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In the original space I had $u=0$ on $ \partial \Omega$ but in this spacetime space, as it includes the variable itself, it is essentially as I written in the above question of having only data only prescribed on the "two sides of the rectangle" rather than the top/bottom where the y-axis is understood as time. For attainability, I assumed some conditions on the Markov transition kernel to attain the expected hitting is finite a.s..
Long story shorter: For a bounded open set $\Omega$ and some stochastic generator $L$ of an autonomous SDE, it is known that solving $Lu=-1$ on $\Omega$ satisfying boundary condition $u = 0$ on $\partial \Omega$ solves $u$ for expected hitting time. In the case of non-autonomous SDE, the Markov transition function is no longer time-homogeneous which traditionally is used to derive the above PDE. To regain the homogeneity, I considered the spacetime coordinates $(t,X_t)$ which is now time-homogeneous. Perhaps surprisingly, $L'u=-1$ for $L'$ on this spacetime space.
@NawafBou-Rabee: Actually, that is essentially the problem I am trying to solve in that I started with a SDE and now trying to solve its PDE counterpart. I think from what I've done, it is attainable under some assumption on L.
Thank you, Fan Zheng. I suppose the point is that contrasting with well-known literature that one needs $u=0$ (say) on the entire boundary $\partial \Omega$ to ensure uniqueness of solution (in the case of uniformly elliptic operator anyway).
@FanZheng, Yes, I meant $u=0$ rather than $Lu=0$ on the boundary - edited. How did you deduce non-uniqueness from your example -the fact it is not a "complete" Dirichlet boundary?
