Let's say I have a (one-dimensional) diffusion process
$$dX=\mu(X_t)dt+\sigma(X_t)dW.$$
Assume we have fixed $\epsilon > 0$ and $t >0$ Under what conditions is $\mathbb{P}^x(X_t < \epsilon)$ going to be a continuous function of $x$? Here, $\mathbb{P}^x(\cdot)$ denotes $\mathbb{P}(\cdot |X_0=x)$ as usual.
I know I already asked a similar question, however, I demanded the marginal distribution function to be smooth, although I later realized I can relax that condition if I work a bit differently. There are some articles (as I mentioned before) that mention continuous dependence on initial data, specifically this one (http://www.worldscientific.com/doi/pdf/10.1142/S0219025702000870). However, the scope is very general and the article deals with evolution equations, I would very much prefer to be able to find similar results for finite-dimensional diffusions since I am not very well versed in infinite-dimensional SDEs.
What I actually want to have at the end is to have this result not for a fixed $t$, but for a stopping time which also depends on the process. The stopping time is sort of "exponentially distributed with a continuous rate" by which I mean that conditioned on the path of the diffusion it has a density $f(t)=\lambda(X_t)e^{-\int_0^t\lambda(X_s)ds}$, where $\lambda$ is a continuous non-negative function. Perhaps there is a proper name for such a distribution.
In any case, I am interested in anything that can be said (from general statements to particular restrictive examples) about the continuity of the marginal distribution as a function of the initial condition. Thanks!