Stochastic heat equations for infinite strings with values in a manifold
HTML articles powered by AMS MathViewer
- by Xin Chen, Bo Wu, Rongchan Zhu and Xiangchan Zhu PDF
- Trans. Amer. Math. Soc. 374 (2021), 407-452 Request permission
Abstract:
In the paper, we construct conservative Markov processes corresponding to the martingale solutions to the stochastic heat equation on $\mathbb {R}^+$ or $\mathbb {R}$ with values in a general Riemannian manifold, which is only assumed to be complete and stochastic complete. This work is an extension of the previous paper of Röckner and the second, third, and fourth authors [SIAM J. Math. Anal. 52 (2020), pp. 2237–2274] on finite volume case.
Moveover, we also obtain some functional inequalities associated to these Markov processes. This implies that on infinite volume case, the exponential ergodicity of the solution of the Ricci curvature is strictly positive and the non-ergodicity of the process if the sectional curvature is negative.
References
- Shigeki Aida, Logarithmic Sobolev inequalities on loop spaces over compact Riemannian manifolds, Stochastic analysis and applications (Powys, 1995) World Sci. Publ., River Edge, NJ, 1996, pp. 1–19. MR 1453122
- Shigeki Aida, Gradient estimates of harmonic functions and the asymptotics of spectral gaps on path spaces, Interdiscip. Inform. Sci. 2 (1996), no. 1, 75–84. MR 1398102, DOI 10.4036/iis.1996.75
- Shigeki Aida and David Elworthy, Differential calculus on path and loop spaces. I. Logarithmic Sobolev inequalities on path spaces, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 1, 97–102 (English, with English and French summaries). MR 1340091
- Sergio Albeverio, Rémi Léandre, and Michael Röckner, Construction of a rotational invariant diffusion on the free loop space, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), no. 3, 287–292 (English, with English and French summaries). MR 1205201
- S. Albeverio and M. Röckner, Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms, Probab. Theory Related Fields 89 (1991), no. 3, 347–386. MR 1113223, DOI 10.1007/BF01198791
- Lars Andersson and Bruce K. Driver, Finite-dimensional approximations to Wiener measure and path integral formulas on manifolds, J. Funct. Anal. 165 (1999), no. 2, 430–498. MR 1698956, DOI 10.1006/jfan.1999.3413
- Marc Arnaudon, Anton Thalmaier, and Feng-Yu Wang, Harnack inequality and heat kernel estimates on manifolds with curvature unbounded below, Bull. Sci. Math. 130 (2006), no. 3, 223–233. MR 2215664, DOI 10.1016/j.bulsci.2005.10.001
- Dennis Barden and Huiling Le, Some consequences of the nature of the distance function on the cut locus in a Riemannian manifold, J. London Math. Soc. (2) 56 (1997), no. 2, 369–383. MR 1489143, DOI 10.1112/S002461079700553X
- Y. Bruned, F. Gabriel, M. Hairer, and L. Zambotti, Geometric stochastic heat equations, arXiv: 1902.02884.
- Y. Bruned, M. Hairer, and L. Zambotti, Algebraic renormalisation of regularity structures, Invent. Math. 215 (2019), no. 3, 1039–1156. MR 3935036, DOI 10.1007/s00222-018-0841-x
- Zhen-Qing Chen and Masatoshi Fukushima, Symmetric Markov processes, time change, and boundary theory, London Mathematical Society Monographs Series, vol. 35, Princeton University Press, Princeton, NJ, 2012. MR 2849840
- Mireille Capitaine, Elton P. Hsu, and Michel Ledoux, Martingale representation and a simple proof of logarithmic Sobolev inequalities on path spaces, Electron. Comm. Probab. 2 (1997), 71–81. MR 1484557, DOI 10.1214/ECP.v2-986
- A. Chandra and M. Hairer, An analytic BPHZ theorem for regularity structures arXiv:1612.08138, pages 1-113, 2016.
- Isaac Chavel, Riemannian geometry, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 98, Cambridge University Press, Cambridge, 2006. A modern introduction. MR 2229062, DOI 10.1017/CBO9780511616822
- Xin Chen, Xue-Mei Li, and Bo Wu, A concrete estimate for the weak Poincaré inequality on loop space, Probab. Theory Related Fields 151 (2011), no. 3-4, 559–590. MR 2851693, DOI 10.1007/s00440-010-0308-5
- X. Chen, X.-M. Li, and B. Wu, Small time gradient and Hessian estimates for logarithmic heat kernel on a general complete manifold, Preprint.
