Kolmogorov equations in infinite dimensions: well-posedness and regularity of solutions, with applications to stochastic generalized Burgers equations. (English) Zbl 1106.35127

The paper develops a new method to uniquely solve a large class of heat equations \[ \frac{du}{dt}=Lu \] in infinitely many variables. The equations are analyzed in spaces of sequentially weakly continuous functions on the underlying infinite-dimensional Banach space weighted by proper (Lyapunov type) functions. This way for the first time the solutions are constructed everywhere without exceptional sets for equations with possibly nonlocally Lipschitz drifts.
Apart from general analytic interest, the main motivation of the authors was to apply this to uniquely solve martingale problems in the sense of Strook-Varadan given by SPDEs from hydrodynamics, such as the stochastic Navier-Stokes equations. In this paper this is done in the case of the stochastic generalized Burgers equation. Uniqueness is shown in the sense of Markov flows.


35R15 PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables)
60J60 Diffusion processes
35Q53 KdV equations (Korteweg-de Vries equations)
35J70 Degenerate elliptic equations
47D06 One-parameter semigroups and linear evolution equations
47D08 Schrödinger and Feynman-Kac semigroups
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60J35 Transition functions, generators and resolvents
Full Text: DOI arXiv


[1] Albeverio, S. and Ferrario, B. (2002). Uniqueness results for the generators of the two-dimensional Euler and Navier–Stokes flows. The case of Gaussian invariant measures. J. Funct. Anal. 193 77–93. · Zbl 1034.35086
[2] 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
[3] Barbu, V. and Da Prato, G. (2002). A phase field system perturbed by noise. Nonlinear Anal. 51 1087–1099. · Zbl 1034.35165
[4] Barbu, V., Da Prato, G. and Debusche, A. (2004). The Kolmogorov equation associated to the stochastic Navier–Stokes equations in 2D. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 2 163–182. · Zbl 1085.60041
[5] Bauer, H. (1974). Wahrscheinlichkeitstheorie und Grundzüge der Masstheorie 2 . Erweiterte Auflage . de Gruyter, Berlin. · Zbl 0272.60001
[6] Bertini, L., Cancrini, N. and Jona-Lasinio, G. (1994). The stochastic Burgers equation. Comm. Math. Phys. 165 211–232. · Zbl 0807.60062
[7] Blumenthal, R. M. and Getoor, R. K. (1968). Markov Processes and Potential Theory . Academic Press, New York. · Zbl 0169.49204
[8] Bogachev, V. and Röckner, M. (2000). A generalization of Khasminski’s theorem on the existence of invariant measures for locally integrable drifts. Theory Probab. Appl. 45 363–378. · Zbl 1004.60061
[9] Bogachev, V. and Röckner, M. (2001). Elliptic equations for measures on infinite dimensional spaces and applications. Probab. Theory Related Fields 120 445–496. · Zbl 1086.35114
[10] Brick, P., Funaki, T. and Woyczynsky, W., eds. (1996). Nonlinear Stochastic PDEs : Hydrodynamic Limit and Burgers ’ Turbulence . Springer, Berlin.
[11] Burgers, J. M. (1984). A mathematical model illustrating the theory of turbulence. In Advances in Applied Mathematics (R. von Mises and Th. von Karman, eds.) 1 171–199.
