I am trying to understand the article "On boundary conditions for multidimensional Diffusion processes" of A. D. Ventzell (or Wentzell). I copy the images for greater convenience:
In the footnote the author says:
I don't understand what are those transformations.
Is anyone familiar with this technique? Could you outline the main steps to obtain the boundary conditions?
This is not as complicated as it seems, just write $$F(x) = F(x) - F(x_0) - \sum_{i=1}^{n-1} \frac{\partial F(x_0)}{\partial t_i(x)} t_i(x ) + F(x_0) + \sum_{i=1}^{n-1} \frac{\partial F(x_0)}{\partial t_i(x)} t_i(x ) $$
the first two terms appear naturally, to the last term it suffices to write
$$\int F(x) - F(x_0) - \sum_{i=1}^{n-1} \frac{\partial F(x_0)}{\partial t_i(x)} t_i(x ){l(t)} d P(t,x_0,x)) =\\\int\frac{F(x) - F(x_0) - \sum_{i=1}^{n-1} \frac{\partial F(x_0)}{\partial t_i(x)} t_i(x ){l(t)}}{ \sum_i t^2_i(x) + n(x)}d\bigg( l(t) (\sum_i t^2_i(x) + n(x)) P(t,x_0,dx)\bigg) $$