# Convergence of Feller processes implies convergence of the resolvent operators of their generators

Let

• $$E$$ be a locally compact separable metric space
• $$(T_n(t))_{t\ge0}$$ and $$(T(t))_{t\ge0}$$ be strongly continuous contraction semigroups on $$C_0(E)$$ with generator $$(\mathcal D(A_n),A_n)$$ and $$(\mathcal D(A),A)$$, respectively, for $$n\in\mathbb N$$
• $$(\Omega,\mathcal A,\operatorname P)$$ be a probability space
• $$(X^n_t)_{t\ge0}$$ and $$(X_t)_{t\ge0}$$ be $$E$$-valued càdlàg processes on $$(\Omega,\mathcal A,\operatorname P)$$ with $$\operatorname E\left[f(X^n_t)\mid(X^n_r)_{r\le s}\right]=(T_n(t-s)f)(X^n_s)\;\;\;\text{almost surely}\tag1$$ and $$\operatorname E\left[f(X_t)\mid(X_r)_{r\le s}\right]=(T(t-s)f)(X_s)\;\;\;\text{almost surely}\tag2$$ for all $$f\in C_0(E)$$ and $$t\ge s\ge0$$

Suppose we know that $$X^n\xrightarrow{n\to\infty}X$$ weakly (wrt the Skorohod topology) whenever $$X^n_0\xrightarrow{n\to\infty}X_0$$. Let $$f\in C_0(E)$$, $$\lambda>0$$, $$(x_n)_{n\in\mathbb N}\subseteq E$$ and $$x\in E$$ with $$x_n\xrightarrow{n\to\infty}x$$. Are we able to conclude that $$(R_\lambda(A_n)f)(x_n)\xrightarrow{n\to\infty}(R_\lambda(A)f)(x),\tag3$$ where $$R_\lambda(A_n)$$ and $$R_\lambda(A)$$ denote the resolvent operator of $$A_n$$ and $$A$$, respectively, with respect to the regular value $$\lambda$$.

From general operator theory, we know that $$R_\lambda(A_n)f=\int_0^\infty e^{-\lambda t}T_n(t)f\:{\rm d}t\tag4$$ and $$R_\lambda(A)f=\int_0^\infty e^{-\lambda t}T(t)f\:{\rm d}t.\tag5$$ From $$(1)$$ and $$(2)$$ we conclude that $$R_\lambda(A)f=\operatorname E\left[\int_0^\infty e^{-\lambda t}f(X^n_t)\:{\rm d}t\mid X^n_0=\;\cdot\;\right]\;\;\;(X^n_0)_\ast\operatorname P\text{-almost surely}\tag6$$ and $$R_\lambda(A)f=\operatorname E\left[\int_0^\infty e^{-\lambda t}f(X_t)\:{\rm d}t\mid X_0=\;\cdot\;\right]\;\;\;(X_0)_\ast\operatorname P\text{-almost surely}.\tag7$$ We may note that $$g:D([0,\infty),E)\to\mathbb R\;,\;\;\;y\mapsto\int_0^\infty e^{-\lambda t}f(y(t))\:{\rm d}t$$ is bounded and continuous (where $$D([0,\infty),E)$$ denotes the space of càdlàg functions from $$[0,\infty)$$ to $$E$$ equipped with the Skorohod topology). Let $$\delta$$ denote the Dirac kernel on $$(E,\mathcal B(E))$$, we may note that $$\delta_{x_n}\xrightarrow{n\to\infty}\delta_x\tag8$$ weakly.

By $$(8)$$ and boundedness and continuity of $$g$$, we easily see that if $$(X^n_0)_\ast\operatorname P=\delta_{x_n}$$ and $$(X_0)_\ast\operatorname P=\delta_x$$, then $$(3)$$ holds true. However, what if $$X^n$$ and $$X$$ have arbitrary initial distributions? Can we reduce the problem somehow to the former case?