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Accepted
Reference: hitting time of Gaussian process
Since they are independent, lets start with treating $W_{t}$ as a deterministic Holder-continuous function (and so lets interpret it as the t-dependent mean of the first term).
Indeed, in "Hitting tim …
1
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Small noise limits with irregular drift
Indeed as a follow-up to the comment, there is a wide literature of studying ODEs
$$\frac{dx(t)}{dt}=b(t,x(t))$$
which can have existence but not uniqueness eg. for $b(x):=2sgn(x)\sqrt{|x|}$, whereas …