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A brief and personal history of stochastic partial differential equations

The author is supported by the grant ANR-15-CE40-0020 - LSD - Large Stochastic Dynamical Models in Non-Equilibrium Statistical Physics (2015)
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  • We trace the evolution of the theory of stochastic partial differential equations from the foundation to its development, until the recent solution of long-standing problems on well-posedness of the KPZ equation and the stochastic quantization in dimension three.

    Mathematics Subject Classification: 60H15.


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  • [1] S. Albeverio and S. Kusuoka, The invariant measure and the flow associated to the $\Phi^4_3$-quantum field model, preprint, 2017, arXiv: 1711.07108.
    [2] S. Albeverio and R. Høegh-Krohn, Dirichlet forms and diffusion processes on rigged Hilbert spaces, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 40 (1977), 1-57.  doi: 10.1007/BF00535706.
    [3] G. AmirI. Corwin and J. Quastel, Probability distribution of the free energy of the continuum directed random polymer in $1+1$ dimensions, Comm. Pure Appl. Math., 64 (2011), 466-537.  doi: 10.1002/cpa.20347.
    [4] M. BalázsJ. Quastel and T. Seppäläinen, Fluctuation exponent of the KPZ/stochastic Burgers equation, J. Amer. Math. Soc., 24 (2011), 683-708.  doi: 10.1090/S0894-0347-2011-00692-9.
    [5] A. Bensoussan and R. Temam, Équations aux dérivées partielles stochastiques non linéaires. I, Israel J. Math., 11 (1972), 95-129.  doi: 10.1007/BF02761449.
    [6] A. Bensoussan and R. Temam, Équations stochastiques du type Navier-Stokes, J. Functional Analysis, 13 (1973), 195-222.  doi: 10.1016/0022-1236(73)90045-1.
    [7] L. Bertini and N. Cancrini, The stochastic heat equation: Feynman-Kac formula and intermittence, J. Statist. Phys., 78 (1995), 1377-1401.  doi: 10.1007/BF02180136.
    [8] L. Bertini and G. Giacomin, Stochastic Burgers and KPZ equations from particle systems, Comm. Math. Phys., 183 (1997), 571-607.  doi: 10.1007/s002200050044.
    [9] J. BricmontA. Kupiainen and R. Lefevere, Ergodicity of the 2D Navier-Stokes equations with random forcing, Comm. Math. Phys., 224 (2001), 65-81.  doi: 10.1007/s002200100510.
    [10] J. BricmontA. Kupiainen and R. Lefevere, Exponential mixing of the 2D stochastic Navier-Stokes dynamics, Comm. Math. Phys., 230 (2002), 87-132.  doi: 10.1007/s00220-002-0708-1.
    [11] Y. Bruned, A. Chandra, I. Chevyrev and M. Hairer, Renormalising SPDEs in regularity structures, to appear in J. Eur. Math. Soc. (JEMS).
    [12] Y. BrunedM. Hairer and L. Zambotti, Algebraic renormalisation of regularity structures, Invent. Math., 215 (2019), 1039-1156.  doi: 10.1007/s00222-018-0841-x.
    [13] E. Cabaña, The vibrating string forced by white noise, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 15 (1970), 111-130.  doi: 10.1007/BF00531880.
    [14] E. Cépa, Problème de Skorohod multivoque, Ann. Probab., 26 (1998), 500-532.  doi: 10.1214/aop/1022855642.
    [15] S. Cerrai, Second Order PDE's in Finite and Infinite Dimension, Lecture Notes in Mathematics, Vol. 1762, Springer-Verlag, Berlin, 2001. doi: 10.1007/b80743.
    [16] A. Chandra and M. Hairer, An analytic BPHZ theorem for Regularity Structures, preprint, 2016, arXiv: 1612.08138.
    [17] K.-T. Chen, Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula, Ann. of Math. (2), 65 (1957), 163-178.  doi: 10.2307/1969671.
