Firstly we consider a finite dimensional Markov semigroup generated by Dunkl Laplacian with drift terms. For this semigroup we prove gradient bounds involving a symmetrised carré-du-champ operator, and we show that for small coefficients this semigroup has a unique invariant measure which satisfies ergodicity properties. We then extend this analysis to an infinite dimensional model on \((\mathbb {R}^{N})^{\mathbb {Z}^{d}}\), consisting of interacting finite dimensional models. We construct an associated Markov semigroup for this model using gradient bounds, and finally we study the existence of invariant measures and ergodicity properties.

首先，我们考虑由Dunkl Laplacian生成的具有漂移项的有限维马尔可夫半群。对于这个半群，我们证明了一个对称对称carré-du-champ算子的梯度边界，并且我们证明了对于小系数，这个半群具有满足人体工程学性质的唯一不变性度量。然后，我们将分析扩展到\（（\ mathbb {R} ^ {N}）^ {\ mathbb {Z} ^ {d}} \）上的无穷维模型，该模型由相互作用的有限维模型组成。我们使用梯度边界为该模型构造了一个关联的马尔可夫半群，最后我们研究了不变测度和遍历性的存在。