Consider a 3D Bessel bridge $\rho_t$ connecting $(x,t)=(0,0)$ and $(x,t)=(0,T)$, whose SDE is given by $$d\rho_t = \left(\frac{1}{\rho_t} - \frac{\rho_t}{T-t}\right)dt + dB_t,$$ where $B_t$ is a standard Brownian motion. The corresponding Fokker-Planck (FP) equation for its density $p(t,x)$ is $$\partial_t p = \frac{1}{2}\partial_x^2 p - \partial_x\left[p\left(\frac{1}{x}-\frac{x}{T-t}\right)\right].$$ Now, I have another process with the following SDE ($a$ is a constant): $$dX_t = \left(\frac{1}{X_t} - \frac{X_t}{T-t}+a\right)dt + dB_t.$$ The process $d\tilde{B}_t = dB_t + adt$ is a Brownian motion with a drift. By changing to a new measure (allowed by the Girsanov's theorem), $d\tilde{B}_t$ becomes a new standard Brownian motion in this new measure. Then we can still interpret the later SDE as the SDE for a 3D Bessel bridge between $(x,t)=(0,0)$ and $(x,t)=(0,T)$. The corresponding FP for the later SDE is $$\partial_t p = \frac{1}{2}\partial_x^2 p - \partial_x\left[p\left(\frac{1}{x}-\frac{x}{T-t} +a \right)\right].$$ My questions are:
- If I know the exact solution for the former FP, how can I derive the solution for the later FP?
- Is there any way to map the second FP to the first FP?
