Abstract
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.
| Original language | English |
|---|---|
| Pages (from-to) | 602-638 |
| Number of pages | 37 |
| Journal | Journal of Functional Analysis |
| Volume | 262 |
| Issue number | 2 |
| DOIs | |
| Publication status | Published - 2012 Jan 15 |
| Externally published | Yes |
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Keywords
- Dirichlet operator
- Essential self-adjointness
- Exp(φ)-quantum fields
- Gibbs measure
- L-uniqueness
- Logarithmic Sobolev inequality
- Path space
- SPDE
- Strong uniqueness
ASJC Scopus subject areas
- Analysis
Cite this
Strong uniqueness for both Dirichlet operators and stochastic dynamics to Gibbs measures on a path space with exponential interactions. / Albeverio, Sergio; Kawabi, Hiroshi; Röckner, Michael.
In: Journal of Functional Analysis, Vol. 262, No. 2, 15.01.2012, p. 602-638.Research output: Contribution to journal › Article
}
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
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
UR - http://www.scopus.com/inward/record.url?scp=80955130700&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=80955130700&partnerID=8YFLogxK
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 -