- X. Chen, X.-M. Li, and B. Wu, Stochastic analysis on loop space over general Riemannian manifold, Preprint.
- X. Chen, X.-M. Li, and B. Wu, Analysis on Free Riemannian Loop Space, Preprint.
- Xin Chen and Bo Wu, Functional inequality on path space over a non-compact Riemannian manifold, J. Funct. Anal. 266 (2014), no. 12, 6753–6779. MR 3198853, DOI 10.1016/j.jfa.2014.03.017
- Kai Lai Chung and Zhong Xin Zhao, From Brownian motion to Schrödinger’s equation, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 312, Springer-Verlag, Berlin, 1995. MR 1329992, DOI 10.1007/978-3-642-57856-4
- E. B. Davies, Heat kernel bounds, conservation of probability and the Feller property, J. Anal. Math. 58 (1992), 99–119. Festschrift on the occasion of the 70th birthday of Shmuel Agmon. MR 1226938, DOI 10.1007/BF02790359
- Bruce K. Driver and Michael Röckner, Construction of diffusions on path and loop spaces of compact Riemannian manifolds, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), no. 5, 603–608 (English, with English and French summaries). MR 1181300
- Bruce K. Driver, A Cameron-Martin type quasi-invariance theorem for Brownian motion on a compact Riemannian manifold, J. Funct. Anal. 110 (1992), no. 2, 272–376. MR 1194990, DOI 10.1016/0022-1236(92)90035-H
- Bruce K. Driver, A Cameron-Martin type quasi-invariance theorem for pinned Brownian motion on a compact Riemannian manifold, Trans. Amer. Math. Soc. 342 (1994), no. 1, 375–395. MR 1154540, DOI 10.1090/S0002-9947-1994-1154540-2
- K. D. Elworthy and Xue-Mei Li, A class of integration by parts formulae in stochastic analysis. I, Itô’s stochastic calculus and probability theory, Springer, Tokyo, 1996, pp. 15–30. MR 1439515
- K. D. Elworthy, Y. Le Jan, and Xue-Mei Li, On the geometry of diffusion operators and stochastic flows, Lecture Notes in Mathematics, vol. 1720, Springer-Verlag, Berlin, 1999. MR 1735806, DOI 10.1007/BFb0103064
- K. D. Elworthy, X.-M. Li, and Steven Rosenberg, Curvature and topology: spectral positivity, Methods and applications of global analysis, Novoe Global. Anal., Voronezh. Univ. Press, Voronezh, 1993, pp. 45–60, 156 (English, with Russian summary). MR 1278304
- Shizan Fang and Paul Malliavin, Stochastic analysis on the path space of a Riemannian manifold. I. Markovian stochastic calculus, J. Funct. Anal. 118 (1993), no. 1, 249–274. MR 1245604, DOI 10.1006/jfan.1993.1145
- Shizan Fang and Feng-Yu Wang, Analysis on free Riemannian path spaces, Bull. Sci. Math. 129 (2005), no. 4, 339–355. MR 2134125, DOI 10.1016/j.bulsci.2004.11.003
- Shizan Fang, Feng-Yu Wang, and Bo Wu, Transportation-cost inequality on path spaces with uniform distance, Stochastic Process. Appl. 118 (2008), no. 12, 2181–2197. MR 2474347, DOI 10.1016/j.spa.2008.01.004
- Shizan Fang and Bo Wu, Remarks on spectral gaps on the Riemannian path space, Electron. Commun. Probab. 22 (2017), Paper No. 19, 13. MR 3627008, DOI 10.1214/17-ECP51
- Tadahisa Funaki, On diffusive motion of closed curves, Probability theory and mathematical statistics (Kyoto, 1986) Lecture Notes in Math., vol. 1299, Springer, Berlin, 1988, pp. 86–94. MR 935980, DOI 10.1007/BFb0078464
- Tadahisa Funaki, A stochastic partial differential equation with values in a manifold, J. Funct. Anal. 109 (1992), no. 2, 257–288. MR 1186323, DOI 10.1016/0022-1236(92)90019-F
- Tadahisa Funaki and Masato Hoshino, A coupled KPZ equation, its two types of approximations and existence of global solutions, J. Funct. Anal. 273 (2017), no. 3, 1165–1204. MR 3653951, DOI 10.1016/j.jfa.2017.05.002
