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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