initial version
This commit is contained in:
Alessio Rondelli
parent
d9e0d8c334
commit
6ee7ac8b4b
27
nota5.aux
Normal file
27
nota5.aux
Normal file
@ -0,0 +1,27 @@
|
||||
\relax
|
||||
\providecommand\hyper@newdestlabel[2]{}
|
||||
\providecommand\HyField@AuxAddToFields[1]{}
|
||||
\providecommand\HyField@AuxAddToCoFields[2]{}
|
||||
\citation{Kolokoltsov}
|
||||
\citation{Kolokoltsov}
|
||||
\citation{Kolokoltsov}
|
||||
\newlabel{e1}{{1}{1}{}{equation.0.1}{}}
|
||||
\newlabel{e2}{{2}{1}{}{equation.0.2}{}}
|
||||
\newlabel{e3}{{3}{1}{}{equation.0.3}{}}
|
||||
\newlabel{e4}{{4}{1}{}{equation.0.4}{}}
|
||||
\newlabel{e4-1}{{5}{2}{}{equation.0.5}{}}
|
||||
\newlabel{f1}{{6}{2}{}{equation.0.6}{}}
|
||||
\citation{LucePagliaPascu}
|
||||
\newlabel{a1}{{1}{3}{}{theorem.1}{}}
|
||||
\citation{Kolokoltsov}
|
||||
\newlabel{t1}{{5}{4}{}{theorem.5}{}}
|
||||
\citation{Villani}
|
||||
\citation{Bogachev}
|
||||
\bibcite{Bogachev}{1}
|
||||
\bibcite{Kolokoltsov}{2}
|
||||
\bibcite{LucePagliaPascu}{3}
|
||||
\bibcite{YAOZHONG}{4}
|
||||
\bibcite{Villani}{5}
|
||||
\newlabel{e10}{{7}{6}{}{equation.0.7}{}}
|
||||
\newlabel{e11}{{8}{6}{}{equation.0.8}{}}
|
||||
\gdef \@abspage@last{6}
|
||||
686
nota5.log
Normal file
686
nota5.log
Normal file
@ -0,0 +1,686 @@
|
||||
This is pdfTeX, Version 3.141592653-2.6-1.40.25 (TeX Live 2023/nixos.org) (preloaded format=pdflatex 1980.1.1) 18 NOV 2024 15:45
|
||||
entering extended mode
|
||||
restricted \write18 enabled.
|
||||
%&-line parsing enabled.
|
||||
**nota5.tex
|
||||
(./nota5.tex
|
||||
LaTeX2e <2023-11-01> patch level 1
|
||||
L3 programming layer <2024-02-20>
|
||||
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/base/article.cls
|
||||
Document Class: article 2023/05/17 v1.4n Standard LaTeX document class
|
||||
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/base/size10.clo
|
||||
File: size10.clo 2023/05/17 v1.4n Standard LaTeX file (size option)
|
||||
)
|
||||
\c@part=\count188
|
||||
\c@section=\count189
|
||||
\c@subsection=\count190
|
||||
\c@subsubsection=\count191
|
||||
\c@paragraph=\count192
|
||||
\c@subparagraph=\count193
|
||||
\c@figure=\count194
|
||||
\c@table=\count195
|
||||
\abovecaptionskip=\skip48
|
||||
\belowcaptionskip=\skip49
|
||||
\bibindent=\dimen140
|
||||
)
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/base/inputenc.sty
|
||||
Package: inputenc 2021/02/14 v1.3d Input encoding file
|
||||
\inpenc@prehook=\toks17
|
||||
\inpenc@posthook=\toks18
|
||||
)
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/amsmath/amsmath.sty
|
||||
Package: amsmath 2023/05/13 v2.17o AMS math features
|
||||
\@mathmargin=\skip50
|
||||
|
||||
For additional information on amsmath, use the `?' option.
