Analytic Theory of Itô-Stochastic Differential Equations with Non-smooth Coefficients

Haesung Lee, Wilhelm Stannat, Gerald Trutnau (Autoren)

Buch | Softcover

126 Seiten

2022 | 1st ed. 2022
Springer Verlag, Singapore
978-981-19-3830-6 (ISBN)

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This book provides analytic tools to describe local and global behavior of solutions to Itô-stochastic differential equations with non-degenerate Sobolev diffusion coefficients and locally integrable drift. Regularity theory of partial differential equations is applied to construct such solutions and to obtain strong Feller properties, irreducibility, Krylov-type estimates, moment inequalities, various types of non-explosion criteria, and long time behavior, e.g., transience, recurrence, and convergence to stationarity. The approach is based on the realization of the transition semigroup associated with the solution of a stochastic differential equation as a strongly continuous semigroup in the Lp-space with respect to a weight that plays the role of a sub-stationary or stationary density. This way we obtain in particular a rigorous functional analytic description of the generator of the solution of a stochastic differential equation and its full domain. The existence of such a weight is shown under broad assumptions on the coefficients. A remarkable fact is that although the weight may not be unique, many important results are independent of it. Given such a weight and semigroup, one can construct and further analyze in detail a weak solution to the stochastic differential equation combining variational techniques, regularity theory for partial differential equations, potential, and generalized Dirichlet form theory. Under classical-like or various other criteria for non-explosion we obtain as one of our main applications the existence of a pathwise unique and strong solution with an infinite lifetime. These results substantially supplement the classical case of locally Lipschitz or monotone coefficients.We further treat other types of uniqueness and non-uniqueness questions, such as uniqueness and non-uniqueness of the mentioned weights and uniqueness in law, in a certain sense, of the solution.

Dr. Haesung Lee is working at Department of Mathematics and Computer Science, Korea Science Academy of KAIST.Professor Wilhelm Stannat is working at Institut für Mathematik, Technische Universität Berlin. Professor Gerald Trutnau is a full-professor at Department of Mathematical Sciences, Seoul National University.

Chapter 1. Introduction.- Chapter 2. The abstract Cauchy problem in Lr-spaces with weights.- Chapter 3.Stochastic differential equations.- Chapter 4. Conclusion and outlook.

Erscheinungsdatum	02.09.2022
Reihe/Serie	SpringerBriefs in Probability and Mathematical Statistics
Zusatzinfo	1 Illustrations, black and white; XV, 126 p. 1 illus.
Verlagsort	Singapore
Sprache	englisch
Maße	155 x 235 mm
Themenwelt	Mathematik / Informatik ► Mathematik ► Analysis
Themenwelt	Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
ISBN-10	981-19-3830-X / 981193830X
ISBN-13	978-981-19-3830-6 / 9789811938306
Zustand	Neuware