Abstract
In this paper, we prove a variational formula for Dirichlet forms generated by general symmetric Markov processes. As its applications, we obtain lower bound estimates of the bottom of spectrum for symmetric Markov processes. Moreover, for a positive measure μ charging no set of zero capacity, we give a new proof of an L 2(μ)-estimate of functions in Dirichlet spaces. Finally, we calculate the principal eigenvalues for absorbing and time changed α-stable processes and obtain conditions for some measures being gaugeable.
| Original language | English |
|---|---|
| Pages (from-to) | 135-151 |
| Number of pages | 17 |
| Journal | Potential Analysis |
| Volume | 23 |
| Issue number | 2 |
| DOIs | |
| Publication status | Published - Sep 2005 |
| Externally published | Yes |
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Keywords
- Dirichlet form
- Principal eigenvalue
- Symmetric α-stable process
- Time change
- Variational formula
ASJC Scopus subject areas
- Mathematics(all)
- Analysis
Cite this
Variational formula for dirichlet forms and estimates of principal eigenvalues for symmetric α-stable processes. / Shiozawa, Yuichi; Takeda, Masayoshi.
In: Potential Analysis, Vol. 23, No. 2, 09.2005, p. 135-151.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Variational formula for dirichlet forms and estimates of principal eigenvalues for symmetric α-stable processes
AU - Shiozawa, Yuichi
AU - Takeda, Masayoshi
PY - 2005/9
Y1 - 2005/9
N2 - In this paper, we prove a variational formula for Dirichlet forms generated by general symmetric Markov processes. As its applications, we obtain lower bound estimates of the bottom of spectrum for symmetric Markov processes. Moreover, for a positive measure μ charging no set of zero capacity, we give a new proof of an L 2(μ)-estimate of functions in Dirichlet spaces. Finally, we calculate the principal eigenvalues for absorbing and time changed α-stable processes and obtain conditions for some measures being gaugeable.
AB - In this paper, we prove a variational formula for Dirichlet forms generated by general symmetric Markov processes. As its applications, we obtain lower bound estimates of the bottom of spectrum for symmetric Markov processes. Moreover, for a positive measure μ charging no set of zero capacity, we give a new proof of an L 2(μ)-estimate of functions in Dirichlet spaces. Finally, we calculate the principal eigenvalues for absorbing and time changed α-stable processes and obtain conditions for some measures being gaugeable.
KW - Dirichlet form
KW - Principal eigenvalue
KW - Symmetric α-stable process
KW - Time change
KW - Variational formula
UR - http://www.scopus.com/inward/record.url?scp=17144428083&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=17144428083&partnerID=8YFLogxK
U2 - 10.1007/s11118-004-5392-7
DO - 10.1007/s11118-004-5392-7
M3 - Article
AN - SCOPUS:17144428083
VL - 23
SP - 135
EP - 151
JO - Potential Analysis
JF - Potential Analysis
SN - 0926-2601
IS - 2
ER -