- Tadahisa Funaki and Jeremy Quastel, KPZ equation, its renormalization and invariant measures, Stoch. Partial Differ. Equ. Anal. Comput. 3 (2015), no. 2, 159–220. MR 3350451, DOI 10.1007/s40072-015-0046-x
- Tadahisa Funaki and Bin Xie, A stochastic heat equation with the distributions of Lévy processes as its invariant measures, Stochastic Process. Appl. 119 (2009), no. 2, 307–326. MR 2493992, DOI 10.1016/j.spa.2008.02.007
- Masatoshi Fukushima, Y\B{o}ichi Ōshima, and Masayoshi Takeda, Dirichlet forms and symmetric Markov processes, De Gruyter Studies in Mathematics, vol. 19, Walter de Gruyter & Co., Berlin, 1994. MR 1303354, DOI 10.1515/9783110889741
- Shizan Fang and Feng-Yu Wang, Analysis on free Riemannian path spaces, Bull. Sci. Math. 129 (2005), no. 4, 339–355. MR 2134125, DOI 10.1016/j.bulsci.2004.11.003
- R. E. Greene and H. Wu, Function theory on manifolds which possess a pole, Lecture Notes in Mathematics, vol. 699, Springer, Berlin, 1979. MR 521983, DOI 10.1007/BFb0063413
- Mathieu Gourcy and Liming Wu, Logarithmic Sobolev inequalities of diffusions for the $L^2$ metric, Potential Anal. 25 (2006), no. 1, 77–102. MR 2238937, DOI 10.1007/s11118-006-9009-1
- M. Hairer, A theory of regularity structures, Invent. Math. 198 (2014), no. 2, 269–504. MR 3274562, DOI 10.1007/s00222-014-0505-4
- M. Hairer, The motion of a random string, arXiv:1605.02192, pages 1–20, 2016.
- Elton P. Hsu, Logarithmic Sobolev inequalities on path spaces over Riemannian manifolds, Comm. Math. Phys. 189 (1997), no. 1, 9–16. MR 1478528, DOI 10.1007/s002200050188
- Elton P. Hsu, Integration by parts in loop spaces, Math. Ann. 309 (1997), no. 2, 331–339. MR 1474195, DOI 10.1007/s002080050115
- Elton P. Hsu, Stochastic analysis on manifolds, Graduate Studies in Mathematics, vol. 38, American Mathematical Society, Providence, RI, 2002. MR 1882015, DOI 10.1090/gsm/038
- Atsushi Inoue and Yoshiaki Maeda, On integral transformations associated with a certain Lagrangian—as a prototype of quantization, J. Math. Soc. Japan 37 (1985), no. 2, 219–244. MR 780661, DOI 10.2969/jmsj/03720219
- R. Léandre, Integration by parts formulas and rotationally invariant Sobolev calculus on free loop spaces, J. Geom. Phys. 11 (1993), no. 1-4, 517–528. Infinite-dimensional geometry in physics (Karpacz, 1992). MR 1230447, DOI 10.1016/0393-0440(93)90075-P
- R. Leandre, Invariant Sobolev calculus on the free loop space, Acta Appl. Math. 46 (1997), no. 3, 267–350. MR 1440476, DOI 10.1023/A:1005730013728
- R. Léandre and J. R. Norris, Integration by parts and Cameron-Martin formulas for the free path space of a compact Riemannian manifold, Séminaire de Probabilités, XXXI, Lecture Notes in Math., vol. 1655, Springer, Berlin, 1997, pp. 16–23. MR 1478712, DOI 10.1007/BFb0119288
- Jörg-Uwe Löbus, A class of processes on the path space over a compact Riemannian manifold with unbounded diffusion, Trans. Amer. Math. Soc. 356 (2004), no. 9, 3751–3767. MR 2055753, DOI 10.1090/S0002-9947-04-03439-7
- Zhi Ming Ma and Michael Röckner, Introduction to the theory of (nonsymmetric) Dirichlet forms, Universitext, Springer-Verlag, Berlin, 1992. MR 1214375, DOI 10.1007/978-3-642-77739-4
- Zhi-Ming Ma and Michael Röckner, Construction of diffusions on configuration spaces, Osaka J. Math. 37 (2000), no. 2, 273–314. MR 1772834
- A. Naber, Characterizations of bounded Ricci curvature on smooth and nonsmooth spaces, arXiv: 1306.6512v4.
