Geometric stochastic heat equations

Bruned, Y.; Gabriel, F.; Hairer, M.; Zambotti, L.

doi:10.1090/jams/977

Geometric stochastic heat equations
HTML articles powered by AMS MathViewer

by Y. Bruned, F. Gabriel, M. Hairer and L. Zambotti

J. Amer. Math. Soc. 35 (2022), 1-80

DOI: https://doi.org/10.1090/jams/977

Published electronically: April 30, 2021

PDF

Abstract:

We consider a natural class of ${\mathbf {R}}^d$-valued one-dimensional stochastic partial differential equations (PDEs) driven by space-time white noise that is formally invariant under the action of the diffeomorphism group on ${\mathbf {R}}^d$. This class contains in particular the Kardar-Parisi-Zhang (KPZ) equation, the multiplicative stochastic heat equation, the additive stochastic heat equation, and rough Burgers-type equations. We exhibit a one-parameter family of solution theories with the following properties:

For all stochastic partial differential equations (SPDEs) in our class for which a solution was previously available, every solution in our family coincides with the previously constructed solution, whether that was obtained using Itô calculus (additive and multiplicative stochastic heat equation), rough path theory (rough Burgers-type equations), or the Hopf-Cole transform (KPZ equation).
Every solution theory is equivariant under the action of the diffeomorphism group, i.e. identities obtained by formal calculations treating the noise as a smooth function are valid.
Every solution theory satisfies an analogue of Itô’s isometry.
The counterterms leading to our solution theories vanish at points where the equation agrees to leading order with the additive stochastic heat equation.

In particular, points (2) and (3) show that, surprisingly, our solution theories enjoy properties analogous to those holding for both the Stratonovich and Itô interpretations of stochastic differential equations (SDEs) simultaneously. For the natural noisy perturbation of the harmonic map flow with values in an arbitrary Riemannian manifold, we show that all these solution theories coincide. In particular, this allows us to conjecturally identify the process associated to the Markov extension of the Dirichlet form corresponding to the $L^2$-gradient flow for the Brownian loop measure.

