Convergence of non-symmetric forms with changing reference measures
- 1( JMT) Aalto University , Department of Mathematics and Systems Analysis , PO Box 11100 ( Otakaari 1 , Espoo ), 00076 Aalto , Finland
- 2Department of Mathematics and Statistics
Abstract
We develop a functional-analytic theory of convergence for non-symmetric generalized Dirichlet forms defined on a sequence of Hilbert spaces Hn converging to a limit space H in the sense of Kuwae-Shioya, with the principal applications concerning L2-spaces over varying reference measures μn → μ. Extending Mosco's variational convergence for Dirichlet forms to the non-symmetric setting via a functional controlling the off diagonal values of the forms, we introduce a notion of generalized convergence of non-symmetric Dirichlet forms and show it to be equivalent to strong convergence of the associated resolvents, to weak convergence of the co-resolvents, and to strong graph convergence of the infinitesimal generators. A version of the Trotter-Kato theorem in the Kuwae-Shioya framework yields the corresponding equivalence for C0-contraction semigroups. As applications, we give sufficient conditions for the generalized convergence of elliptic and sectorial ai,j-forms with changing speed measures, both on ℝd and in infinite dimensions over a Gelfand triple, and deduce weak convergence of the laws of the associated right processes under tightness of the path measures.
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BibTeX
@misc{tolle2026convergence,
title = {{Convergence of non-symmetric forms with changing reference measures}},
author = {Jonas M. Tölle},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2611.00975},
eprint = {2611.00975},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00975},
}
RIS
TY - UNPB
TI - Convergence of non-symmetric forms with changing reference measures
AU - Jonas M. Tölle
PY - 2026
DA - 2026/10/31
PB - arXiv
DO - 10.48550/arxiv.2611.00975
UR - https://arxiv.org/abs/2611.00975
AB - We develop a functional-analytic theory of convergence for non-symmetric generalized Dirichlet forms defined on a sequence of Hilbert spaces H_n converging to a limit space H in the sense of Kuwae-Shioya, with the principal applications concerning L^2-spaces over varying reference measures μ_n → μ. Extending Mosco's variational convergence for Dirichlet forms to the non-symmetric setting via a functional controlling the off diagonal values of the forms, we introduce a notion of generalized convergence of non-symmetric Dirichlet forms and show it to be equivalent to strong convergence of the associated resolvents, to weak convergence of the co-resolvents, and to strong graph convergence of the infinitesimal generators. A version of the Trotter-Kato theorem in the Kuwae-Shioya framework yields the corresponding equivalence for C_0-contraction semigroups. As applications, we give sufficient conditions for the generalized convergence of elliptic and sectorial a_i,j-forms with changing speed measures, both on ℝ^d and in infinite dimensions over a Gelfand triple, and deduce weak convergence of the laws of the associated right processes under tightness of the path measures.
SN - 2331-8422
ER -