Given a diffusion process $ X_t $ on a Riemannian manifold $(M,g)$, with an infinitesimal generator $\mathcal{G}=\Delta_g/2 + b$, the Onsager-Machlup function is well-known to be: $$ \mathcal{L}(x,v) = \frac{1}{2}||v-b(x)||_{g(x)}^2 +\frac{1}{2}\text{div}_g\, b(x) - \frac{1}{12}R(x) $$

where $(x,v)\in TM$ and $R$ is the scalar curvature of $M$.

My question is **what is known about $\mathcal{L}$ for a killed diffusion process?**

In other words, if there is a random time (which may or may not depend on where the process goes) at which the process is killed, how does $\mathcal{L}$ change? Or, more specifically, how would the expression for most probable path $$\lim_{\epsilon\rightarrow 0}\frac{P(d_g(X_t,\phi_1(t))\leq \epsilon\;\forall\; t\in[0,T])}{P(d_g(X_t,\phi_2(t))\leq \epsilon\;\forall\; t\in[0,T])} =\exp\left( J_T[\phi_2] - J_T[\phi_1] \right) $$ where $$ J_T[\phi] = \int_0^T \mathcal{L}(\phi(t),\dot{\phi}(t))\,dt $$ for Riemannian distance $d_g$, be altered by $X_t$ having a killing time $\tau$ (such that, it seems, $T$ becomes $\tau\land T$ )? Forgive me if the question is silly; I am not familiar with killed diffusions.