Fokker–Planck–Kolmogorov Equations

Abstract

Mathematics Subject Classification. Primary 35-02, 35J15, 35K10, 60J35, 60J60. For additional information and updates on this book, visit www.ams.org/bookpages/surv-207 Library of Congress Cataloging-in-Publication Data Fokker-Planck-Kolmogorov equations /Vladimir I. Bogachev, Nicolai V. Krylov, Michael Röckner, Stanislav V. Shaposhnikov. pages cm. -(Mathematical surveys and monographs ; volume 207) Includes bibliographical references and index. ISBN 978-1-4704-2558-6 (alk. paper) 1. Fokker-Planck equation. 2. Stochastic differential equations. I. Bogachev, V. I. (Vladimir Igorevich), 1961-II. Krylov, N. V. (Nicolai Vladimirovich). III. Röckner, Michael, 1956-IV. Shaposhnikov, Stanislav V. QA274.23.F65 2015 515 .353-dc23 2015024922Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given.

Variational approach to coarse-graining of generalized gradient flows

“…Proposition C. 13 Let g be the solution of the variational problem (117) (in the sense of Corollary C.7) with U = 0 and with initial datum g 0 ∈ X. If g 0 ∈ C 3 (R 2d ) ∩ X, then g satisfies…”

Quasi regular Dirichlet forms and the stochastic quantization problem

“…There are many counterexamples known (see e.g. [BRSt00] or [BKRS14]). In Section 5 we shall give an explicit (Gaussian) counterexample with state space being a Hilbert space, in which even the underlying martingale problem is well-posed.…”

“…In Section 5 we shall give an explicit (Gaussian) counterexample with state space being a Hilbert space, in which even the underlying martingale problem is well-posed. For references on invariant and infinitesimally invariant measures see, e.g., [ABR99], [AFe04], [AFe08], [AKR97a], [AKR97b], [AKR98], [AKR02], [ARW01], [ARü02], [MR10], [MZ10], [ARZ93b], [BoRöZh00], [DPZ92], [E99], [BKRS14]. To the above "inverse problem" there exists a direct problem: given a (Markov) stochastic process M , find (if possible) a probability measure µ s.t.…”

“…Below, we want to briefly describe an important subclass thereof, where one is just given an operator L and a measure µ, intrinsically related to L (see Definition 5.1 below), that will serve as a "reference measure". The underlying idea has been first put forward systematically in [ABR99] and [R98], and was one motivation that has eventually led to the recent monograph [BKRS14].…”

Loss of Energy Concentration in Nonlinear Evolution Beam Equations

“…4) On the other hand, the McKean dynamics (1.3) and the corresponding McKeanVlasov-Fokker-Planck equation (1.4) can have more than one invariant measures, for nonconvex confining potentials and at sufficiently low temperatures (Dawson 1983;Tamura 1984). This is not surprising, since the McKean-Vlasov equation is a nonlinear, nonlocal PDE and the standard uniqueness of solutions for the linear (stationary) Fokker-Planck equation does not apply (Bogachev et al 2015).…”