coordinates could be numerically simulated by implementing the midstep algorithm of Eq. (2.238). Evaluation of the required drift velocities would, however, require the evaluation of sums of derivatives of B or whose values will depend on the decomposition of the mobility used to dehne these quantities. This provides a worse starting point for numerical simulation than the forward Euler algorithm interpretation. 

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. 

A completely unambiguous formulation of the Langevin equation may, however, be constructed by relating the random Langevin forces to random functions /m(f) of the type used in our discussion of the Stratonovich SDE, by taking... 

When supplemented by Eq. (2.269) for t p(r), the Langevin equation becomes equivalent to a standard Stratonovich SDE with a drift coefficient and a coefficient... 

The drift velocity may then be derived by using Eq. (2.243) for the drift velocity of a Stratonovich SDE. We consider the cases of singular and nonsingular mobility matrices separately. 

We now instead calculate the drift velocity and diffusivity by directly integrating the traditional formulation of the Langevin equation in terms of random forces, and compare the results to those obtained above by rewriting the Langevin equation as a standard Stratonovich SDE. As in the analysis of the Stratonovich SDE, we calculate the first and second moments of an increment AX (f) = Z (f) — X (0) by integrating Eq. (2.262) from a known initial condition at f = 0. 

Stratonovich SDEs with a nonsingular mobility yields a drift velocity... 

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. 

Here, we have borrowed Hiitter and Ottinger s use of a o symbol to indicate a kinetic interpretation of the stochastic term, but adopted a more explicit notation for its use. The o is used here to indicate that the function to its left should be evaluated at a midstep position, as in a Stratonovich SDE, but that the random quantity to the right of the diamond should be evaluated by evaluating the function Cp (X) at the beginning of each timestep, as in an Ito SDE. This notation is similar to that used by Peters [13] to denote a mixed interpretation that is identical to the kinetic interpretation defined above, which Peters indicates by using a Stratonovich circle in the position where we use a diamond. 

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. 

Expressions for the drift velocity produced by Ito (1), Stratonovich (S), Eangevin (L), and kinetic (K) SDEs, and for the drift coefficient required for each type of SDE to produce the predicted drift velocity of Eq. (2.344), in a generic system of coordinates. VL is as defined in... 

Ito and Stratonovich SDEs are defined as different interpretations of equations of the form... 

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