How is this finite difference expression derived? In this $\text{paper}^1$, the following equation:
$$\frac{\partial O}{\partial t}(p,x;t) = \left[p\frac{\partial}{\partial x} - \left(\gamma p + \frac{dv(x)}{dx} \right)\frac{\partial}{\partial p} + \frac{\gamma}{\beta}\frac{\partial^2}{\partial p^2}\right]O(p,x;t) \\ + \int_{-\infty}^\infty dp' w(p-p',x)O(p',x,t) $$
is rewritten (for short times) as the following finite difference expression (FDE):
$$O(p-(\gamma p +  dv_x(x))\Delta t, x + p \Delta t; t ) = O(p,x; t+\Delta t) + \Delta t\left[\frac{\gamma}{\beta}\frac{\partial^2}{\partial p^2}O(p,x; t+\Delta t)  + \int_{-\infty}^\infty dp' w(p-p',x, t')O(p',x;t + \Delta t) \right] \\ + \text{order}(\Delta t^2)$$
I am confused how this is derived from the first. I see that the left hand side of the FDE can be rewritten as $O(p+\frac{\partial p}{\partial t}\Delta t, x + \frac{\partial x}{\partial t}\Delta t, t)$, but I don't know how this term appears. Also, any derivation I attempt has a term $O(x,p,t)$, but the above FDE does not.
Any hints understanding this would be appreciated.
P.S. These equations corresponds to Equations (2.27) and (A1) in the paper:
1.Zhang, Shesheng, and Eli Pollak. "Quantum dynamics for dissipative systems: A numerical study of the Wigner–Fokker–Planck equation." The Journal of chemical physics 118.10 (2003): 4357-4364.
 A: As discussed in the comments, I think the signs are wrong – here’s how the equation makes sense with different signs. (I’m assuming that $\mathrm dv_x(x)$ is meant to be $\frac{\mathrm dv(x)}{\mathrm dx}$, and that there’s no significance to the fact that the time argument is delimited by a semicolon in the first equation but by a comma in the second.)
Taylor-expanding the terms with $\Delta t$ in the arguments in the second equation to first order yields
$$
O(p,x,t)-\left(\gamma p+\frac{\mathrm dv(x)}{\mathrm dt}\right)\Delta t\frac{\partial O}{\partial p}(p,x,t)+p\Delta t\frac{\partial O}{\partial x}(p,x,t)=O(p,x,t)+\Delta t\frac{\partial O}{\partial t}(p,x,t)+ \Delta t\left[\frac{\gamma}{\beta}\frac{\partial^2}{\partial p^2}O(p,x, t+\Delta t)  + \int_{-\infty}^\infty dp' w(p-p',x, t')O(p',x,t + \Delta t) \right] \\ + \text{order}(\Delta t^2)\;.
$$
Replacing $t+\Delta t$ by $t$ only yields terms of second order (since those terms are already multiplied by $\Delta t$). Then cancelling $O(p,x,t)$, dividing through by $\Delta t$ and rearranging yields the first equation up to signs.