    [18] Y.-T. Chen, Pathwise nonuniqueness for the SPDEs of some super-Brownian motions with immigration, Ann. Probab., 43 (2015), 3359-3467.  doi: 10.1214/14-AOP962.
    [19] Y. M. Chen, On scattering of waves by objects imbedded in random media: Stochastic linear partial differential equations and scattering of waves by conducting sphere imbedded in random media, J. Mathematical Phys., 5 (1964), 1541-1546.  doi: 10.1063/1.1931186.
    [20] I. Corwin, Kardar-Parisi-Zhang universality, Notices Amer. Math. Soc., 63 (2016), 230-239.  doi: 10.1090/noti1334.
    [21] G. Da Prato and J. Zabczyk, Ergodicity for Infinite-Dimensional Systems, London Mathematical Society Lecture Note Series, vol. 229, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511662829.
    [22] G. Da Prato and A. Debussche, Strong solutions to the stochastic quantization equations, Ann. Probab., 31 (2003), 1900-1916.  doi: 10.1214/aop/1068646370.
    [23] G. Da Prato, M. Iannelli and L. Tubaro, Stochastic differential equations in Banach spaces, variational formulation, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8), 61 (1976), 168–176 (1977).
    [24] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, Vol. 44, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.
    [25] G. Da Prato and J. Zabczyk, Second Order Partial Differential Equations in Hilbert spaces, London Mathematical Society Lecture Note Series, vol. 293, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511543210.
    [26] R. C. DalangC. Mueller and L. Zambotti, Hitting properties of parabolic s.p.d.e.'s with reflection, Ann. Probab., 34 (2006), 1423-1450.  doi: 10.1214/009117905000000792.
    [27] J. L. Daleckiĭ, Differential equations with functional derivatives and stochastic equations for generalized random processes, Dokl. Akad. Nauk SSSR, 166 (1966), 1035-1038. 
    [28] D. A. Dawson, Stochastic evolution equations, Math. Biosci., 15 (1972), 287-316.  doi: 10.1016/0025-5564(72)90039-9.
    [29] D. A. Dawson, Measure-valued Markov processes, in École d'Été de Probabilités de Saint-Flour XXI-1991 doi: 10.1007/BFb0084190.
    [30] D. A. Dawson and K. J. Hochberg, The carrying dimension of a stochastic measure diffusion, Ann. Probab., 7 (1979), 693–703. http://links.jstor.org/sici?sici=0091-1798(197908)7:4<693:TCDOAS>2.0.CO;2-E&origin=MSN. doi: 10.1214/aop/1176994991.
    [31] H. Doss, Liens entre équations différentielles stochastiques et ordinaires, Ann. Inst. H. Poincaré Sect. B (N.S.), 13 (1977), 99-125. 
    [32] W. EJ. C. Mattingly and Y. Sinai, Gibbsian dynamics and ergodicity for the stochastically forced Navier-Stokes equation, Comm. Math. Phys., 224 (2001), 83-106.  doi: 10.1007/s002201224083.
    [33] K. D. Elworthy and X.-M. Li, Formulae for the derivatives of heat semigroups, J. Funct. Anal., 125 (1994), 252-286.  doi: 10.1006/jfan.1994.1124.
    [34] A. M. Etheridge and C. Labbé, Scaling limits of weakly asymmetric interfaces, Comm. Math. Phys., 336 (2015), 287-336.  doi: 10.1007/s00220-014-2243-2.
    [35] F. FlandoliM. Gubinelli and E. Priola, Well-posedness of the transport equation by stochastic perturbation, Invent. Math., 180 (2010), 1-53.  doi: 10.1007/s00222-009-0224-4.
    [36] F. Flandoli and B. Maslowski, Ergodicity of the 2-D Navier-Stokes equation under random perturbations, Comm. Math. Phys., 172 (1995), 119–141. http://projecteuclid.org/euclid.cmp/1104273961 doi: 10.1007/BF02104513.