[12] Cardon-Weber, C. (1999). Large deviations for a Burgers’ type SPDE. Stochastic Process. Appl. 84 53–70. · Zbl 0996.60073
[13] Cerrai, S. (2001). Second Order PDE ’ s in Finite and Infinite dimensions : A Probabilistic Approach . Springer, Berlin. · Zbl 0983.60004
[14] Choquet, G. (1969). Lectures on Analysis . I, II, III. Benjamin, Inc., New York. · Zbl 0181.39602
[15] Da Prato, G. (2001). Elliptic operators with unbounded coefficients: Construction of a maximal dissipative extension. J. Evol. Equ. 1 1–18. · Zbl 0984.47037
[16] Da Prato, G. and Debusche, A. (2000). Maximal dissipativity of the Dirichlet operator corresponding to the Burgers equation. In Stochastic Processes , Physics and Geometry : New Interplays 1 85–98. Amer. Math. Soc., Providence, RI. · Zbl 0978.60069
[17] Da Prato, G. and Debusche, A. (2003). Ergodicity for the 3D stochastic Navier–Stokes equations. J. Math. Pures Appl. 82 877–947. · Zbl 1109.60047
[18] Da Prato, G., Debussche, A. and Temam, R. (1995). Stochastic Burgers equation. In Nonlinear Differential Equations Appl. 1 389–402. Birkhäuser, Basel. · Zbl 0824.35112
[19] Da Prato, G. and Gatarek, D. (1995). Stochastic Burgers equation with correlated noise. Stochastics Stochastics Rep. 52 29–41. · Zbl 0853.35138
[20] Da Prato, G. and Röckner, M. (2002). Singular dissipative stochastic equations in Hilbert spaces. Probab. Theory Related Fields 124 261–303. · Zbl 1036.47029
[21] Da Prato, G. and Röckner, M. (2004). Invariant measures for a stochastic porous medium equation. In Stochastic Analysis and Related Topics (H. Kunita, Y. Takanashi and S. Watanabe, eds.) 13–29. Math. Soc. Japan, Tokyo. · Zbl 1102.76074
[22] Da Prato, G. and Tubaro, L. (2001). Some results about dissipativity of Kolmogorov operators. Czechoslovak Math. J. 51 685–699. · Zbl 0996.47028
[23] Da Prato, G. and Vespri, V. (2002). Maximal \(L^p\) regularity for elliptic equations with unbounded coefficients. Nonlinear Anal. 49 747–755. · Zbl 1012.35027
[24] Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions . Cambridge Univ. Press. · Zbl 0761.60052
[25] Eberle, A. (1999). Uniqueness and Non-Uniqueness of Semigroups Generated by Singular Diffusion Operators . Springer, Berlin. · Zbl 0957.60002
[26] Flandoli, F. and Gątarek, D. (1995). Martingale and stationary solutions for stochastic Navier–Stokes equations. Probab. Theory Related Fields 102 367–391. · Zbl 0831.60072
[27] Flandoli, F. and Gozzi, F. (1998). Kolmogorov equation associated to a stochastic Navier–Stokes equation. J. Funct. Anal. 160 312–336. · Zbl 0928.60044
[28] Gilbarg, D. and Trudinger, N. S. (1983). Elliptic Partial Differential Equations of Second Order , 2nd ed. Springer, Berlin. · Zbl 0562.35001
[29] Gyöngy, I. (1998). Existence and uniqueness results for semilinear stochastic partial differential equations. Stochastic Process. Appl. 73 271–299. · Zbl 0942.60058
[30] Gyöngy, I. and Nualart, D. (1999). On the stochastic Burgers equation in the real line. Ann. Prob. 27 782–802. · Zbl 0939.60058
[31] Gyöngy, I. and Rovira, C. (1999). On stochastic partial differential equations with polynomial nonlinearities. Stochastics Stochastics Rep. 67 123–146. · Zbl 0945.60052
[32] Gyöngy, I. and Rovira, C. (2000). On \(L^p\)-solutions of semilinear stochastic partial differential equations. Stochastic Process. Appl. 90 83–108. · Zbl 1046.60059
[33] Hopf, E. (1950). The partial differential equation \(u_t + uu_x = \mu u_xx\). Comm. Pure Appl. Math. 3 201–230. · Zbl 0039.10403
[34] Jarchow, H. (1981). Locally Convex Spaces . Teubner, Stuttgart. · Zbl 0466.46001
[35] Krylov, N. V. and Röckner, M. (2005). Strong solutions for stochastic equations with singular time dependent drift. Probab. Theory Related Fields 131 154–196. · Zbl 1072.60050