|
||||
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/amsmath/amstext.sty
|
||||
Package: amstext 2021/08/26 v2.01 AMS text
|
||||
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/amsmath/amsgen.sty
|
||||
File: amsgen.sty 1999/11/30 v2.0 generic functions
|
||||
\@emptytoks=\toks19
|
||||
\ex@=\dimen141
|
||||
))
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/amsmath/amsbsy.sty
|
||||
Package: amsbsy 1999/11/29 v1.2d Bold Symbols
|
||||
\pmbraise@=\dimen142
|
||||
)
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/amsmath/amsopn.sty
|
||||
Package: amsopn 2022/04/08 v2.04 operator names
|
||||
)
|
||||
\inf@bad=\count196
|
||||
LaTeX Info: Redefining \frac on input line 234.
|
||||
\uproot@=\count197
|
||||
\leftroot@=\count198
|
||||
LaTeX Info: Redefining \overline on input line 399.
|
||||
LaTeX Info: Redefining \colon on input line 410.
|
||||
\classnum@=\count199
|
||||
\DOTSCASE@=\count266
|
||||
LaTeX Info: Redefining \ldots on input line 496.
|
||||
LaTeX Info: Redefining \dots on input line 499.
|
||||
LaTeX Info: Redefining \cdots on input line 620.
|
||||
\Mathstrutbox@=\box51
|
||||
\strutbox@=\box52
|
||||
LaTeX Info: Redefining \big on input line 722.
|
||||
LaTeX Info: Redefining \Big on input line 723.
|
||||
LaTeX Info: Redefining \bigg on input line 724.
|
||||
LaTeX Info: Redefining \Bigg on input line 725.
|
||||
\big@size=\dimen143
|
||||
LaTeX Font Info: Redeclaring font encoding OML on input line 743.
|
||||
LaTeX Font Info: Redeclaring font encoding OMS on input line 744.
|
||||
\macc@depth=\count267
|
||||
LaTeX Info: Redefining \bmod on input line 905.
|
||||
LaTeX Info: Redefining \pmod on input line 910.
|
||||
LaTeX Info: Redefining \smash on input line 940.
|
||||
LaTeX Info: Redefining \relbar on input line 970.
|
||||
LaTeX Info: Redefining \Relbar on input line 971.
|
||||
\c@MaxMatrixCols=\count268
|
||||
\dotsspace@=\muskip16
|
||||
\c@parentequation=\count269
|
||||
\dspbrk@lvl=\count270
|
||||
\tag@help=\toks20
|
||||
\row@=\count271
|
||||
\column@=\count272
|
||||
\maxfields@=\count273
|
||||
\andhelp@=\toks21
|
||||
\eqnshift@=\dimen144
|
||||
\alignsep@=\dimen145
|
||||
\tagshift@=\dimen146
|
||||
\tagwidth@=\dimen147
|
||||
\totwidth@=\dimen148
|
||||
\lineht@=\dimen149
|
||||
\@envbody=\toks22
|
||||
\multlinegap=\skip51
|
||||
\multlinetaggap=\skip52
|
||||
\mathdisplay@stack=\toks23
|
||||
LaTeX Info: Redefining \[ on input line 2953.
|
||||
LaTeX Info: Redefining \] on input line 2954.
|
||||
)
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/amsfonts/amsfonts.sty
|
||||
Package: amsfonts 2013/01/14 v3.01 Basic AMSFonts support
|
||||
\symAMSa=\mathgroup4
|
||||
\symAMSb=\mathgroup5
|
||||
LaTeX Font Info: Redeclaring math symbol \hbar on input line 98.
|
||||
LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold'
|
||||
(Font) U/euf/m/n --> U/euf/b/n on input line 106.
