Recent progress on the dirichlet forms associated with stochastic quantization problems

Rongchan Zhu; Xiangchan Zhu

doi:10.1007/978-3-319-74929-7_39

Recent progress on the dirichlet forms associated with stochastic quantization problems

Rongchan Zhu^*, Xiangchan Zhu

^*Corresponding author for this work

School of Mathematics and Statistics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

1 Citation (Scopus)

Abstract

In this paper we present recent progress on the Dirichlet forms associated with stochastic quantization problems obtained in Röckner et al. (J Funct Anal, 272(10):4263–4303, 2017, [23]), Röckner et al. (Commun Math Phys, 352(3):1061–1090, 2017, [24]), Zhu and Zhu (Dirichlet form associated with the Φ₃ ⁴ model, 2017, [27]). In the two dimensional case we have obtained the equivalence of the two notions of solutions, the restricted Markov uniqueness and the uniqueness of martingale problem. In the three dimensional case we construct the Dirichlet form associated with the dynamical Φ⁴ ₃ model obtained in Catellier and Chouk (Paracontrolled distributions and the 3-dimensional stochastic quantization equation, [6]), Hairer (Invent Math, 198:269–504, 2014, [14]), Mourrat and Weber (Global well-posedness of the dynamic Φ⁴ ₃ model on the torus, [20].

Original language	English
Title of host publication	Stochastic Partial Differential Equations and Related Fields - In Honor of Michael Röckner SPDERF, 2016
Editors	Gerald Trutnau, Andreas Eberle, Walter Hoh, Moritz Kassmann, Martin Grothaus, Wilhelm Stannat
Publisher	Springer New York LLC
Pages	561-571
Number of pages	11
ISBN (Print)	9783319749280
DOIs	https://doi.org/10.1007/978-3-319-74929-7_39
Publication status	Published - 2018
Event	International conference on Stochastic Partial Differential Equations and Related Fields, SPDERF 2016 - Bielefeld, Germany Duration: 10 Oct 2016 → 14 Oct 2016

Publication series

Name	Springer Proceedings in Mathematics and Statistics
Volume	229
ISSN (Print)	2194-1009
ISSN (Electronic)	2194-1017

Conference

Conference	International conference on Stochastic Partial Differential Equations and Related Fields, SPDERF 2016
Country/Territory	Germany
City	Bielefeld
Period	10/10/16 → 14/10/16

Keywords

Dirichlet form
Paracontrolled distributions
Regularity structures
Renormalisation
Space-time white noise
ϕ model

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10.1007/978-3-319-74929-7_39

Cite this

Zhu, R., & Zhu, X. (2018). Recent progress on the dirichlet forms associated with stochastic quantization problems. In G. Trutnau, A. Eberle, W. Hoh, M. Kassmann, M. Grothaus, & W. Stannat (Eds.), Stochastic Partial Differential Equations and Related Fields - In Honor of Michael Röckner SPDERF, 2016 (pp. 561-571). (Springer Proceedings in Mathematics and Statistics; Vol. 229). Springer New York LLC. https://doi.org/10.1007/978-3-319-74929-7_39

Zhu, Rongchan ; Zhu, Xiangchan. / Recent progress on the dirichlet forms associated with stochastic quantization problems. Stochastic Partial Differential Equations and Related Fields - In Honor of Michael Röckner SPDERF, 2016. editor / Gerald Trutnau ; Andreas Eberle ; Walter Hoh ; Moritz Kassmann ; Martin Grothaus ; Wilhelm Stannat. Springer New York LLC, 2018. pp. 561-571 (Springer Proceedings in Mathematics and Statistics).

@inproceedings{119c39749f224f9c8cdad714ab0b39ec,

title = "Recent progress on the dirichlet forms associated with stochastic quantization problems",

abstract = "In this paper we present recent progress on the Dirichlet forms associated with stochastic quantization problems obtained in R{\"o}ckner et al. (J Funct Anal, 272(10):4263–4303, 2017, [23]), R{\"o}ckner et al. (Commun Math Phys, 352(3):1061–1090, 2017, [24]), Zhu and Zhu (Dirichlet form associated with the Φ3 4 model, 2017, [27]). In the two dimensional case we have obtained the equivalence of the two notions of solutions, the restricted Markov uniqueness and the uniqueness of martingale problem. In the three dimensional case we construct the Dirichlet form associated with the dynamical Φ4 3 model obtained in Catellier and Chouk (Paracontrolled distributions and the 3-dimensional stochastic quantization equation, [6]), Hairer (Invent Math, 198:269–504, 2014, [14]), Mourrat and Weber (Global well-posedness of the dynamic Φ4 3 model on the torus, [20].",

keywords = "Dirichlet form, Paracontrolled distributions, Regularity structures, Renormalisation, Space-time white noise, ϕ model",

author = "Rongchan Zhu and Xiangchan Zhu",

note = "Publisher Copyright: {\textcopyright} Springer International Publishing AG, part of Springer Nature 2018.; International conference on Stochastic Partial Differential Equations and Related Fields, SPDERF 2016 ; Conference date: 10-10-2016 Through 14-10-2016",

year = "2018",

doi = "10.1007/978-3-319-74929-7_39",

language = "English",

isbn = "9783319749280",

series = "Springer Proceedings in Mathematics and Statistics",

publisher = "Springer New York LLC",

pages = "561--571",

editor = "Gerald Trutnau and Andreas Eberle and Walter Hoh and Moritz Kassmann and Martin Grothaus and Wilhelm Stannat",

booktitle = "Stochastic Partial Differential Equations and Related Fields - In Honor of Michael R{\"o}ckner SPDERF, 2016",