    [37] H. Föllmer, Calcul d'Itô sans probabilités, in Seminar on Probability, XV (Univ. Strasbourg, Strasbourg, 1979/1980) (French), Lecture Notes in Math., Vol. 850, Springer, Berlin, 1981,143–150.
    [38] P. K. Friz and M. Hairer, A Course on Rough Paths, Universitext, Springer, Cham, 2014. doi: 10.1007/978-3-319-08332-2.
    [39] M. Fukushima, Dirichlet Forms and Markov Processes, North-Holland Mathematical Library, Vol. 23, North-Holland Mathematical Library, Vol. 23, North-Holland Publishing Co., Amsterdam-New York, Kodansha, Ltd., Tokyo, 1980.
    [40] M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, De Gruyter Studies in Mathematics, Vol. 19, Walter de Gruyter & Co., Berlin, 2011.
    [41] T. Funaki and S. Olla, Fluctuations for $\nabla\phi$ interface model on a wall, Stochastic Process. Appl., 94 (2001), 1-27.  doi: 10.1016/S0304-4149(00)00104-6.
    [42] W. E. Gibson, An exact solution for a class of stochastic partial differential equations, SIAM J. Appl. Math., 15 (1967), 1357-1362.  doi: 10.1137/0115118.
    [43] J. Glimm and A. Jaffe, Quantum Physics, 2nd edition, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4612-4728-9.
    [44] P. Gonçalves and M. Jara, Nonlinear fluctuations of weakly asymmetric interacting particle systems, Arch. Ration. Mech. Anal., 212 (2014), 597-644.  doi: 10.1007/s00205-013-0693-x.
    [45] L. Gross, Potential theory on Hilbert space, J. Functional Analysis, 1 (1967), 123-181.  doi: 10.1016/0022-1236(67)90030-4.
    [46] M. Gubinelli, Controlling rough paths, J. Funct. Anal., 216 (2004), 86-140.  doi: 10.1016/j.jfa.2004.01.002.
    [47] M. Gubinelli and M. Jara, Regularization by noise and stochastic Burgers equations, Stoch. Partial Differ. Equ. Anal. Comput., 1 (2013), 325-350.  doi: 10.1007/s40072-013-0011-5.
    [48] M. Gubinelli, P. Imkeller and N. Perkowski, Paracontrolled distributions and singular PDEs, Forum Math. Pi, 3 (2015), e6, 75 pp. doi: 10.1017/fmp.2015.2.
    [49] M. Gubinelli and N. Perkowski, Energy solutions of KPZ are unique, J. Amer. Math. Soc., 31 (2018), 427-471.  doi: 10.1090/jams/889.
    [50] M. Gubinelli and S. Tindel, Rough evolution equations, Ann. Probab., 38 (2010), 1-75.  doi: 10.1214/08-AOP437.
    [51] M. Hairer, Rough stochastic PDEs, Comm. Pure Appl. Math., 64 (2011), 1547-1585.  doi: 10.1002/cpa.20383.
    [52] M. Hairer, Solving the KPZ equation, Ann. of Math. (2), 178 (2013), 559-664.  doi: 10.4007/annals.2013.178.2.4.
    [53] M. Hairer, A theory of regularity structures, Invent. Math., 198 (2014), 269-504.  doi: 10.1007/s00222-014-0505-4.
    [54] M. Hairer and J. C. Mattingly, Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing, Ann. of Math. (2), 164 (2006), 993-1032.  doi: 10.4007/annals.2006.164.993.
    [55] G. Jona-Lasinio and P. K. Mitter, On the stochastic quantization of field theory, Comm. Math. Phys., 101 (1985), 409-436.  doi: 10.1007/BF01216097.
    [56] M. Kardar, G. Parisi and Y.-C. Zhang, Dynamic scaling of growning interfaces, Phys. Rev. Lett., 56 (1986), 4 pp. doi: 10.1103/PhysRevLett.56.889.
