# Gluing Markov processes

I am looking for a reference on the gluing together of strong Markov processes to get a new one.

Here is an example of what I have in mind. Let $B^1, B^2, \ldots$ be independent one-dimensional Brownian motions started at $0$ and $\tau_i$ be the first time $B^i$ hits $1$. Construct a new process $Y$ as follows. Let $Y(t)=B^1(t)$ for $t<\tau_1$, $Y(t)=1-B^2(t-\tau_1)$ for $\tau_1 \leq t<\tau_1+\tau_2$, $Y(t)=B^3(t-\tau_1-\tau_2)$ for $\tau_1+\tau_2\leq t <\tau_1+\tau_2+\tau_3$ and so on. I would like to conclude that the this construction gives a strong Markov process (in fact its a Brownian motion).

More generally, suppose I have a suitably nice topological space $E$ and open sets $U_1, \ldots, U_n$ such that $E=\cup U_i$. Suppose for each $\overline{U_i}$ I have a strong Markov process $X_i$ in $\overline{U_i}$ killed at $\overline{U_i}-U_i$ and that started at $x\in U_i \cap U_j$, $X_i$ and $X_j$ are the same up until the first time they leave $U_i \cap U_j$. Is it possible to "glue" these $X_i$'s together (analogous to the example above) to get a strong Markov process on all of $E$?

I think I can make sense of my example by looking at the infinitesimal generator for the process and observing that it is "nice" and therefore is the generator of a strong Markov process. Perhaps I can use the same sort of idea in general if the $X_i$ are "nice" (that is if the $X_i$ have "nice" generators operating on $C_0(U_i)$, I can use them to construct a nice generator on $C_0(E)$ and then use general theory to say there is a process with this generator), but I'm wondering if there is anything in the literature on this sort of construction.

EDIT: I slightly changed my example to try to emphasize that I don't want to a priori assume that I know anything about the process I get in the end.

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Your first example is a realization of Brownian motion because it's null at 0, a.s. continuous, and its independent increments have the right sort of Gaussian distribution (to see this, note the distribution of the sum of two Gaussian-distributed RVs and use a telescoping sum if necessary). In fact your example is basically the reflection principle: see, e.g. theorem 2.7 in people.bath.ac.uk/maspm/tcc-8.pdf – Steve Huntsman Feb 23 '12 at 2:43
You should be able to prove this directly from the strong Markov property. No need to use generators. Use induction to show that $Y^{\tau_i}$ has the required distribution stopped at time $i$. Also, do you have a precise definition of the strong Markov property used here and pathwise properties of the processes $X_i$ (right-continuity, etc)? – George Lowther Feb 23 '12 at 4:13
Perhaps for simplicity I want to restrict to Feller processes. I want my filtrations to be such that the Debut theorem holds and under these conditions I can assume my processes are Cadlag. By strong Markov property then, I mean something like $\mathbf{E}^{x}[\theta_T \eta|\mathcal{F}_t]=\mathbf{E}^{X(T)}[\eta]$, where $\theta_T$ is the shift operator (where everything is suitably measurable with respect to what it needs to be). – ShawnD Feb 23 '12 at 20:49
I believe in my example that I can see that it is Brownian motion by looking at its distributions (but it seems messy because at time $t$ I could be running any of $B^1$, $B^2$, . . . ), but I'm interested in what is known about this sort of procedure when nothing is known about the process I get in end. Namely, if I have a bunch of strong Markov processes on subsets of a space, can I somehow glue them together to get one on the entire space. In my example, $U_1=(-\infty,\1)$ and $U_2(0,\infty)$ and I'm "gluing" Brownian motion on $U_1$ and $U_2$ to get Brownian motion on $\mathbf{R}$ – ShawnD Feb 23 '12 at 20:56