}

Zhu, R & Zhu, X 2018, Recent progress on the dirichlet forms associated with stochastic quantization problems. in G Trutnau, A Eberle, W Hoh, M Kassmann, M Grothaus & W Stannat (eds), Stochastic Partial Differential Equations and Related Fields - In Honor of Michael Röckner SPDERF, 2016. Springer Proceedings in Mathematics and Statistics, vol. 229, Springer New York LLC, pp. 561-571, International conference on Stochastic Partial Differential Equations and Related Fields, SPDERF 2016, Bielefeld, Germany, 10/10/16. https://doi.org/10.1007/978-3-319-74929-7_39

Recent progress on the dirichlet forms associated with stochastic quantization problems. / Zhu, Rongchan; Zhu, Xiangchan.
Stochastic Partial Differential Equations and Related Fields - In Honor of Michael Röckner SPDERF, 2016. ed. / Gerald Trutnau; Andreas Eberle; Walter Hoh; Moritz Kassmann; Martin Grothaus; Wilhelm Stannat. Springer New York LLC, 2018. p. 561-571 (Springer Proceedings in Mathematics and Statistics; Vol. 229).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

TY - GEN

T1 - Recent progress on the dirichlet forms associated with stochastic quantization problems

AU - Zhu, Rongchan

AU - Zhu, Xiangchan

N1 - Publisher Copyright: © Springer International Publishing AG, part of Springer Nature 2018.

PY - 2018

Y1 - 2018

N2 - In this paper we present recent progress on the Dirichlet forms associated with stochastic quantization problems obtained in Röckner et al. (J Funct Anal, 272(10):4263–4303, 2017, [23]), Röckner et al. (Commun Math Phys, 352(3):1061–1090, 2017, [24]), Zhu and Zhu (Dirichlet form associated with the Φ3 4 model, 2017, [27]). In the two dimensional case we have obtained the equivalence of the two notions of solutions, the restricted Markov uniqueness and the uniqueness of martingale problem. In the three dimensional case we construct the Dirichlet form associated with the dynamical Φ4 3 model obtained in Catellier and Chouk (Paracontrolled distributions and the 3-dimensional stochastic quantization equation, [6]), Hairer (Invent Math, 198:269–504, 2014, [14]), Mourrat and Weber (Global well-posedness of the dynamic Φ4 3 model on the torus, [20].

AB - In this paper we present recent progress on the Dirichlet forms associated with stochastic quantization problems obtained in Röckner et al. (J Funct Anal, 272(10):4263–4303, 2017, [23]), Röckner et al. (Commun Math Phys, 352(3):1061–1090, 2017, [24]), Zhu and Zhu (Dirichlet form associated with the Φ3 4 model, 2017, [27]). In the two dimensional case we have obtained the equivalence of the two notions of solutions, the restricted Markov uniqueness and the uniqueness of martingale problem. In the three dimensional case we construct the Dirichlet form associated with the dynamical Φ4 3 model obtained in Catellier and Chouk (Paracontrolled distributions and the 3-dimensional stochastic quantization equation, [6]), Hairer (Invent Math, 198:269–504, 2014, [14]), Mourrat and Weber (Global well-posedness of the dynamic Φ4 3 model on the torus, [20].

KW - Dirichlet form

KW - Paracontrolled distributions

KW - Regularity structures

KW - Renormalisation

KW - Space-time white noise

KW - ϕ model

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U2 - 10.1007/978-3-319-74929-7_39

DO - 10.1007/978-3-319-74929-7_39

M3 - Conference contribution

AN - SCOPUS:85049973009

SN - 9783319749280

T3 - Springer Proceedings in Mathematics and Statistics

SP - 561

EP - 571

BT - Stochastic Partial Differential Equations and Related Fields - In Honor of Michael Röckner SPDERF, 2016

A2 - Trutnau, Gerald

A2 - Eberle, Andreas

A2 - Hoh, Walter

A2 - Kassmann, Moritz

A2 - Grothaus, Martin

A2 - Stannat, Wilhelm

PB - Springer New York LLC

T2 - International conference on Stochastic Partial Differential Equations and Related Fields, SPDERF 2016

Y2 - 10 October 2016 through 14 October 2016

ER -

Zhu R, Zhu X. Recent progress on the dirichlet forms associated with stochastic quantization problems. In Trutnau G, Eberle A, Hoh W, Kassmann M, Grothaus M, Stannat W, editors, Stochastic Partial Differential Equations and Related Fields - In Honor of Michael Röckner SPDERF, 2016. Springer New York LLC. 2018. p. 561-571. (Springer Proceedings in Mathematics and Statistics). doi: 10.1007/978-3-319-74929-7_39

Recent progress on the dirichlet forms associated with stochastic quantization problems

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