Consider the motion of a Brownian particle in two or more dimensions, whose coordinate processes are standard Brownian motions with zero drift initially, and then at some random/unobservable time, one of the coordinate processes gets a non-zero drift governed by an independent finite-state Markov chain. Given that the position of the Brownian particle is being observed in real time, the problem is to detect the time at which a coordinate process gets the drift as accurately as possible. This is the so-called quickest real-time detection problem of a Markovian drift. The motivation for a Markovian drift stems from the consideration of a switching environment in practical problems in finance, engineering, and other areas.  To solve such problem, our main efforts are devoted to solving an equivalent optimal stopping problem with respect to a regime switching diffusion. Our result shows that the optimal stopping boundary can be represented as a unique solution to a non-linear integral equation in some admissible class. This is a joint work with Professor Philip Ernst at Rice University.