    [57] N. Konno and T. Shiga, Stochastic partial differential equations for some measure-valued diffusions, Probab. Theory Related Fields, 79 (1988), 201-225.  doi: 10.1007/BF00320919.
    [58] N. V. Krylov, A $W^n_2$-theory of the Dirichlet problem for SPDEs in general smooth domains, Probab. Theory Related Fields, 98 (1994), 389-421.  doi: 10.1007/BF01192260.
    [59] N. V. Krylov and B. L. Rozovskiĭ, The Cauchy problem for linear stochastic partial differential equations, Izv. Akad. Nauk SSSR Ser. Mat., 41 (1977), 1329–1347, 1448.
    [60] S. Kuksin and A. Shirikyan, Stochastic dissipative PDEs and Gibbs measures, Comm. Math. Phys., 213 (2000), 291-330.  doi: 10.1007/s002200000237.
    [61] S. Kuksin and A. Shirikyan, Ergodicity for the randomly forced 2D Navier-Stokes equations, Math. Phys. Anal. Geom., 4 (2001), 147-195.  doi: 10.1023/A:1011989910997.
    [62] A. Kupiainen, Renormalization group and stochastic PDEs, Ann. Henri Poincaré, 17 (2016), 497-535.  doi: 10.1007/s00023-015-0408-y.
    [63] J.-F. Le Gall, Spatial Branching Processes, Random Snakes and Partial Differential Equations, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1999. doi: 10.1007/978-3-0348-8683-3.
    [64] R. H. Lyon, Response of a nonlinear string to random excitation, J. Acoust. Soc. Amer., 32 (1960), 953-960.  doi: 10.1121/1.1908341.
    [65] T. J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14 (1998), 215-310.  doi: 10.4171/RMI/240.
    [66] Z. M. Ma and M. Röckner, Introduction to the Theory of (Nonsymmetric) Dirichlet Forms, Universitext, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-77739-4.
    [67] J. C. Mattingly, Ergodicity of 2D Navier-Stokes equations with random forcing and large viscosity, Comm. Math. Phys., 206 (1999), 273–288. doi: 10.1007/s002200050706.
    [68] G. Miermont, Aspects of random maps, Saint-Flour Lecture notes, 2014. http://perso.ens-lyon.fr/gregory.miermont/coursSaint-Flour.pdf
    [69] J.-C. Mourrat and H. Weber, Convergence of the two-dimensional dynamic Ising-Kac model to $\Phi^4_2$, Comm. Pure Appl. Math., 70 (2017), 717-812.  doi: 10.1002/cpa.21655.
    [70] C. Mueller, On the support of solutions to the heat equation with noise, Stochastics Stochastics Rep., 37 (1991), 225-245.  doi: 10.1080/17442509108833738.
    [71] C. MuellerL. Mytnik and E. Perkins, Nonuniqueness for a parabolic SPDE with $\frac{3}{4}-\epsilon$-Hölder diffusion coefficients, Ann. Probab., 42 (2014), 2032-2112.  doi: 10.1214/13-AOP870.
    [72] C. MuellerL. Mytnik and J. Quastel, Effect of noise on front propagation in reaction-diffusion equations of KPP type, Invent. Math., 184 (2011), 405-453.  doi: 10.1007/s00222-010-0292-5.
    [73] L. Mytnik, Superprocesses in random environments, Ann. Probab., 24 (1996), 1953-1978.  doi: 10.1214/aop/1041903212.
    [74] L. Mytnik and E. Perkins, Pathwise uniqueness for stochastic heat equations with Hölder continuous coefficients: The white noise case, Probab. Theory Related Fields, 149 (2011), 1-96.  doi: 10.1007/s00440-009-0241-7.
    [75] D. Nualart and É. Pardoux, White noise driven quasilinear SPDEs with reflection, Probab. Theory Related Fields, 93 (1992), 77-89.  doi: 10.1007/BF01195389.
