Let $A$ be an open subset of $\Bbb R^2$, and let us consider a diffusion $\mathrm dX_t = f(X_t)\mathrm dt + g(X_t)\mathrm dW_t$ where $f$ and $g$ are globally Lipschitz continuous maps. Suppose I am given a stochastic kernel $$ T:\partial A\times \mathscr B(A)\to[0,1] $$ and I would like to define a stochastic process that evolves as a diffusion over $A$, and when hitting the boundary $\partial A$ it jumps inside $A$ according to $T$. I guess for this process to be well-defined, the boundary of $A$ has to have some nice properties - e.g. to be piece-wise smooth. I'd be happy if anybody could clarify this point. Perhaps, even for a planar Brownian motion some non-trivial conditions are needed.
It is likely that some additional properties are required for the kernel $T$ as well - that is, perhaps $T(B|x) = 1$ for any $x\in \partial A$ where $B\subset A$ is such that $d(B,\partial A)>0$. However, in this particular question I wonder regarding the conditions for the boundary.