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pr.chance – Convergence of Feller processes Answer

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pr.chance – Convergence of Feller processes

Let $E$ breathe a domestically compact Hausdorff house with countable abject. We deem a sequence of Feller processes $X^n=({X_t^n}_{t ge0 }, {P_x^n}_{x in E})$, $n in mathbb{N}$. That is, every $X^n$ is a càdlàg stochastic course of on $E$. The semigroup ${T_t^n}_{t>0}$ leaves invariant $C_0(E)$, the house of steady features vanishing at infinity.

Let $X$ breathe a Feller course of on $E$. The semigroup is denoted by ${T_t}_{t>0}$.
Suppose that for any $f in C_{0}(E)$ and $t>0$, we’ve
commence{align*}
lim_{n to infty}sup_{x in E}|T_t^nf(x)-T_tf(x)|=0.
aim{align*}

In different phrases, the finite distribution of $X^n$ converges to that of $X$.

My query

We occupy furthermore that $X$ and ${X^n}_{n=1}^infty$ own collectively steady transition density. That is, we’ve for any $t>0$, $x in E$, $n in mathbb{N}$,
commence{align*}
P_x(X_t in dy)=p_t(x,y),dmu(y),
P_x^n(X_t^n in dy)=p_t^n(x,y),dmu_n(y).
aim{align*}

Here, $mu$ and ${mu_n}_{n=1}^infty$ denote Borel measures on $E$. When does the subsequent pretense maintain?

For any $t>0$ and $x,y in E$, we’ve
commence{align*}
(bigstar)quad lim_{n to infty}p_t^n(x,y)=p_t(x,y).
aim{align*}

Is there a route to get $(bigstar)$ with out utilizing an Ascoli-Arzela kind dispute?

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