    [76] D. Nualart, The Malliavin Calculus and Related Topics, 2nd edition, Probability and its Applications (New York), Springer-Verlag, Berlin, 2006.
    [77] E. Pardoux, Sur des équations aux dérivées partielles stochastiques monotones, C. R. Acad. Sci. Paris Sér. A-B, 275 (1972), A101–A103.
    [78] G. Parisi and Y. S. Wu, Perturbation theory without gauge fixing, Sci. Sinica, 24 (1981), 483-496. 
    [79] E. Perkins, Dawson-Watanabe superprocesses and measure-valued diffusions, in Lectures on Probability Theory and Statistics (Saint-Flour, 1999), Lecture Notes in Math., Vol. 1781, Springer, Berlin, 2002,125–324.
    [80] J. Quastel, Introduction to KPZ, in Current Developments in Mathematics, 2011
    [81] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Grundlehren der Mathematischen Wissenschaften, Vol. 293, 3rd edition, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-662-06400-9.
    [82] B. L. Rozovskiĭ, Stochastic differential equations in infinite-dimensional spaces, and filtering problems, in Proceedings of the School and Seminar on the Theory of Random Processes (Druskininkai, 1974), Part Ⅱ (Russian), 1975,147–194.
    [83] B. L. Rozovskiĭ, Stochastic partial differential equations, Mat. Sb. (N.S.), 96(138) (1975), 314–341,344.
    [84] T. Shiga, Diffusion processes in population genetics, J. Math. Kyoto Univ., 21 (1981), 133-151.  doi: 10.1215/kjm/1250522109.
    [85] T. Shiga, Existence and uniqueness of solutions for a class of nonlinear diffusion equations, J. Math. Kyoto Univ., 27 (1987), 195-215.  doi: 10.1215/kjm/1250520714.
    [86] B. SimonThe $P(\phi)_{2}$ Euclidean (Quantum) Field Theory, Princeton University Press, Princeton, N.J., 1974. 
    [87] M. R. Spiegel, The random vibrations of a string, Quart. Appl. Math., 10 (1952), 25-33.  doi: 10.1090/qam/45976.
    [88] D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Classics in Mathematics, Springer-Verlag, Berlin, 2006.
    [89] H. J. Sussmann, On the gap between deterministic and stochastic ordinary differential equations, Ann. Probability, 6 (1978), 19-41.  doi: 10.1214/aop/1176995608.
    [90] J. B. Walsh, An introduction to stochastic partial differential equations, in École d'Été de Probabilités de Saint-Flour, XIV-1984, Lecture Notes in Math., Vol. 1180, Springer, Berlin, 1986,265–439. doi: 10.1007/BFb0074920.
    [91] S. Watanabe, A limit theorem of branching processes and continuous state branching processes, J. Math. Kyoto Univ., 8 (1968), 141-167.  doi: 10.1215/kjm/1250524180.
    [92] M. Zakai, On the optimal filtering of diffusion processes, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 11 (1969), 230-243.  doi: 10.1007/BF00536382.
    [93] L. Zambotti, A reflected stochastic heat equation as symmetric dynamics with respect to the 3-d Bessel bridge, J. Funct. Anal., 180 (2001), 195-209.  doi: 10.1006/jfan.2000.3685.
    [94] L. Zambotti, Integration by parts formulae on convex sets of paths and applications to SPDEs with reflection, Probab. Theory Related Fields, 123 (2002), 579-600.  doi: 10.1007/s004400200203.
    [95] L. Zambotti, Occupation densities for SPDEs with reflection, Ann. Probab., 32 (2004), 191-215.  doi: 10.1214/aop/1078415833.
    [96] L. Zambotti, Random Obstacle Problems, Lecture Notes in Mathematics, Vol. 2181, Springer, Cham, 2017. doi: 10.1007/978-3-319-52096-4.
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