×

Discretisations of rough stochastic PDEs. (English) Zbl 1406.60094

The aim of this paper is to develop a general framework for spatial discretization of the class of parabolic stochastic partial differential equations (PDEs) of the form \[ \partial_tu=Au+F(u,\xi), \] where \(A\) is an elliptic differential operator, \(\xi\) is a rough noise, and \(F\) is a nonlinear function in \(u\) which is affine in \(\xi\). In the discretization, the spatial variable takes values in the dyadic grid with mesh tending to zero. The solutions of the initial PDE are provided in the framework of the theory of regularity structures, and they are functions in time. A particular example, prototypical for the class of the equations under consideration, is the dynamics model which is called \(\Phi^4\) model in dimension 3, or \(\Phi^4_3\), that is the same. This model is considered on the torus, and operator \(A\) contains Laplace operator. It is proved that the dynamical \(\Phi^4_3\) model, discretized to the dyadic grid, converges after renormalization to its continuous counterpart. The convergence is in probability and in the norm of the functional spaces introduced in the main theorem. This result in particular implies that, as expected, the \(\Phi^4_3\) measure with a sufficiently small coupling constant is invariant for this equation and that the lifetime of its solutions is almost surely infinite for almost every initial condition.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H40 White noise theory
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI arXiv Euclid

References:

