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

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,