Let $\{X_t\}_{t \geq 0}$ be a symmetric, nearest-neighbor random walk on $\mathbb{Z}^d$ with exponential holding times of expectation $1/d$, starting at the origin. For a potential $V: \mathbb{Z}^d \to [0, \infty)$ with finite and nonempty support, define transformed path measures by $d \hat{\mathbb{P}}_T \equiv \exp (T^{-1} \int_0^T \int_0^T V(X_s - X_t) ds dt) d \mathbb{P}/Z_T$ for $T > 0$, where $Z_T$ is the normalizing constant. If $d = 1$ or if the self-attraction is sufficiently strong, then $||X_t||_{\infty}$ has an exponential moment under $\mathbb{P}_T$ which is uniformly bounded for $T > 0$ and $t \in [0, T]$. We also prove that ${X_t}_{t \geq 0}$ under suitable subsequences of ${\hat{\mathbb{P}}_T}_{T > 0}$ behaves for large $T$ asymptotically like a mixture of space-inhomogeneous ergodic random walks. For special cases like a sufficiently strong Dirac-type interaction, we even prove convergence of the transformed path measures and the law of $X_T$ as well as of the law of the empirical measure $L_T$ under ${\hat{\mathbb{P}}_T}_{T > 0}$.
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机译：假设$ \ {X_t \} _ {t \ geq 0} $是对称的，最近邻居在$ \ mathbb {Z} ^ d $上的随机行走，其指数保持时间为$ 1 / d $，从原点开始。对于可能的$ V：\ mathbb {Z} ^ d \ to [0，\ infty）$具有有限和非空支持，请通过$ d \ hat {\ mathbb {P}} _ T \ equiv \ exp（对于$ T> 0 $，T ^ {-1} \ int_0 ^ T \ int_0 ^ TV（X_s-X_t）ds dt）d \ mathbb {P} / Z_T $，其中$ Z_T $是归一化常数。如果$ d = 1 $或自我吸引力足够强，则$ ||| X_t || _ {\ infty} $在$ \ mathbb {P} _T $下有一个指数矩，该指数矩对于$ T> 0 $和$ t \ in [0，T] $。我们还证明了$ {\ hat {\ mathbb {P}} _ T} _ {T> 0} $的适当子序列下的$ {X_t} _ {t \ geq 0} $对于大的$ T $表现为渐近混合型空间不均匀的遍历遍历随机游走。对于某些特殊情况，例如足够强的Dirac型相互作用，我们甚至证明了$ {\ hat {\ mathbb {下的变换路径量度和$ X_T $定律以及经验量度定律$ L_T $的收敛。 P}} _ T} _ {T> 0} $。
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