• $E$ 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?


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