Mathematical Analysis, Probability and Applications – Plenary Lectures: ISAAC 2015, Macau, China

335 Pages·2016·6.846 MB·English

by  Tao Qian, Luigi G. Rodino (eds.)

Springer Proceedings in Mathematics & Statistics
Tao Qian
Luigi G. Rodino Editors
Mathematical
Analysis,
Probability and
Applications –
Plenary Lectures
ISAAC 2015, Macau, China
Springer Proceedings in Mathematics & Statistics
Volume 177
Springer Proceedings in Mathematics & Statistics
This book series features volumes composed of selected contributions from
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Tao Qian Luigi G. Rodino
(cid:129)
Editors
Mathematical Analysis,
–
Probability and Applications
Plenary Lectures
ISAAC 2015, Macau, China
123
Editors
TaoQian Luigi G.Rodino
Faculty of Science andTechnology Department ofMathematics
University of Macao University of Turin
Taipa Turin
Macao Italy
ISSN 2194-1009 ISSN 2194-1017 (electronic)
SpringerProceedings in Mathematics& Statistics
ISBN978-3-319-41943-5 ISBN978-3-319-41945-9 (eBook)
DOI 10.1007/978-3-319-41945-9
LibraryofCongressControlNumber:2016945107
MathematicsSubjectClassification(2010): 35QXX,46EXX,60GXX
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Preface
The present volume is a collection of papers devoted to current research topics in
mathematical analysis, probability and applications, including the topics in math-
ematicalphysicsandnumericalanalysis.Itoriginatesfromplenarylecturesgivenat
the 10th International ISAAC Congress, held during 3–8 August 2015 at the
University of Macau, China.
The papers, authored by eminent specialists, aim at presenting to a large audi-
ence some of the attractive and challenging themes of modern analysis:
(cid:129) Partial differential equations of mathematical physics, including study of the
equations of incompressible viscous flows, and of the Tricomi, Klein–Gordon
and Einstein–de Sitter equations. Governing equations offluid membranes are
also considered in this volume.
(cid:129) Fourier analysis and applications, in particular construction of Fourier and
Mellin-type transform pairs for given planar domains, multiplication and com-
position operators for modulation spaces, harmonic analysis of first-order sys-
tems on Lipschitz domains.
(cid:129) Reviewsofresultsonprobability,concerninginparticularthebi-freeextension
ofthefreeprobabilityandasurveyofBrownianmotionbasedontheLangevin
equation with white noise.
(cid:129) Numerical analysis, in particular sparse approximation by greedy algorithms,
and theory of reproducing kernels, with applications to analysis and numerical
analysis.
The volume also includes a contribution on visual exploration of complex
functions:thetechniqueofdomaincolouringallowstorepresentcomplexfunctions
as images, and it draws surprisingly mathematics near the modern arts.
v
vi Preface
Besides plenary talks, about 300 scientific communications were delivered
during the Macau ISAAC Congress. Their texts are published in an independent
volume.Onthewhole,thecongressdemonstrated,inparticular,theincreasingand
major role of Asian countries in several research areas of mathematical analysis.
Taipa, Macao Tao Qian
Turin, Italy Luigi G. Rodino
March 2016
Contents
A Review of Brownian Motion Based Solely on the Langevin
Equation with White Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
L. Cohen
Geometry-Fitted Fourier-Mellin Transform Pairs. . . . . . . . . . . . . . . . . 37
Darren Crowdy
First Order Approach to LP Estimates for the Stokes Operator
on Lipschitz Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Alan McIntosh and Sylvie Monniaux
The Study of Complex Shapes of Fluid Membranes,
the Helfrich Functional and New Applications . . . . . . . . . . . . . . . . . . . 77
Zhong-Can Ou-Yang and Zhan-Chun Tu
Multiplication and Composition in Weighted Modulation Spaces. . . . . . 103
Maximilian Reich and Winfried Sickel
A Reproducing Kernel Theory with Some General Applications. . . . . . 151
Saburou Saitoh
Sparse Approximation by Greedy Algorithms . . . . . . . . . . . . . . . . . . . 183
V. Temlyakov
The Bi-free Extension of Free Probability . . . . . . . . . . . . . . . . . . . . . . 217
Dan-Virgil Voiculescu
Stability of the Prandtl Boundary Layers. . . . . . . . . . . . . . . . . . . . . . . 235
Y.-G. Wang
