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Dirichlet forms and polymer models based on stable processes

Author

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  • Li, Liping
  • Li, Xiaodan

Abstract

In this paper, we are concerned with polymer models based on α-stable processes, where α∈(d2,d∧2) and d stands for dimension. They are attached with a delta potential at the origin and the associated Gibbs measures are parametrized by a constant γ∈R∪{−∞} playing the role of inverse temperature. Phase transition exhibits with critical value γcr=0. Our first object is to formulate the associated Dirichlet form of the canonical Markov process X(γ) induced by the Gibbs measure for a globular state γ>0 or the critical state γ=0. Approach of Dirichlet forms also leads to deeper descriptions of their probabilistic counterparts. Furthermore, we will characterize the behaviour of polymer near the critical point from probabilistic viewpoint by showing that X(γ) is convergent to X(0) as γ↓0 in a certain meaning.

Suggested Citation

  • Li, Liping & Li, Xiaodan, 2020. "Dirichlet forms and polymer models based on stable processes," Stochastic Processes and their Applications, Elsevier, vol. 130(10), pages 5940-5972.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:10:p:5940-5972
    DOI: 10.1016/j.spa.2020.04.011
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    References listed on IDEAS

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    1. Kim, Panki, 2006. "Weak convergence of censored and reflected stable processes," Stochastic Processes and their Applications, Elsevier, vol. 116(12), pages 1792-1814, December.
    2. Blount, Douglas & Kouritzin, Michael A., 2010. "On convergence determining and separating classes of functions," Stochastic Processes and their Applications, Elsevier, vol. 120(10), pages 1898-1907, September.
    3. Chen, Zhen-Qing & Fukushima, Masatoshi, 2015. "One-point reflection," Stochastic Processes and their Applications, Elsevier, vol. 125(4), pages 1368-1393.
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