Consider two state spaces $X$ and $Y$ and infinitesimal generators of Markov processes $(A_y)_{y\in Y}$ and $B$, on $X$ and $Y$ respectively. We assume that $A_y$ share the same domain $D(A)$, and denote the domain of $B$ by $ D(B)$.
Now, define the operator $$ Lf(x,y):= A_yf(\cdot ,y ) (x)+ Bf(x,\cdot)(y)\,, $$ say for $f$ in the sub algebra generated by $D(A)$ and $D(B)$. In the literature this goes under the name of "random evolution" (but I've also encountered names such as controlled/modulated Markov process or doubly stochastic process).
Question: When is $L$ the generator of a Markov process on the product state space $X\times Y$?
I've seen generators of the above form in the book by Ethier and Kurtz, Chapter 12, and the following two articles
[1] Trotter, H. F. (1959). On the Product of Semi-Groups of Operators. Proceedings of the American Mathematical Society, 10(4), 545–551.
[2] Kurtz, T.G. (1973). A limit theorem for perturbed operator semigroups with applications to random evolutions. Journal of Functional Analysis, 12, 55-67.
If I understood the literature correctly, the answer to my question is affirmative when $B$ is bounded or multiplied with a small constant, for example. Or if $B$ is "controlled" in some sense by $A$.
But what if $B$ is just an arbitrary well-behaved Markov process (buy $Y$ uncountable), which does not do anything crazy broadly speaking? If $A_y$ depends on $y$ in a regular enough way, I'd expect the operator $L$ still to define a Markov operator. For instance, for a system of SDEs this is standard.
But I could not find any literature on this.. so my guess
- either I am looking at the wrong place, in which case I happy about any pointers
- or this is a difficult problem?
(In application, $L$ was often studied in terms of limit theorems when replacing $ B$ with $ aB $, $a>0$, and sending $ a$ to infinity. See e.g. the book by Ethier-Kurtz, however, they always assume that the operator $ A+aB $ is a generator to begin with.)
Thanks.