|
||||
)
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/amsfonts/amssymb.sty
|
||||
Package: amssymb 2013/01/14 v3.01 AMS font symbols
|
||||
)
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/doublestroke/dsfont.sty
|
||||
Package: dsfont 1995/08/01 v0.1 Double stroke roman fonts
|
||||
)
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/fancyhdr/fancyhdr.sty
|
||||
Package: fancyhdr 2022/11/09 v4.1 Extensive control of page headers and footers
|
||||
|
||||
\f@nch@headwidth=\skip53
|
||||
\f@nch@O@elh=\skip54
|
||||
\f@nch@O@erh=\skip55
|
||||
\f@nch@O@olh=\skip56
|
||||
\f@nch@O@orh=\skip57
|
||||
\f@nch@O@elf=\skip58
|
||||
\f@nch@O@erf=\skip59
|
||||
\f@nch@O@olf=\skip60
|
||||
\f@nch@O@orf=\skip61
|
||||
)
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/tools/indentfirst.sty
|
||||
Package: indentfirst 2023/07/02 v1.03 Indent first paragraph (DPC)
|
||||
)
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/graphics/graphicx.sty
|
||||
Package: graphicx 2021/09/16 v1.2d Enhanced LaTeX Graphics (DPC,SPQR)
|
||||
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/graphics/keyval.sty
|
||||
Package: keyval 2022/05/29 v1.15 key=value parser (DPC)
|
||||
\KV@toks@=\toks24
|
||||
)
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/graphics/graphics.sty
|
||||
Package: graphics 2022/03/10 v1.4e Standard LaTeX Graphics (DPC,SPQR)
|
||||
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/graphics/trig.sty
|
||||
Package: trig 2021/08/11 v1.11 sin cos tan (DPC)
|
||||
)
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/graphics-cfg/graphics.cfg
|
||||
File: graphics.cfg 2016/06/04 v1.11 sample graphics configuration
|
||||
)
|
||||
Package graphics Info: Driver file: pdftex.def on input line 107.
|
||||
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/graphics-def/pdftex.def
|
||||
File: pdftex.def 2022/09/22 v1.2b Graphics/color driver for pdftex
|
||||
))
|
||||
\Gin@req@height=\dimen150
|
||||
\Gin@req@width=\dimen151
|
||||
)
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/base/newlfont.sty
|
||||
Package: newlfont 1998/08/17 v2.2m Standard LaTeX package
|
||||
LaTeX Info: Redefining \em on input line 57.
|
||||
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/base/latexsym.sty
|
||||
Package: latexsym 1998/08/17 v2.2e Standard LaTeX package (lasy symbols)
|
||||
\symlasy=\mathgroup6
|
||||
LaTeX Font Info: Overwriting symbol font `lasy' in version `bold'
|
||||
(Font) U/lasy/m/n --> U/lasy/b/n on input line 52.
|
||||
))
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/amscls/amsthm.sty
|
||||
Package: amsthm 2020/05/29 v2.20.6
|
||||
\thm@style=\toks25
|
||||
\thm@bodyfont=\toks26
|
||||
\thm@headfont=\toks27
|
||||
\thm@notefont=\toks28
|
||||
\thm@headpunct=\toks29
|
||||
\thm@preskip=\skip62
|
||||
\thm@postskip=\skip63
|
||||
\thm@headsep=\skip64
|
||||
\dth@everypar=\toks30
|
||||
)
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/mathtools/mathtools.sty
|
||||
Package: mathtools 2024/03/11 v1.30 mathematical typesetting tools
|
||||
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/tools/calc.sty
|
||||
Package: calc 2023/07/08 v4.3 Infix arithmetic (KKT,FJ)
|
||||
\calc@Acount=\count274
|
||||
\calc@Bcount=\count275
|
||||
\calc@Adimen=\dimen152
|
||||
\calc@Bdimen=\dimen153
|
||||
\calc@Askip=\skip65
|
||||
\calc@Bskip=\skip66
|
||||
LaTeX Info: Redefining \setlength on input line 80.