[1] Albeverio, S. and Röckner, M. (1991). Stochastic differential equations in infinite dimensions: Solutions via Dirichlet forms. Probab. Theory Related Fields 89 347-386. · Zbl 0725.60055
[2] Bahouri, H., Chemin, J.-Y. and Danchin, R. (2011). Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 343. Springer, Heidelberg. · Zbl 1227.35004
[3] Bertini, L. and Giacomin, G. (1997). Stochastic Burgers and KPZ equations from particle systems. Comm. Math. Phys.183 571-607. · Zbl 0874.60059
[4] Bourgain, J. (1994). Periodic nonlinear Schrödinger equation and invariant measures. Comm. Math. Phys.166 1-26. · Zbl 0822.35126
[5] Brydges, D. C., Fröhlich, J. and Sokal, A. D. (1983). A new proof of the existence and nontriviality of the continuum \(φ^{4}_{2}\) and \(φ^{4}_{3}\) quantum field theories. Comm. Math. Phys.91 141-186.
[6] Catellier, R. and Chouk, K. (2013). Paracontrolled distributions and the 3-dimensional stochastic quantization equation. · Zbl 1433.60048
[7] Daubechies, I. (1988). Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math.41 909-996. · Zbl 0644.42026
[8] Daubechies, I. (1992). Ten Lectures on Wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics 61. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA. · Zbl 0776.42018
[9] Da Prato, G. and Debussche, A. (2003). Strong solutions to the stochastic quantization equations. Ann. Probab.31 1900-1916. · Zbl 1071.81070
[10] Da Prato, G. and Zabczyk, J. (2014). Stochastic Equations in Infinite Dimensions, 2nd ed. Encyclopedia of Mathematics and Its Applications 152. Cambridge Univ. Press, Cambridge. · Zbl 1317.60077
[11] Feldman, J. (1974). The \(λφ^{4}_{3}\) field theory in a finite volume. Comm. Math. Phys.37 93-120.
[12] Giacomin, G., Lebowitz, J. L. and Presutti, E. (1999). Deterministic and stochastic hydrodynamic equations arising from simple microscopic model systems. In Stochastic Partial Differential Equations: Six Perspectives. Math. Surveys Monogr.64 107-152. Amer. Math. Soc., Providence, RI. · Zbl 0927.60060
[13] Gubinelli, M. (2004). Controlling rough paths. J. Funct. Anal.216 86-140. · Zbl 1058.60037
[14] Gubinelli, M., Imkeller, P. and Perkowski, N. (2015). Paracontrolled distributions and singular PDEs. Forum Math. Pi 3 e6, 75. · Zbl 1333.60149
[15] Gubinelli, M. and Perkowski, N. (2017). KPZ reloaded. Comm. Math. Phys.349 165-269. · Zbl 1388.60110
[16] Guerra, F., Rosen, L. and Simon, B. (1975). The \(\mathbf{P}(ϕ)_{2}\) Euclidean quantum field theory as classical statistical mechanics. I, II. Ann. of Math. (2) 101 111-189; ibid. (2) 101 (1975), 191-259.
[17] Hairer, M. (2011). Rough stochastic PDEs. Comm. Pure Appl. Math.64 1547-1585. · Zbl 1229.60079
[18] Hairer, M. (2013). Solving the KPZ equation. Ann. of Math. (2) 178 559-664. · Zbl 1281.60060
[19] Hairer, M. (2014). A theory of regularity structures. Invent. Math.198 269-504. · Zbl 1332.60093
[20] Hairer, M. (2016). Regularity structures and the dynamical \(Φ^{4}_{3}\) model. In Current Developments in Mathematics 2014 1-49. Int. Press, Somerville, MA.
[21] Hairer, M. and Labbé, C. (2015). Multiplicative stochastic heat equations on the whole space.
[22] Hairer, M., Maas, J. and Weber, H. (2014). Approximating rough stochastic PDEs. Comm. Pure Appl. Math.67 776-870. · Zbl 1302.60095
[23] Hairer, M. and Matetski, K. (2016). Optimal rate of convergence for stochastic Burgers-type equations. Stoch. Partial Differ. Equ., Anal. Computat.4 402-437. · Zbl 1357.60069
[24] Has’minskiĭ, R. Z. (1980). Stochastic Stability of Differential Equations. Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis 7. Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md.. Translated from the Russian by D. Louvish.
[25] Hörmander, L. (1955). On the theory of general partial differential operators. Acta Math.94 161-248.
[26] Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes, 2nd ed. North-Holland Mathematical Library 24. North-Holland, Amsterdam; Kodansha, Ltd., Tokyo. · Zbl 0684.60040
[27] Jona-Lasinio, G. and Mitter, P. K. (1985). On the stochastic quantization of field theory. Comm. Math. Phys.101 409-436. · Zbl 0588.60054
[28] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York. · Zbl 0996.60001
[29] Kardar, M., Parisi, G. and Zhang, Y.-C. (1986). Dynamic scaling of growing interfaces. Phys. Rev. Lett.56 889-892. · Zbl 1101.82329
[30] Kupiainen, A. (2016). Renormalization group and stochastic PDEs. Ann. Henri Poincaré17 497-535. · Zbl 1347.81063
[31] Lui, S. H. (2011). Numerical Analysis of Partial Differential Equations. Wiley, Hoboken, NJ. · Zbl 1239.65001
[32] Lyons, T. J. (1998). Differential equations driven by rough signals. Rev. Mat. Iberoam.14 215-310. · Zbl 0923.34056
[33] Meyer, Y. (1992). Wavelets and Operators. Cambridge Studies in Advanced Mathematics 37. Cambridge Univ. Press, Cambridge. Translated from the 1990 French original by D. H. Salinger.
[34] Mourrat, J.-C. and Weber, H. (2017). Convergence of the two-dimensional dynamic Ising-Kac model to \(Φ^{4}_{2}\). Comm. Pure Appl. Math.70 717-812. · Zbl 1364.82013
[35] Nelson, E. (1973). The free Markoff field. J. Funct. Anal.12 211-227. · Zbl 0273.60079
[36] Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd ed. Springer, Berlin. · Zbl 1099.60003
[37] Park, Y. M. (1975). Lattice approximation of the \((λϕ^{4}-μϕ)_{3}\) field theory in a finite volume. J. Math. Phys.16 1065-1075.
[38] Park, Y. M. (1977). Convergence of lattice approximations and infinite volume limit in the \((λϕ^{4}-σϕ^{2}-τϕ)_{3}\) field theory. J. Math. Phys.18 354-366.
[39] Simon, B. and Griffiths, R. B. (1973). The \((ϕ^{4})_{2}\) field theory as a classical Ising model. Comm. Math. Phys.33 145-164.
[40] Zhu, R. and Zhu, X. (2018). Lattice approximation to the dynamical \(Φ_{3}^{4}\) model. Ann. Probab.46 397-455. · Zbl 1393.82014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.