January  2021, 41(1): 471-487. doi: 10.3934/dcds.2020264

A brief and personal history of stochastic partial differential equations

Laboratoire de Probabilités, Statistique et Modélisation, Sorbonne Université, Université de Paris, CNRS, 4, Place Jussieu, 75005 Paris, France

Received  December 2019 Published  January 2021 Early access  July 2020

Fund Project: The author is supported by the grant ANR-15-CE40-0020 - LSD - Large Stochastic Dynamical Models in Non-Equilibrium Statistical Physics (2015)

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.

Citation: Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264
References:
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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.

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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.

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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.

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Y. BrunedM. Hairer and L. Zambotti, Algebraic renormalisation of regularity structures, Invent. Math., 215 (2019), 1039-1156.  doi: 10.1007/s00222-018-0841-x.

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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.

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show all references

References:
[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]

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