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Let $U\in C^{\infty}(\mathbb R^n;\mathbb R)$ and consider a continuos time Markov chain on the scaled integer lattice $\delta\mathbb Z^n$ with jump rates given by

$r_{\delta}(x,y) := \begin{cases} \exp\{-[U(\frac{x+y}{2})-U(x)]\} \ \ \ \ & \text{ if } x-y \in \delta\mathbb Z^n \text{ and } x ,y \text{ are nearest neighbours} \\ 0 &\text{ otherwise } \end{cases}$

So it jumps only to nearest neighbours and it is reversible with respect to $e^{-U}$.

Assume that $U$ grows sufficiently fast at infinity such that $\sum_{x\in\mathbb \delta Z^n} e^{-U(x)}<\infty$.

Question: what are nice conditions on $U$ such that this chain is nonexplosive (at least for small $\delta$)?

Can it be that the continuous version (diffusion reversible w.r.t. $e^{-U}dx$, satisfying the SDE

$ dX_t = -\nabla U(X_t) dt + dB_t$

(with $B_t$ Brownian motion)

does not explode, but the discrete does for every small $\delta$?

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