set theory - Which very large cardinals are preserved under Woodin's forcing for $mathsf{AC}$?

pr.likelihood – Harnack inequalities for Markov chains Answer

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pr.likelihood – Harnack inequalities for Markov chains

We deem a (steady time) Markov train $X=({X_t}_{t ge 0},{P_x}_{x in V})$ on a finite clique $V$. We occupy furthermore that $V$ is embedded into $mathbb{R}^d$. The generator $mathcal{L}$ of $X$ is outlined by
commence{align*}
mathcal{L}f(x)=sum_{y in V}P(x,y)f(y)-f(x),quad x in V,, f in mathbb{R}^V.
aim{align*}

Here, $P(x,y)$ denotes the transition likelihood of $X$. Let $A subset V$ and $h colon A to mathbb{R}$ a duty on $A$. We say that $h$ is harmonic on $A$ if $mathcal{L}h(x)=0$ for $x in A$.

I’m involved with a Harnack inequality for these harmonic features. The inequality signifies that
commence{align*}
sup_{y in B(x,r/2)}h(y) le C inf_{y in B(x,r/2)}h(y)
aim{align*}

for a harmonic duty $h$ on $B(x,r)=y-x$. Here, $|cdot|$ denotes the Euclidean distance, and $C$ denotes a constructive ceaseless relying on $x$ and $r$.

What assumptions can breathe made for $V$ and $P$ to secure this kindly of inequality?

If $V$ is a finite clique, I cerebrate it ought to breathe obtained underneath a gentle assumption, however is that grievance?

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