References

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
Y. Bruned, A. Chandra, I. Chevyrev, and M. Hairer, Renormalising SPDEs in regularity structures, J. Eur. Math. Soc. 23 (2021), no. 3, 869–947. arXiv:1711.10239. https://dx.doi.org/10.4171/JEMS/1025.
Y. Bruned, M. Hairer, and L. Zambotti, Algebraic renormalisation of regularity structures, Invent. Math. 215 (2019), no. 3, 1039–1156. arXiv:1610.08468. https://dx.doi.org/10.1007/s00222-018-0841-x.
V. I. Bogachev, Measure theory. Vol. I, II, Springer-Verlag, Berlin, 2007. MR 2267655, DOI 10.1007/978-3-540-34514-5
G. Cébron, A. Dahlqvist, and C. Male, Traffic distributions and independence ii: universal constructions for traffic spaces, 2020. arXiv:1601.00168.
K. Chouk, J. Gairing, and N. Perkowski, An invariance principle for the two-dimensional parabolic Anderson model with small potential, Stoch. Partial Differ. Equ. Anal. Comput. 5 (2017), no. 4, 520–558. arXiv:1609.02471. https://dx.doi.org/10.1007/s40072-017-0096-3.
A. Chandra and M. Hairer, An analytic BPHZ theorem for regularity structures, ArXiv e-prints, 2016. arXiv:1612.08138v5.
K. Cheng, Quantization of a general dynamical system by Feynman’s path integration formulation, J. Math. Phys. 13 (1972), no. 11, 1723–1726. https://dx.doi.org/10.1063/1.1665897.
F. Chapoton and M. Livernet, Pre-Lie algebras and the rooted trees operad, Int. Math. Res. Not. 2001 (2001), no. 8, 395–408. https://dx.doi.org/10.1155/S1073792801000198.
R. W. R. Darling, On the convergence of Gangolli processes to Brownian motion on a manifold, Stochastics 12 (1984), no. 3–4, 277–301. https://dx.doi.org/10.1080/17442508408833305.
H. Dekker, On the path integral for diffusion in curved spaces, Phys. A 103 (1980), no. 3, 586–596. MR 593102, DOI 10.1016/0378-4371(80)90027-8
Bryce S. Dewitt, Dynamical theory in curved spaces. I. A review of the classical and quantum action principles, Rev. Mod. Phys. 29 (1957), 377–397. MR 0095691, DOI 10.1103/revmodphys.29.377
Bryce DeWitt, Supermanifolds, 2nd ed., Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1992. MR 1172996, DOI 10.1017/CBO9780511564000
G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications, vol. 152, 2nd ed., Cambridge University Press, Cambridge, 2014, xviii+493. https://dx.doi.org/10.1017/CBO9781107295513.
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
D. Erhard and M. Hairer, Discretisation of regularity structures, Ann. Inst. Henri Poincaré Probab. Stat. 55 (2019), no. 4, 2209–2248. arXiv:1705.02836. https://dx.doi.org/10.1214/18-AIHP947.
J. Eells, Jr., and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109–160. https://dx.doi.org/10.2307/2373037.
Ognian Enchev and Daniel W. Stroock, Pinned Brownian motion and its perturbations, Adv. Math. 119 (1996), no. 2, 127–154. MR 1390796, DOI 10.1006/aima.1996.0029
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
R. Graham, Path integral formulation of general diffusion processes, Z. Phys. B 26 (1977), no. 3, 281–290. https://dx.doi.org/10.1007/BF01312935.
M. Hairer, Rough stochastic PDEs, Comm. Pure Appl. Math. 64 (2011), no. 11, 1547–1585. arXiv:1008.1708. https://dx.doi.org/10.1002/cpa.20383.
M. Hairer, Solving the KPZ equation, Ann. Math. (2) 178 (2013), no. 2, 559–664. arXiv:1109.6811. https://dx.doi.org/10.4007/annals.2013.178.2.4.
M. Hairer, A theory of regularity structures, Invent. Math. 198 (2014), no. 2, 269–504. arXiv:1303.5113. https://dx.doi.org/10.1007/s00222-014-0505-4.
M. Hairer, The motion of a random string, ArXiv e-prints, 2016, Proceedings of the XVIII ICMP. arXiv:1605.02192.
M. Hairer and K. Matetski, Discretisations of rough stochastic PDEs, Ann. Probab. 46 (2018), no. 3, 1651–1709. arXiv:1511.06937. https://dx.doi.org/10.1214/17-AOP1212.
M. Hairer and J. Mattingly, The strong Feller property for singular stochastic PDEs, Ann. Inst. Henri Poincaré Probab. Stat. 54 (2018), no. 3, 1314–1340. arXiv:1610.03415. https://dx.doi.org/10.1214/17-AIHP840.
M. Hairer and É. Pardoux, A Wong-Zakai theorem for stochastic PDEs, J. Math. Soc. Japan 67 (2015), no. 4, 1551–1604. arXiv:1409.3138. https://dx.doi.org/10.2969/jmsj/06741551.
M. Hairer and J. Quastel, A class of growth models rescaling to KPZ, Forum Math. Pi 6 (2018), e3. arXiv:1512.07845. https://dx.doi.org/10.1017/fmp.2018.2.
M. Hairer and H. Shen, A central limit theorem for the KPZ equation, Ann. Probab. 45 (2017), no. 6B, 4167–4221. arXiv:1507.01237. https://dx.doi.org/10.1214/16-AOP1162.
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
A. Inoue and Y. Maeda, On integral transformations associated with a certain Lagrangian—as a prototype of quantization, J. Math. Soc. Japan 37 (1985), no. 2, 219–244. https://dx.doi.org/10.2969/jmsj/03720219.
André Joyal and Ross Street, The geometry of tensor calculus. I, Adv. Math. 88 (1991), no. 1, 55–112. MR 1113284, DOI 10.1016/0001-8708(91)90003-P
P. Li and L.-F. Tam, The heat equation and harmonic maps of complete manifolds, Invent. Math. 105 (1991), no. 1, 1–46. https://dx.doi.org/10.1007/BF01232256.
M. Markl, S. Merkulov, and S. Shadrin, Wheeled PROPs, graph complexes and the master equation, J. Pure Appl. Algebra 213 (2009), no. 4, 496–535. arXiv:math/0610683. https://dx.doi.org/10.1016/j.jpaa.2008.08.007.
J. Nash, The imbedding problem for Riemannian manifolds, Ann. of Math. (2) 63 (1956), 20–63. https://dx.doi.org/10.2307/1969989.
J. R. Norris, Ornstein-Uhlenbeck processes indexed by the circle, Ann. Probab. 26 (1998), no. 2, 465–478. https://dx.doi.org/10.1214/aop/1022855640.
Roger Penrose, Applications of negative dimensional tensors, Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) Academic Press, London, 1971, pp. 221–244. MR 0281657
Peter Petersen, Riemannian geometry, 2nd ed., Graduate Texts in Mathematics, vol. 171, Springer, New York, 2006. MR 2243772
M. Rockner, B. Wu, R. Zhu, and X. Zhu, Stochastic heat equations with values in a manifold via Dirichlet forms, SIAM J. Math. Anal. 52 (2020), no. 3, 2237–2274. arXiv:1711.09570. https://dx.doi.org/10.1137/18M1211076.
D. W. Stroock and S. R. S. Varadhan, On the support of diffusion processes with applications to the strong maximum principle, Proc. Sixth Berkeley Symp. on Math. Statist. and Prob., vol. 3, University of California Press, Berkeley, Calif., 1972, pp. 333–359. https://dx.doi.org/euclid.bsmsp/1200514345.
Y. Takahashi and S. Watanabe, The probability functionals (Onsager-Machlup functions) of diffusion processes, Stochastic Integrals (Proc. Sympos., Univ. Durham, Durham, 1980), Lecture Notes in Math., vol. 851 Springer, Berlin, 1981, pp. 433–463. https://dx.doi.org/10.1007/BFb0088735.
G. S. Um, On normalization problems of the path integral method, J. Math. Phys. 15 (1974), no. 2, 220–224. https://dx.doi.org/10.1063/1.1666626.
Kentaro Yano, The theory of Lie derivatives and its applications, North-Holland Publishing Co., Amsterdam; P. Noordhoff Ltd., Groningen; Interscience Publishers Inc., New York, 1957. MR 0088769