[36] Kühnemund, F. and van Neerven, J. (2004). A Lie–Trotter product formula for Ornstein–Uhlenbeck semigroups in infinite dimensions. J. Evol. Equ. 4 53–73. · Zbl 1060.35155
[37] Ladyzhenskaya, O. A., Solonnikov, N. A. and Ural’tseva, N. N. (1968). Linear and Quasi-Linear Equations of Parabolic Type . Amer. Math. Soc., Providence, RI. · Zbl 0174.15403
[38] Lanjri, X. and Nualart, D. (1999). The stochastic Burgers equation: Absolute continuity of the density. Stochastics Stochastics Rep. 66 273–292. · Zbl 0936.60055
[39] Léon, J. A., Nualart, D. and Pettersson, R. (2000). The stochastic Burgers equation: Finite moments and smoothness of the density. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 3 363–385. · Zbl 0968.60057
[40] Lindenstrauss, J. and Tzafriri, L. (1979). Classical Banach Spaces 2 . Function Spaces . Springer, Berlin, · Zbl 0403.46022
[41] Liu, T. P. and Yu, S. H. (1997). Propagation of stationary viscous Burgers shock under the effect of boundary. Arch. Rational Mech. Anal. 139 57–82. · Zbl 0895.76039
[42] Long, H. and Simão, I. (2000). Kolmogorov equations in Hilbert spaces with application to essential self-adjointness of symmetric diffusion operators. Osaka J. Math. 37 185–202. · Zbl 0956.60065
[43] Ma, Z.-M. and Röckner, M. (1992). Introduction to the Theory of ( Nonsymmetric ) Dirichlet Forms . Springer, Berlin.
[44] Matsumura, M. and Nishihara, K. (1994). Asymptotic stability of travelling waves for scalar viscous conservation laws with non-convex nonlinearity. Comm. Math. Phys. 165 83–96. · Zbl 0811.35080
[45] Nualart, D. and Viens, F. (2000). Evolution equation of a stochastic semigroup with white-noise drift. Ann. Probab. 28 36–73. · Zbl 1044.60052
[46] Röckner, M. (1999). \(L^p\)-analysis of finite and infinite dimensional diffusion operators. Stochastic PDE ’ s and Kolmogorov ’ s Equations in Infinite Dimensions . Lecture Notes in Math. 1715 65–116. Springer, Berlin.
[47] Röckner, M. and Sobol, Z. (2003). Markov solutions for martingale problems: Method of Lyapunov functions.
[48] Röckner, M. and Sobol, Z. (2004). A new approach to Kolmogorov equations in infinite dimensions and applications to stochastic generalized Burgers equations. C. R. Acad. Sci. Paris 338 945–949. · Zbl 1109.60050
[49] Röckner, M. and Sobol, Z. (2004). \(L^1\)-theory for the Kolmogorov operators of stochastic generalized Burgers equations. In Quantum Information and Complexity : Proceedings of the 2003 Meijo Winter School and Conference (T. Hida, K. Saitô and Si Si, eds.) 87–105. World Scientific, Singapore.
[50] Sadovnichaya, I. V. (2000). The direct and the inverse Kolmogorov equation for the stochastic Schrödinger equation. Vestnik Moskov. Univ. Ser. I Mat. Mekh. 2000 15–20, 86. [Translation in Moscow Univ. Math. Bull. (2000) 55 15–19.] · Zbl 0987.35149
[51] Schwartz, L. (1973). Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures . Oxford Univ. Press. · Zbl 0298.28001
[52] Stannat, W. (1999). The theory of generalized Dirichlet forms and its applications in analysis and stochastics. Mem. Amer. Math. Soc. 142 1–101. · Zbl 1230.60006
[53] Stannat, W. (1999). (Nonsymmetric) Dirichlet operators on \(L^1\) : Existence, uniqueness and associated Markov processes. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 28 99–140. · Zbl 0946.31003
[54] Stroock, D. W. and Varadhan, S. R. S. (1979). Multidimensional Diffusion Processes . Springer, Berlin. · Zbl 0426.60069
[55] Truman, A. and Zhao, H. Z. (1996). On stochastic diffusion equations and stochastic Burgers’ equations. J. Math. Phys. 37 283–307. · Zbl 0866.35149
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