|
||||
LaTeX Info: Redefining \addtolength on input line 81.
|
||||
\calc@Ccount=\count276
|
||||
\calc@Cskip=\skip67
|
||||
)
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/mathtools/mhsetup.sty
|
||||
Package: mhsetup 2021/03/18 v1.4 programming setup (MH)
|
||||
)
|
||||
\g_MT_multlinerow_int=\count277
|
||||
\l_MT_multwidth_dim=\dimen154
|
||||
\origjot=\skip68
|
||||
\l_MT_shortvdotswithinadjustabove_dim=\dimen155
|
||||
\l_MT_shortvdotswithinadjustbelow_dim=\dimen156
|
||||
\l_MT_above_intertext_sep=\dimen157
|
||||
\l_MT_below_intertext_sep=\dimen158
|
||||
\l_MT_above_shortintertext_sep=\dimen159
|
||||
\l_MT_below_shortintertext_sep=\dimen160
|
||||
\xmathstrut@box=\box53
|
||||
\xmathstrut@dim=\dimen161
|
||||
)
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/units/nicefrac.sty
|
||||
Package: nicefrac 1998/08/04 v0.9b Nice fractions
|
||||
\L@UnitsRaiseDisplaystyle=\skip69
|
||||
\L@UnitsRaiseTextstyle=\skip70
|
||||
\L@UnitsRaiseScriptstyle=\skip71
|
||||
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/base/ifthen.sty
|
||||
Package: ifthen 2022/04/13 v1.1d Standard LaTeX ifthen package (DPC)
|
||||
))
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/epstopdf-pkg/epstopdf.sty
|
||||
Package: epstopdf 2020-01-24 v2.11 Conversion with epstopdf on the fly (HO)
|
||||
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/generic/infwarerr/infwarerr.sty
|
||||
Package: infwarerr 2019/12/03 v1.5 Providing info/warning/error messages (HO)
|
||||
)
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/grfext/grfext.sty
|
||||
Package: grfext 2019/12/03 v1.3 Manage graphics extensions (HO)
|
||||
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/generic/kvdefinekeys/kvdefinekeys.sty
|
||||
Package: kvdefinekeys 2019-12-19 v1.6 Define keys (HO)
|
||||
))
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/kvoptions/kvoptions.sty
|
||||
Package: kvoptions 2022-06-15 v3.15 Key value format for package options (HO)
|
||||
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/generic/ltxcmds/ltxcmds.sty
|
||||
Package: ltxcmds 2023-12-04 v1.26 LaTeX kernel commands for general use (HO)
|
||||
)
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/kvsetkeys/kvsetkeys.sty
|
||||
Package: kvsetkeys 2022-10-05 v1.19 Key value parser (HO)
|
||||
))
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/generic/pdftexcmds/pdftexcmds.sty
|
||||
Package: pdftexcmds 2020-06-27 v0.33 Utility functions of pdfTeX for LuaTeX (HO
|
||||
)
|
||||
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/generic/iftex/iftex.sty
|
||||
Package: iftex 2022/02/03 v1.0f TeX engine tests
|
||||
)
|
||||
Package pdftexcmds Info: \pdf@primitive is available.
|
||||
Package pdftexcmds Info: \pdf@ifprimitive is available.
|
||||
Package pdftexcmds Info: \pdfdraftmode found.
|
||||
)
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/epstopdf-pkg/epstopdf-base.sty
|
||||
Package: epstopdf-base 2020-01-24 v2.11 Base part for package epstopdf
|
||||
Package epstopdf-base Info: Redefining graphics rule for `.eps' on input line 4
|
||||
85.
|
||||
Package grfext Info: Graphics extension search list:
|
||||
(grfext) [.pdf,.png,.jpg,.mps,.jpeg,.jbig2,.jb2,.PDF,.PNG,.JPG,.JPE
|
||||
G,.JBIG2,.JB2,.eps]
|
||||
(grfext) \AppendGraphicsExtensions on input line 504.
