Strong uniqueness for both Dirichlet operators and stochastic dynamics to Gibbs measures on a path space with exponential interactions
Strong uniqueness for both Dirichlet operators and stochastic dynamics to Gibbs measures on a path space with exponential interactions
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DOI：
10.1016/j.jfa.2011.09.023
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发表时间：
2010-06
影响因子：
1.7
通讯作者：
S. Albeverio;Hiroshi Kawabi;M. Rockner
中科院分区：
文献类型：
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作者：
S. Albeverio;Hiroshi Kawabi;M. Rockner
通过推广前两位作者提出的SPDE方法，证明了无穷体积中指数型相互作用下路径空间C（R，Rd）上Dirichlet算子对Gibbs测度的Lp-唯一性.我们也给出了相应的动力学的SPDE表征。特别地，我们第一次证明了SPDE强解的存在性和唯一性，尽管自相互作用势不是可微的，因此漂移可能是不连续的。作为例子，我们的结果适用，我们提到的随机量子化的P（λ）1-，exp（λ）1-，和三角量子场在无限体积。特别是，我们的研究结果意味着这些模型的随机动力学的发电机基本自伴。最后，作为SPDE强唯一性结果的应用，我们证明了由上述Dirichlet算子生成的扩散半群的一些泛函不等式。
We prove Lp-uniqueness of Dirichlet operators for Gibbs measures on the path space C(R,Rd) associated with exponential type interactions in infinite volume by extending an SPDE approach presented in previous work by the last two named authors. We also give an SPDE characterization of the corresponding dynamics. In particular, for the first time, we prove existence and uniqueness of a strong solution for the SPDE, though the self-interaction potential is not assumed to be differentiable, hence the drift is possibly discontinuous. As examples, to which our results apply, we mention the stochastic quantization of P(ϕ)1-, exp(ϕ)1-, and trigonometric quantum fields in infinite volume. In particular, our results imply essential self-adjointness of the generator of the stochastic dynamics for these models. Finally, as an application of the strong uniqueness result for the SPDE, we prove some functional inequalities for diffusion semigroups generated by the above Dirichlet operators.