- J. R. Norris, Ornstein-Uhlenbeck processes indexed by the circle, Ann. Probab. 26 (1998), no. 2, 465–478. MR 1626166, DOI 10.1214/aop/1022855640
- Michael Röckner, Bo Wu, Rongchan Zhu, and Xiangchan Zhu, Stochastic heat equations with values in a manifold via Dirichlet forms, SIAM J. Math. Anal. 52 (2020), no. 3, 2237–2274. MR 4093593, DOI 10.1137/18M1211076
- Daniel W. Stroock, An introduction to the analysis of paths on a Riemannian manifold, Mathematical Surveys and Monographs, vol. 74, American Mathematical Society, Providence, RI, 2000. MR 1715265, DOI 10.1090/surv/074
- Anton Thalmaier, On the differentiation of heat semigroups and Poisson integrals, Stochastics Stochastics Rep. 61 (1997), no. 3-4, 297–321. MR 1488139, DOI 10.1080/17442509708834123
- Anton Thalmaier and Feng-Yu Wang, Gradient estimates for harmonic functions on regular domains in Riemannian manifolds, J. Funct. Anal. 155 (1998), no. 1, 109–124. MR 1622800, DOI 10.1006/jfan.1997.3220
- Fengyu Wang, Spectral gap on path spaces with infinite time-interval, Sci. China Ser. A 42 (1999), no. 6, 600–604. MR 1716986, DOI 10.1007/BF02880078
- Feng-Yu Wang, Weak Poincaré inequalities on path spaces, Int. Math. Res. Not. 2 (2004), 89–108. MR 2040325, DOI 10.1155/S1073792804130882
- F. Y. Wang, Functional Inequalities, Markov Semigroup and Spectral Theory, Chinese Sciences Press, Beijing (2005)
- Feng-Yu Wang and Bo Wu, Quasi-regular Dirichlet forms on Riemannian path and loop spaces, Forum Math. 20 (2008), no. 6, 1084–1096. MR 2479291, DOI 10.1515/FORUM.2008.049
- Feng-Yu Wang and Bo Wu, Quasi-regular Dirichlet forms on free Riemannian path spaces, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 12 (2009), no. 2, 251–267. MR 2541396, DOI 10.1142/S0219025709003628
- Fengyu Wang and Bo Wu, Pointwise characterizations of curvature and second fundamental form on Riemannian manifolds, Sci. China Math. 61 (2018), no. 8, 1407–1420. MR 3833743, DOI 10.1007/s11425-017-9296-8
- B. Wu, Characterizations of the upper bound of Bakry-Emery curvature, to appear in J. Geom. Anal..
Additional Information
- Xin Chen
- Affiliation: School of Mathematical Sciences, Shanghai Jiaotong University, Shanghai 200240, People’s Republic of China
- Email: chenxin217@sjtu.edu.cn
- Bo Wu
- Affiliation: School of Mathematical Sciences, Fudan University, Shanghai 200433, People’s Republic of China
- Email: wubo@fudan.edu.cn
- Rongchan Zhu
- Affiliation: Department of Mathematics, Beijing Institute of Technology, Beijing 100081, People’s Republic of China
- Email: zhurongchan@126.com
- Xiangchan Zhu
- Affiliation: Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
- Email: zhuxiangchan@126.com
- Received by editor(s): July 25, 2019
- Received by editor(s) in revised form: May 4, 2020
- Published electronically: November 2, 2020
- Additional Notes: The third author is the corresponding author.
This research was supported in part by NSFC (11671035, 11771037, 11922103, 12071085, 11871338). Financial support by the DFG through the CRC 1283” Taming uncertainty and profiting from randomness and low regularity in analysis, stochastics and their applications" and support by key Lab of Random Complex Structures and Data Science, Chinese Academy of Science are gratefully acknowledged. - © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 374 (2021), 407-452
- MSC (2020): Primary 37A25, 39B62, 60H15
- DOI: https://doi.org/10.1090/tran/8193
- MathSciNet review: 4188188