Consider the partial differential equation: $$\frac{\partial u}{\partial t}(x,t) + \mu(x,t) \frac{\partial u}{\partial x}(x,t) + \frac{1}{2} \sigma^2(x,t) \frac{\partial^2 u}{\partial x^2}(x,t) - V(x,t) u(x,t) + f(x,t) = 0$$ defined for all $x \in \mathbb{R}_+$ and $t \in [0,T]$, subject to the terminal condition: $$u(x,T) = \psi(x)$$ where $\mu, \sigma, \psi, V, f$ are known functions, $T$ is a parameter, and $u: \mathbb{R}_+\times[0,T] \rightarrow \mathbb{R}$ is the unknown. The pde admits a Feynman-Kac representation, which is standard and can be found for instance here.
When the coefficients $\mu, \sigma, V, f$ do not depend on time and are piecewise constants over the partition $[0,c), [c, \infty)$, for instance of the form $\mu(x) = \mu_1 1_{\{x > c\}} + \mu_2 1_{\{x \leq c\}}$ ($1_{\{x>c\}}$ takes value 1 if $x>c$ and zero otherwise), one could solve the pde piecewisely by finding two different solutions and then pasting them together. An example of the above can be found in this post.
Is the same true for the FK representation? Is it possible to find piecewisely two representations separately for $x \in [0,c)$ and $x \in [c, \infty)$ and then put them together?