Harnack inequalities for stochastic partial differential equations.

*(English)*Zbl 1332.60006
SpringerBriefs in Mathematics. New York, NY: Springer (ISBN 978-1-4614-7933-8/pbk; 978-1-4614-7934-5/ebook). x, 125 p. (2013).

This book gives an overview of recent results of Harnack inequalities and their applications to solutions of stochastic partial differential equations (SPDEs). Classical Harnack inequalities for deterministic elliptic and parabolic partial differential equations (PDEs) relate the values of a positive harmonic function at two points and the results are dimension dependent and not applicable for equations in infinite dimensions. In the SPDEs case, the associated Fokker-Planck equations are PDEs in infinite-dimensional spaces. The main contribution of the book is the investigation of the dimension-free Harnack inequalities. These results have several applications to SPDEs. The authors introduce a new coupling by change of measure method that is used rather than the usual maximum principle in the literature of PDEs.

The book consists of four chapters. In Chapter 1, the authors introduce a general theory concerning dimension-free Harnack inequalities, which includes the main idea of establishing Harnack inequalities and derivative formulas using coupling by change of measure, derivative formulas using the Malliavin calculus, links of Harnack inequalities to gradient estimates, and various applications of Harnack inequalities. Chapter 2 presents a Harnack inequality with power together with the log-Harnack inequality for the semigroup associated to a class of nonlinear SPDEs, which include the stochastic generalized porous media equation or fast-diffusion model. In Chapter 3, the authors investigate gradient estimates and Harnack inequalities for semilinear SPDEs using coupling by change of measure method, by gradient estimates, and by finite-dimensional approximations. Chapter 4 is devoted to gradient estimates and Harnack inequalities for the segment solution of stochastic functional differential equations. Again, the coupling by change of measure method and the Malliavin calculus are used.

The book consists of four chapters. In Chapter 1, the authors introduce a general theory concerning dimension-free Harnack inequalities, which includes the main idea of establishing Harnack inequalities and derivative formulas using coupling by change of measure, derivative formulas using the Malliavin calculus, links of Harnack inequalities to gradient estimates, and various applications of Harnack inequalities. Chapter 2 presents a Harnack inequality with power together with the log-Harnack inequality for the semigroup associated to a class of nonlinear SPDEs, which include the stochastic generalized porous media equation or fast-diffusion model. In Chapter 3, the authors investigate gradient estimates and Harnack inequalities for semilinear SPDEs using coupling by change of measure method, by gradient estimates, and by finite-dimensional approximations. Chapter 4 is devoted to gradient estimates and Harnack inequalities for the segment solution of stochastic functional differential equations. Again, the coupling by change of measure method and the Malliavin calculus are used.

Reviewer: Jan Pospíšil (Plzeň)

##### MSC:

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |

60H07 | Stochastic calculus of variations and the Malliavin calculus |

35R60 | PDEs with randomness, stochastic partial differential equations |