The continuum directed polymer in Lévy noise

Berger, Quentin; Lacoin, Hubert

Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 213-280.

Nous présentons dans cet article la construction d’un modèle de polymère dirigé continu dans un environnement donné par un bruit de Lévy en espace-temps. L’un des principaux objectifs de cette construction est de décrire la limite d’échelle d’un polymère dirigé discret dans un environnement à queue lourde et, pour cette raison, nous mettons plus particulièrement en avant le cas de bruits $\alpha$-stables avec $\alpha \in (1,2)$. Notre construction peut être réalisée en n’importe quelle dimension, pourvu que la mesure de Lévy satisfasse certaines conditions (qui dépendent de la dimension). Nous examinons aussi quelques propriétés de base du polymère continu, ainsi que la relation entre ce modèle et l’équation de la chaleur stochastique avec bruit multiplicatif de Lévy.

We present in this paper the construction of a continuum directed polymer model in an environment given by space-time Lévy noise. One of the main objectives of this construction is to describe the scaling limit of a discrete directed polymer in a heavy-tail environment and for this reason we put special emphasis on the case of $\alpha$-stable noises with $\alpha \in (1,2)$. Our construction can be performed in arbitrary dimension, provided that the Lévy measure satisfies specific (and dimension dependent) conditions. We also discuss a few basic properties of the continuum polymer and the relation between this model and the stochastic heat equation with multiplicative Lévy noise.

Classification : 82B44, 60G57, 60K35, 60H15
Keywords: Disordered polymer model, scaling limit, Lévy noise, stochastic heat equation
Mot clés : Modèle de polymère désordonné, limite d’échelle, bruit de Lévy, équation de la chaleur stochastique

[1] Alberts, Tom; Khanin, Konstantin; Quastel, Jeremy Intermediate disorder regime for directed polymers in dimension $1+1$, Phys. Rev. Lett., Volume 105 (2010) no. 9, 090603 | DOI

[2] Alberts, Tom; Khanin, Konstantin; Quastel, Jeremy The continuum directed random polymer, J. Statist. Phys., Volume 154 (2014) no. 1-2, pp. 305-326 | DOI | MR | Zbl

[3] Alexander, Kenneth; Yıldırım, Gökhan Directed polymers in a random environment with a defect line, Electron. J. Probab., Volume 20 (2015), 6, 20 pages | DOI | MR | Zbl

[4] Auffinger, Antonio; Louidor, Oren Directed polymers in a random environment with heavy tails, Comm. Pure Appl. Math., Volume 64 (2011) no. 2, pp. 183-204 | DOI | MR | Zbl

[5] Berestycki, Nathanaël An elementary approach to Gaussian multiplicative chaos, Electron. Comm. Probab., Volume 22 (2017), 27, 12 pages | DOI | MR | Zbl

[6] Berger, Quentin; Chong, Carsten; Lacoin, Hubert The stochastic heat equation with multiplicative Lévy noise: Existence, Moments, and Intermittency, 2021 | arXiv

[7] Berger, Quentin; Lacoin, Hubert The high-temperature behavior for the directed polymer in dimension $1+2$, Ann. Inst. H. Poincaré Probab. Statist., Volume 53 (2017) no. 1, pp. 430-450 | DOI | MR | Zbl

[8] Berger, Quentin; Lacoin, Hubert Pinning on a defect line: characterization of marginal relevance and sharp asymptotics for the critical point shift, J. Inst. Math. Jussieu, Volume 17 (2018) no. 2, pp. 305-346 | DOI | MR | Zbl

[9] Berger, Quentin; Lacoin, Hubert The scaling limit of the directed polymer with power-law tail disorder, Comm. Math. Phys., Volume 386 (2021) no. 2, pp. 1051-1105 | DOI | MR | Zbl

[10] Berger, Quentin; Torri, Niccolò Directed polymers in heavy-tail random environment, Ann. Probab., Volume 47 (2019) no. 6, pp. 4024-4076 | DOI | MR | Zbl

[11] Bertini, Lorenzo; Cancrini, Nicoletta The stochastic heat equation: Feynman-Kac formula and intermittence, J. Statist. Phys., Volume 78 (1995) no. 5-6, pp. 1377-1401 | DOI | MR | Zbl

