Stochastic Partial Differential Equations
随机偏微分方程
－E. Pardoux－(Marseille, France)
复旦大学
英文版




Lectures given in Fudan University, Shangha¨ı, April 2007

Contents
1 Introduction and Motivation 5
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 Turbulence . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.2 Population dynamics, population genetics . . . . . . . 7
1.2.3 Neurophysiology . . . . . . . . . . . . . . . . . . . . . 8
1.2.4 Evolution of the curve of interest rate . . . . . . . . . . 8
1.2.5 Non Linear Filtering . . . . . . . . . . . . . . . . . . . 8
1.2.6 Movement by mean curvature in random environment . 9
1.2.7 Hydrodynamic limit of particle systems . . . . . . . . . 10
1.2.8 Fluctuations of an interface on a wall . . . . . . . . . . 11
2 SPDEs as infinite dimensional SDEs 13
2.1 Itˆo calculus in Hilbert space . . . . . . . . . . . . . . . . . . . 13
2.2 SPDE with additive noise . . . . . . . . . . . . . . . . . . . . 16
2.2.1 The semi–group approach to linear parabolic PDEs . . 17
2.2.2 The variational approach to linear and nonlinear
parabolic PDEs . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Variational approach to SPDEs . . . . . . . . . . . . . . . . . 25
2.3.1 Monotone – coercive SPDEs . . . . . . . . . . . . . . . 25
2.3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.3 Coercive SPDEs with compactness . . . . . . . . . . . 37
2.4 Semilinear SPDEs . . . . . . . . . . . . . . . . . . . . . . . . . 43
3 SPDEs driven by space–time white noise 49
3.1 Restriction to one–dimensional space variable . . . . . . . . . 49
3.2 A general existence–uniqueness result . . . . . . . . . . . . . . 51
3.3 More general existence and uniqueness result . . . . . . . . . . 59
3.4 Positivity of the solution . . . . . . . . . . . . . . . . . . . . . 59
3.5 Applications of Malliavin calculus to SPDEs . . . . . . . . . . 60
3.6 SPDEs and the super Brownian motion . . . . . . . . . . . . . 66
3.6.1 The case
= 1/2 . . . . . . . . . . . . . . . . . . . . . 66
3.6.2 Other values of
< 1 . . . . . . . . . . . . . . . . . . . 73
3.7 SPDEs with singular drift, and reflected SPDEs . . . . . . . . 79
3.7.1 Reflected SPDE . . . . . . . . . . . . . . . . . . . . . . 80
3.7.2 SPDE with critical singular drift . . . . . . . . . . . . 82