|
||||
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/latexconfig/epstopdf-sys.cfg
|
||||
File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv
|
||||
e
|
||||
)))
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/caption/caption.sty
|
||||
Package: caption 2023/08/05 v3.6o Customizing captions (AR)
|
||||
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/caption/caption3.sty
|
||||
Package: caption3 2023/07/31 v2.4d caption3 kernel (AR)
|
||||
\caption@tempdima=\dimen162
|
||||
\captionmargin=\dimen163
|
||||
\caption@leftmargin=\dimen164
|
||||
\caption@rightmargin=\dimen165
|
||||
\caption@width=\dimen166
|
||||
\caption@indent=\dimen167
|
||||
\caption@parindent=\dimen168
|
||||
\caption@hangindent=\dimen169
|
||||
Package caption Info: Standard document class detected.
|
||||
)
|
||||
\c@caption@flags=\count278
|
||||
\c@continuedfloat=\count279
|
||||
)
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/caption/subcaption.sty
|
||||
Package: subcaption 2023/07/28 v1.6b Sub-captions (AR)
|
||||
Package caption Info: New subtype `subfigure' on input line 238.
|
||||
\c@subfigure=\count280
|
||||
Package caption Info: New subtype `subtable' on input line 238.
|
||||
\c@subtable=\count281
|
||||
)
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/graphics/color.sty
|
||||
Package: color 2022/01/06 v1.3d Standard LaTeX Color (DPC)
|
||||
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/graphics-cfg/color.cfg
|
||||
File: color.cfg 2016/01/02 v1.6 sample color configuration
|
||||
)
|
||||
Package color Info: Driver file: pdftex.def on input line 149.
|
||||
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/graphics/mathcolor.ltx))
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/comment/comment.sty
|
||||
\CommentStream=\write3
|
||||
Excluding comment 'comment')
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/geometry/geometry.sty
|
||||
Package: geometry 2020/01/02 v5.9 Page Geometry
|
||||
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/generic/iftex/ifvtex.sty
|
||||
Package: ifvtex 2019/10/25 v1.7 ifvtex legacy package. Use iftex instead.
|
||||
)
|
||||
\Gm@cnth=\count282
|
||||
\Gm@cntv=\count283
|
||||
\c@Gm@tempcnt=\count284
|
||||
\Gm@bindingoffset=\dimen170
|
||||
\Gm@wd@mp=\dimen171
|
||||
\Gm@odd@mp=\dimen172
|
||||
\Gm@even@mp=\dimen173
|
||||
\Gm@layoutwidth=\dimen174
|
||||
\Gm@layoutheight=\dimen175
|
||||
\Gm@layouthoffset=\dimen176
|
||||
\Gm@layoutvoffset=\dimen177
|
||||
\Gm@dimlist=\toks31
|
||||
)
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/hyperref/hyperref.sty
|
||||
Package: hyperref 2024-01-20 v7.01h Hypertext links for LaTeX
|
||||
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/generic/pdfescape/pdfescape.sty
|
||||
Package: pdfescape 2019/12/09 v1.15 Implements pdfTeX's escape features (HO)
|
||||
)
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/hycolor/hycolor.sty
|
||||
Package: hycolor 2020-01-27 v1.10 Color options for hyperref/bookmark (HO)
|
||||
)
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/auxhook/auxhook.sty
|
||||
Package: auxhook 2019-12-17 v1.6 Hooks for auxiliary files (HO)
|
||||
)
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/hyperref/nameref.sty
|
||||
Package: nameref 2023-11-26 v2.56 Cross-referencing by name of section
|
||||
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/refcount/refcount.sty
|
||||
Package: refcount 2019/12/15 v3.6 Data extraction from label references (HO)
|
||||
)
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/generic/gettitlestring/gettitlestring.sty
|
||||
Package: gettitlestring 2019/12/15 v1.6 Cleanup title references (HO)
|
||||
)
|
||||
\c@section@level=\count285
|
||||
)
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/etoolbox/etoolbox.sty
|
||||
Package: etoolbox 2020/10/05 v2.5k e-TeX tools for LaTeX (JAW)
|
||||
\etb@tempcnta=\count286
|
||||
)
|
||||
\@linkdim=\dimen178
|
||||
\Hy@linkcounter=\count287
|
||||
\Hy@pagecounter=\count288
|
||||
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/hyperref/pd1enc.def
|
||||
File: pd1enc.def 2024-01-20 v7.01h Hyperref: PDFDocEncoding definition (HO)
|
||||
Now handling font encoding PD1 ...