[12] Birkner, Matthias A condition for weak disorder for directed polymers in random environment, Electron. Comm. Probab., Volume 9 (2004), pp. 22-25 | DOI | MR | Zbl

[13] Bolthausen, Erwin A note on the diffusion of directed polymers in a random environment, Comm. Math. Phys., Volume 123 (1989) no. 4, pp. 529-534 http://projecteuclid.org/euclid.cmp/1104178982 | DOI | MR | Zbl

[14] Bolthausen, Erwin; den Hollander, Frank Localization transition for a polymer near an interface, Ann. Probab., Volume 25 (1997) no. 3, pp. 1334-1366 | DOI | MR | Zbl

[15] Bovier, Anton Statistical mechanics of disordered systems: a mathematical perspective, Camb. Ser. Stat. Probab. Math., 18, Cambridge University Press, Cambridge, 2006 | DOI

[16] Caravenna, Francesco; Giacomin, Giambattista The weak coupling limit of disordered copolymer models, Ann. Probab., Volume 38 (2010) no. 6, pp. 2322-2378 | DOI | MR | Zbl

[17] Caravenna, Francesco; Sun, Rongfeng; Zygouras, Nikos The continuum disordered pinning model, Probab. Theory Relat. Fields, Volume 164 (2016) no. 1-2, pp. 17-59 | DOI | MR | Zbl

[18] Caravenna, Francesco; Sun, Rongfeng; Zygouras, Nikos Scaling limits of disordered systems and disorder relevance, 2016 to appear in Proceedings for XVIII International Congress on Math. Physics (Santiago de Chile, 2015) | arXiv

[19] Caravenna, Francesco; Sun, Rongfeng; Zygouras, Nikos Polynomial chaos and scaling limits of disordered systems, J. Eur. Math. Soc. (JEMS), Volume 19 (2017) no. 1, pp. 1-65 | DOI | MR | Zbl

[20] Caravenna, Francesco; Sun, Rongfeng; Zygouras, Nikos Universality in marginally relevant disordered systems, Ann. Appl. Probab., Volume 27 (2017) no. 5, pp. 3050-3112 | DOI | MR | Zbl

[21] Caravenna, Francesco; Sun, Rongfeng; Zygouras, Nikos On the moments of the $(2+1)$-dimensional directed polymer and stochastic heat equation in the critical window, Comm. Math. Phys., Volume 372 (2019) no. 2, pp. 385-440 | DOI | MR | Zbl

[22] Caravenna, Francesco; Sun, Rongfeng; Zygouras, Nikos The critical 2d stochastic heat flow, 2021 | arXiv

[23] Chong, Carsten Stochastic PDEs with heavy-tailed noise, Stochastic Process. Appl., Volume 127 (2017) no. 7, pp. 2262-2280 | DOI | MR | Zbl

[24] Clark, Jeremy Weak-disorder limit at criticality for directed polymers on hierarchical graphs, Comm. Math. Phys., Volume 386 (2021) no. 2, pp. 651-710 | DOI | MR | Zbl

[25] Comets, Francis Weak disorder for low dimensional polymers: the model of stable laws, Markov Process. Related Fields, Volume 13 (2007) no. 4, pp. 681-696 | MR | Zbl

[26] Comets, Francis Directed polymers in random environments, Lect. Notes in Math., 2175, Springer, Cham, 2017 École d’été de probabilités (Saint-Flour, 2016) | DOI

[27] Comets, Francis; Shiga, Tokuzo; Yoshida, Nobuo Directed Polymers in a random environment: strong disorder and path localization, Bernoulli, Volume 9 (2003) no. 4, pp. 705-723 | DOI | Zbl

[28] Comets, Francis; Vargas, Vincent Majorizing multiplicative cascades for directed polymers in random media, ALEA Lat. Am. J. Probab. Math. Stat., Volume 2 (2006), pp. 267-277 | MR | Zbl

[29] Comets, Francis; Yoshida, Nobuo Directed polymers in random environment are diffusive at weak disorder, Ann. Probab., Volume 34 (2006) no. 5, pp. 1746-1770 | DOI | MR | Zbl

