The backward Kolmogorov equation (BKE): $$\frac{du}{dt} = A(x,t) \cdot \nabla_x u(x,t) -\frac12 \text{Tr}(BB^t(x,t) \text{Hess}_x u(x,t) - f(t,x,u, B,\nabla g), \;\;\; t<T$$
If $f\equiv 0$ then this is a linear BKE. I read that if a BKE is linear, then there exists a corresponding stochastic process $X(t)$ whose probability density is the solution of the BKE, and $X(t)$ solves the stochastic differential equation $$dX(t) = A(X(t),t) dt + B(X(t)),t)dW(t).$$
Now, if the BKE is non-linear, when does a corresponding stochastic process exist? For example, the Allen-Cahn equation is a non-linear BKE. How do we find the corresponding stochastic process?