\[ \begin{array} { l l } { { \overline { { { f } } } ( t _ { n } ) = \left\langle \int f ( x + \Delta x ) P _ { n - 1 } ( x ) d ^ { D } x \right\rangle _ { \mathrm { N s } } } } \\ { { \qquad = \left\langle \int \left( f ( x ) + f _ { \ast } ( x ) \Delta x ^ { i } + ( 1 / 2 ) f _ { \ast j } ( x ) \Delta x ^ { i } \Delta x ^ { j } + . . . \right) P _ { n - 1 } ( x ) d ^ { D } x \right\rangle _ { \mathrm { N s } } } } \\ { { \qquad = \displaystyle \int f ( x ) \left\langle \hat { 1 } - \frac { \partial } { \partial x ^ { i } } \Delta x ^ { i } + ( 1 / 2 ) \frac { \partial ^ { 2 } } { \partial x ^ { i } \partial x ^ { j } } \Delta x ^ { i } \Delta x ^ { j } + . . . \right\rangle _ { \mathrm { N s } } P _ { n - 1 } ( x ) d ^ { D } x . } } & { { \mathrm { ( A 8 ) } } } \end{array} \]

Here, \( f _ { / j } \, \equiv \, \partial f / \partial x ^ { i } \) and similar for \( f _ { { ' } i j } \, = \, \partial ^ { 2 } f / \partial x ^ { i } \partial x ^ { j } \), and the partial integration has been used.

The next step is to Taylor expand \( \Delta x \) up to second order in \( \Delta t \) and substitute this expansion into the above expression. Using Equation (A1), one has

\[ \Delta x ^ { i } = \mathcal { F } _ { n } ^ { i } \Delta t + \alpha ( \mathcal { F } _ { n } ^ { i } ) , \mathcal { F } _ { n } ^ { j } \Delta t ^ { 2 } + \ldots \eqno ( \mathrm { A 9 } ) \]

Substituting this expression into Equation (A8), using Equation (A2), and performing the stochastic averaging over \( \xi _ { n } \) with the help of Equation (7), one arrives at

\[ \overline { { f } } ( t _ { n } ) = \int f ( x ) ( \widehat { 1 } - \Delta t \hat { H } _ { \alpha } ^ { ( D ) } + \ldots ) P _ { n - 1 } ( x ) d ^ { D } x , \eqno ( \mathrm { A 1 0 } ) \]

with the FP operator being

\[ \hat { H } _ { \alpha } ^ { ( D ) } = - \frac { \partial } { \partial x ^ { i } } F _ { \alpha } ^ { i } ( x ) - \Theta \frac { \partial } { \partial x ^ { i } } e _ { a } ^ { i } ( x ) \frac { \partial } { \partial x ^ { j } } e _ { a } ^ { j } ( x ) \eqno ( \mathrm { A 1 1 } ) \]

and with the \( \alpha \)-dependent flow vector field from Equation (91).

In the above derivation of the FP operator, SdE (A9) was used as a formal equation defining \( \Delta x \). One can take an alternative view on stochastic dynamics in which the dynamics is continuous in time and the noise is piece-wise constant, as given in Figure 1. For a fixed noise configuration, one has a continuous trajectory x(t), defined by \( \dot { x } \, = \, \mathcal { F } _ { n } ( x ( t ) ) \) with the initial condition \( x ( t _ { n - 1 } ) = x _ { n _ { 1 } } \). Now, there is no freedom in choosing \( \alpha \) because \( \Delta x \) is uniquely defined by the evolution according to the Picard-Lindelo¨f theorem. In particular, \( \Delta x \) has a unique Taylor expansion in \( \Delta t \):

\[ \Delta x ^ { i } = \left. \frac { \partial x ^ { i } } { \partial t } \right| _ { \Delta t = 0 } \Delta t + \frac { 1 } { 2 } \left. \frac { \partial ^ { 2 } x ^ { i } } { \partial ^ { 2 } t } \right| _ { \Delta t = 0 } \Delta t ^ { 2 } + . . . \eqno ( \mathrm { A 1 2 } ) \]

The first coefficient here is determined from the SDE itself,

\[ \left. \frac { \partial x ^ { i } } { \partial t } \right| _ { \Lambda t = 0 } = \mathcal { F } _ { n } ^ { i } ( x ) , \eqno { ( \mathrm { A 1 3 } ) } \]

whereas the second coefficient is obtained via one differentiation of the SDE over time

\[ \left. \frac { \partial ^ { 2 } x ^ { i } } { \partial ^ { 2 } t } \right| _ { \Delta t = 0 } = \left. \frac { \partial \mathcal { F } _ { n } ^ { i } ( x ) } { \partial t } \right| _ { \Delta t = 0 } = \mathcal { F } _ { n ^ { \prime } j } ^ { i } ( x ) \left. \frac { \partial x ^ { i } } { \partial t } \right| _ { \Delta t = 0 } = \mathcal { F } _ { n ^ { \prime } j } ^ { i } ( x ) \mathcal { F } _ { n } ^ { j } ( x ) ; \eqno { ( \mathrm { A 1 4 } ) } \]

thus, the quantity in Equation (A12) becomes

\[ \Delta x ^ { i } = \mathcal { F } _ { n } ^ { i } ( x ) \Delta t + \frac { 1 } { 2 } \mathcal { F } _ { n ^ { \prime } j } ^ { i } ( x ) \mathcal { F } _ { n } ^ { j } ( x ) \Delta t ^ { 2 } + \ldots \eqno ( \mathrm { A 1 5 } ) \]

Comparing this equation with Equation (A9), one concludes that the Stratonovich choice of \( \alpha = 1 / \tau \) 2 must always be used for the continuous-time picture of temporal evolution.

Concerning the Ito interpretation of SDEs, it is often said that, unlike all other interpre-tations, the Ito approach respects the Markovian property in the sense that the increment \( \Delta x _ { n } \) or, equivalently, the final point \( x _ { n } \, = \, x _ { n - 1 } + \Delta x _ { n } \) is a function of only \( x _ { n - 1 } \) and not of \( x _ { n } \). This advantage of Ito SDEs, however, is a misinterpretation. Indeed, the very state-ment that \( x _ { n } \) is a function of itself for \( \alpha \) 0 does not make sense from the point of view of functional dependence. This sentence only tells us that \( x _ { n } \) as a function of \( x _ { n - 1 } \) is given only implicitly by Equation (90). For a fixed noise variable \( \xi _ { n } \), the final point \( x _ { n } \) together with the increment \( \Delta x _ { n } \) is always a function of \( x _ { n - 1 } \) only. Its explicit expression is given by Equation (A9) up to second order in \( \Delta t \), the only accuracy relevant in the continuous-time limit.

Furthermore, the Markovian property of stochastic processes is concerned not with the trajectories (the variables \( x _ { n } \) and \( x _ { n - 1 } \)) but rather with the temporal evolution of TPDs. In application to the SdEs (90), the Markovian property means that the TPD at time moment \( t _ { n } \) depends on the TPD at the previous time moment \( t _ { n - 1 } \) only and not on the TPD at earlier time moments. As clearly observed from Equation (A6), which is correct for all \( \alpha \), all the interpretations of SDEs satisfy this requirement of Markovianity. In other words, Ito SDEs are just as Markovian as SDEs in all the other interpretations.

In other words, the only advantage of the Ito interpretation is the relative ease of its numerical implementation because the increment as a function of \( x _ { n - 1 } \) is given explicitly by the Ito SdE. This convenience for numerical implementations, however, does not have any significance from the mathematical point of view.