Visual Exploration of Complex Functions . . . . . . . . . . . . . . . . . . . . . . 253
Elias Wegert
Integral Transform Approach to Time-Dependent
Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
Karen Yagdjian
vii
A Review of Brownian Motion Based Solely
on the Langevin Equation with White Noise
L.Cohen
Abstract WegiveahistoricalandmathematicalreviewofBrownianmotionbased
solely on the Langevin equation. We derive the main statistical properties with-
outbringinginexternalandsubsidiaryissues,suchastemperature,Focker-Planck
equations, the Maxwell–Boltzmann distribution, spectral analysis, the fluctuation-
dissipationtheorem,amongmanyothertopicsthataretypicallyintroducedindiscus-
sionsoftheLangevinequation.Themethodweuseistheformalsolutionapproach,
whichwasthestandardmethoddevisedbythefoundersofthefield.Inaddition,we
givesomerelevanthistoricalcomments.
· · · ·
Keywords Brownianmotion Langevinequation History Einstein Whitenoise
1 Introduction
Thetwoseeminglysimpleequations(asoriginallywritten)
∂f(x,t) ∂2f(x,t)
=D (1)
∂t ∂x2
and
d2x dx
m =−6πμa +X (2)
dt2 dt
revolutionized our understanding of the of the universe and ushered an incredible
numberofphysicalandmathematicalideas[11].ThefirstequationisduetoEinstein
[13],whoseaimwastoshowthatatomsexist;thesecondisduetoLangevin[26],
whobroughtforthanewperspectiveregardingboththephysicsandmathematicsof
Einstein’sidea.
B
L.Cohen ( )
CityUniversityofNewYork,695ParkAve.,NewYork,NY10065,USA
e-mail:[email protected]
©SpringerInternationalPublishingSwitzerland2016 1
T.QianandL.G.Rodino(eds.),MathematicalAnalysis,Probability
andApplications–PlenaryLectures,SpringerProceedings
inMathematics&Statistics177,DOI10.1007/978-3-319-41945-9_1
2 L.Cohen
ItisoftensaidthatBrown[3]discoveredBrownianmotion,Einsteinexplainedit,
Langevinsimplifiedit,andPerrin[31]provedit;thislistingmissestotallythemoti-
vationsandwonderfulhistoryofthesubject[11].BrowndidnotdiscoverBrownian
motion, but did study it extensively. Einstein was not aware of Brownian motion;
hepredicted Brownianmotiontoobtainamacroscopicmanifestationofatomsthat
could be measured. Equation(1) is the equation for the probability density for the
Brownianparticleatpositionxandtimet.Hederivedthestandarddeviationofthe
visibleBrownianparticlethatcouldbeexperimentallyverifiedifindeedatomsexist.
Hesolvedexplicitlyforthestandarddeviationofposition,
(cid:2)
√
λ = x2 = 2Dt (3)
x
and connected the parameters with the temperature of the medium, and the yet
unnamed Avogadro number, a number that few believed in, and had never been
measured or estimated at that time. Of course, Eq.(1) was known for 100 years
beforeEinstein;itisthefamousheatequationfirstderivedbyFourier.Howeverthat
isnotrelevant.WhatisimportantisthatEinsteinderivedtheprobabilitydensityfor
positionoftheBrownianparticle.Perrinhadalreadybeenworkingontheissueof
the existence of atoms, and his motivation was certainly heightened by Einstein’s
results.HeexperimentallyverifiedEq.(3),andhenceverifiedtheEinsteinideathat
therandom“invisible”microscopicatomscanmanifestamacroscopiceffectwhich
canbemeasured[31].
Equation(2)wasthestartofthefieldofrandomdifferentialequationandisnow
calledtheLangevinequation.Thewayitstands,itisNewton’sequationofmotion
wherethelefthandismasstimestheacceleration,thefirsttermontherightisthe
forceof“friction”whichisproportionaltothevelocity,andthesecondterm,X,is
anadditionalforce.InLangevin’swords:“X isindifferentlypositiveandnegative”.
X is what we now call the random force. Langevin’s insight was to realize that to
obtainthemainresultofEinstein,Eq.(3),onedoesnothavetosolveandderivethe
probability density, but one can obtain the second moment simply from Newton’s
equation and moreover that one can obtain it in a relatively simple manner. The
Langevin equation has been applied to numerous fields and to a wide variety of
physical situations. Random differential equations have become standard in many
branchesofscienceandhasproducedrichmathematics[7, 21–23, 27, 28, 35, 36,
39, 45].
1.1 TheAimofThisArticle
Theauthor’sinvolvementwithBrownianmotion[1,2,8]startedwithhisattemptto
understandandapplythetheoryofChandrasekharandvonNeumann[4–6]regarding
therandommotionofstars,asubjectthatisfundamentalinstellardynamics,because
itistherandommotionthatisimportantintheevolutionofacollectionofstars,such