TY - JOUR
T1 - Strong uniqueness for both Dirichlet operators and stochastic dynamics to Gibbs measures on a path space with exponential interactions
AU - Albeverio, Sergio
AU - Kawabi, Hiroshi
AU - Röckner, Michael
N1 - Funding Information:
and thus µ is supported by a much smaller subset of C(R,R) than C.
Funding Information:
The authors are grateful to Masao Hirokawa for useful discussions on the paper [15], and to Volker Betz and Martin Hairer for providing helpful comments on Remark 2.4. They were partially supported by the DFG–JSPS joint research project “Dirichlet Forms, Stochastic Analysis and Interacting Systems” (2007–2008), and CRC 701, as well as by the project NEST of the Provincia Autonoma di Trento, at University of Trento and by HCM at University of Bonn. The second named author was also partially supported by Grant-in-Aid for Young Scientists (Start-up), No. 18840034 and (B), No. 20740076 of the Ministry of Education, Culture, Sports, Science and Technology, Japan. This work was completed while the authors were visiting Isaac Newton Institute for Mathematical Sciences at University of Cambridge. They would like to thank the institute for its warm hospitality.
PY - 2012/1/15
Y1 - 2012/1/15
N2 - 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.
AB - 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.
KW - Dirichlet operator
KW - Essential self-adjointness
KW - Exp(φ)-quantum fields
KW - Gibbs measure
KW - L-uniqueness
KW - Logarithmic Sobolev inequality
KW - Path space
KW - SPDE
KW - Strong uniqueness
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U2 - 10.1016/j.jfa.2011.09.023
DO - 10.1016/j.jfa.2011.09.023
M3 - Article
AN - SCOPUS:80955130700
VL - 262
SP - 602
EP - 638
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
SN - 0022-1236
IS - 2
ER -