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Let $X$ and $Y$ be two diffusion processes. Suppose they have generators $G_X$ and $G_Y$ with domains $D(G_X)$ and $D(G_Y)$ and cores $C(G_X)$ and $C(G_Y)$. Let $Z$ be the product diffusion with generator $G_Z := G_X + G_Y$. It is known that $D(G_X) \cap D(G_Y)$ is a core for $G_Z$.

Is $C(G_X) \widehat{\otimes} C(G_Y)$ a core for $G_Z$? If not, is it possible to produce a core for $G_Z$ from $C(G_X)$ and $C(G_Y)$ alone?

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  • $\begingroup$ I'm a little confused by your notation. Can you be more explicit about the ambient Banach spaces where the generators live, and what $\widehat\otimes$ means? $\endgroup$
    – Nate Eldredge
    Commented Dec 17, 2016 at 1:37
  • $\begingroup$ Let $U_X$ and $U_Y$ be the state spaces of $X$ and $Y$. I view the generators as linear maps $D(G_X) \to C_0(U_X)$ and similarly for Y, where $C_0$ denotes the space of continuous functions vanishing at infinity on a locally compact space. I'm being deliberately vague with the completion; I'm happy to allow the uncompleted tensor product or its Banach space completion in $C_0(U_X \times U_Y)$. $\endgroup$
    – ysys
    Commented Dec 17, 2016 at 1:44
  • $\begingroup$ So is $D(G_X)$ a subspace of $C_0(U_X)$, or some other space of functions on $U_X$? In that case, how does it make sense to sum the operators $G_X, G_Y$, when their domains live on different spaces (and hence have empty intersection)? $\endgroup$
    – Nate Eldredge
    Commented Dec 17, 2016 at 1:47
  • $\begingroup$ Yes, it's a subspace. My apologies, by the sum I mean $G_X \otimes 1 + 1 \otimes G_Y$, where I view $C_0(U_X \times U_Y)$ as a completion of $C_0(U_X) \otimes C_0(U_Y)$. $\endgroup$
    – ysys
    Commented Dec 17, 2016 at 1:51
  • $\begingroup$ I think you are right that the intersection does not make sense in this abstract setting (I have Neumann conditions in mind). $\endgroup$
    – ysys
    Commented Dec 17, 2016 at 1:53

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