[30] Dey, Partha S.; Zygouras, Nikos High temperature limits for $(1+1)$-dimensional directed polymer with heavy-tailed disorder, Ann. Probab., Volume 44 (2016) no. 6, pp. 4006-4048 | DOI | MR | Zbl

[31] Fageot, Julien; Unser, Michael; Ward, John Paul On the Besov regularity of periodic Lévy noises, Appl. Comput. Harmon. Anal., Volume 42 (2017) no. 1, pp. 21 -36 | DOI | Zbl

[32] Giacomin, Giambattista Random polymer models, Imperial College Press, London, 2007 | DOI

[33] Giacomin, Giambattista Disorder and critical phenomena through basic probability models, Lect. Notes in Math., 2025, Springer-Verlag, Heidelberg, 2011 École d’été de probabilités (Saint-Flour, 2010) | DOI

[34] Giacomin, Giambattista; Lacoin, Hubert; Toninelli, Fabio Lucio Disorder relevance at marginality and critical point shift, Ann. Inst. H. Poincaré Probab. Statist., Volume 47 (2011) no. 1, pp. 148-175 | DOI | Numdam | MR | Zbl

[35] Gu, Yu; Quastel, Jeremy; Tsai, Li-Cheng Moments of the 2D SHE at criticality, Probab. Math. Phys., Volume 2 (2021) no. 1, pp. 179-219 | DOI

[36] Huse, David A.; Henley, Christopher L. Pinning and roughening of domain walls in Ising systems due to random impurities, Phys. Rev. Lett., Volume 54 (1985) no. 25, pp. 2708-2711 | DOI

[37] Imbrie, John Z.; Spencer, Thomas Diffusion of directed polymers in a random environment, J. Statist. Phys., Volume 52 (1988) no. 3-4, pp. 609-626 | DOI | MR | Zbl

[38] Kalinauskaitė, N. Certain expansions of multidimensional symmetric stable densities with parameter $a>1$, Litovsk. Mat. Sb., Volume 10 (1970) no. 3, pp. 491-495 (in Russian) | DOI | MR | Zbl

[39] Lacoin, Hubert New bounds for the free energy of directed polymer in dimension $1+1$ and $1+2$, Comm. Math. Phys., Volume 294 (2010) no. 2, pp. 471-503 | DOI | MR | Zbl

[40] Lacoin, Hubert; Sohier, Julien Disorder relevance without Harris criterion: the case of pinning model with $\gamma$-stable environment, Electron. J. Probab., Volume 22 (2017), 50, 26 pages | DOI | MR | Zbl

[41] Last, Günter; Penrose, Mathew Lectures on the Poisson process, Institute of Math. Stat. Textbooks, 7, Cambridge University Press, Cambridge, 2018

[42] Levin, David A.; Peres, Yuval Markov chains and mixing times: second edition, American Mathematical Society, Providence, RI, 2017 | DOI

[43] Mueller, Carl The heat equation with Lévy noise, Stochastic Process. Appl., Volume 74 (1998) no. 1, pp. 67-82 | DOI | MR | Zbl

[44] Nakashima, Makoto Free energy of directed polymers in random environment in $1+1$ dimension at high temperature, Electron. J. Probab., Volume 24 (2019), 50, 43 pages | DOI | MR | Zbl

[45] Saint Loubert Bié, Erwan Étude d’une EDPS conduite par un bruit poissonnien, Probab. Theory Relat. Fields, Volume 111 (1998) no. 2, pp. 287-321 | DOI | MR | Zbl

[46] Samorodnitsky, Gennady; Taqqu, Murad Stable non-Gaussian random processes: stochastic models with infinite variance, Chapman & Hall, New York, NY, 1994

[47] Shi, Zhan Branching random walks, Lect. Notes in Math., 2151, Springer, Cham, 2015 École d’été de probabilités (Saint-Flour, 2012) | DOI

[48] Viveros, Roberto Directed polymer in $\gamma$-stable random environments, Ann. Inst. H. Poincaré Probab. Statist., Volume 57 (2021) no. 2, pp. 1081-1102 | DOI | MR | Zbl

[49] Wei, Ran On the long-range directed polymer model, J. Statist. Phys., Volume 165 (2016) no. 2, pp. 320-350 | DOI | MR | Zbl