 # 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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