Kolmogorov Operators and Their Applications

Kolmogorov equations are a fundamental bridge between the theory of partial differential equations and that of stochastic differential equations that arise in several research fields.

This volume collects a selection of the talks given at the Cortona meeting by experts in both fields, who presented the most recent developments of the theory. Particular emphasis has been given to degenerate partial differential equations, Itô processes, applications to kinetic theory and to finance.

Chapter 1. Local Regularity for the Landau Equation (with Coulomb Interaction Potential).- Chapter 2. L 2 Hypocoercivity methods for kinetic Fokker-Planck equations with factorised Gibbs states.- Chapter 3. New Perspectives on recent trends for Kolmogorov operators.- Chapter 4. Schauder estimates for Kolmogorov-Fokker-Planck operators with coefficients measurable in time and Holder continuous in space.-Chapter 5. A new proof of the geometric Soboleva embedding for generalised Kolmogorov operators.- Chapter 6. Intrinsic Taylor formula for non-homogeneous Kolmogorov-type Lie groups.- Chapter 7. Form-boundedness and sdes with singular drift.- Chapter 8. About the regularity of degenerate non-local Kolmogorov operators under diffusive perturbations.- Chapter 9. Integration by parts formula for exit times of one dimensional diffusions.- Chapter 10. On averaged control and iteration improvement for a class of multidimensional ergodicdiffusions.

Stéphane Menozzi is Full Professor at Université d'Évry Val d'Essonne-Paris Saclay. His research concerns degenerate and/or singular Stochastic Differential Equations, regularity, heat-kernel estimates, approximation. Those equations can be viewed as the probabilistic counterpart to the corresponding Kolmogorov operators.

Andrea Pascucci is Full Professor of Probability and Statistics at the Alma Mater Studiorum - Università di Bologna. His expertise lies in Stochastic Partial Differential Equations, particularly of degenerate parabolic type. He has contributed to the field, focusing on applications in mathematical finance, including American options, Asian/path-dependent options, and volatility modeling.

Sergio Polidoro is Full professor of Mathematical Analysis at the University of Modena and Reggio Emilia. His research activity mainly concerns regularity theory for second order partial differential equations with non-negative characteristic form. His main contributions in this field are regularity results and heat-kernel estimates for degenerate Kolmogorov equations.