|
||||
... no UTF-8 mapping file for font encoding PD1
|
||||
)
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/generic/intcalc/intcalc.sty
|
||||
Package: intcalc 2019/12/15 v1.3 Expandable calculations with integers (HO)
|
||||
)
|
||||
\Hy@SavedSpaceFactor=\count289
|
||||
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/hyperref/puenc.def
|
||||
File: puenc.def 2024-01-20 v7.01h Hyperref: PDF Unicode definition (HO)
|
||||
Now handling font encoding PU ...
|
||||
... no UTF-8 mapping file for font encoding PU
|
||||
)
|
||||
Package hyperref Info: Hyper figures OFF on input line 4179.
|
||||
Package hyperref Info: Link nesting OFF on input line 4184.
|
||||
Package hyperref Info: Hyper index ON on input line 4187.
|
||||
Package hyperref Info: Plain pages OFF on input line 4194.
|
||||
Package hyperref Info: Backreferencing OFF on input line 4199.
|
||||
Package hyperref Info: Implicit mode ON; LaTeX internals redefined.
|
||||
Package hyperref Info: Bookmarks ON on input line 4446.
|
||||
\c@Hy@tempcnt=\count290
|
||||
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/url/url.sty
|
||||
\Urlmuskip=\muskip17
|
||||
Package: url 2013/09/16 ver 3.4 Verb mode for urls, etc.
|
||||
)
|
||||
LaTeX Info: Redefining \url on input line 4784.
|
||||
\XeTeXLinkMargin=\dimen179
|
||||
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/generic/bitset/bitset.sty
|
||||
Package: bitset 2019/12/09 v1.3 Handle bit-vector datatype (HO)
|
||||
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/generic/bigintcalc/bigintcalc.sty
|
||||
Package: bigintcalc 2019/12/15 v1.5 Expandable calculations on big integers (HO
|
||||
)
|
||||
))
|
||||
\Fld@menulength=\count291
|
||||
\Field@Width=\dimen180
|
||||
\Fld@charsize=\dimen181
|
||||
Package hyperref Info: Hyper figures OFF on input line 6063.
|
||||
Package hyperref Info: Link nesting OFF on input line 6068.
|
||||
Package hyperref Info: Hyper index ON on input line 6071.
|
||||
Package hyperref Info: backreferencing OFF on input line 6078.
|
||||
Package hyperref Info: Link coloring OFF on input line 6083.
|
||||
Package hyperref Info: Link coloring with OCG OFF on input line 6088.
|
||||
Package hyperref Info: PDF/A mode OFF on input line 6093.
|
||||
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/base/atbegshi-ltx.sty
|
||||
Package: atbegshi-ltx 2021/01/10 v1.0c Emulation of the original atbegshi
|
||||
package with kernel methods
|
||||
)
|
||||
\Hy@abspage=\count292
|
||||
\c@Item=\count293
|
||||
\c@Hfootnote=\count294
|
||||
)
|
||||
Package hyperref Info: Driver (autodetected): hpdftex.
|
||||
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/hyperref/hpdftex.def
|
||||
File: hpdftex.def 2024-01-20 v7.01h Hyperref driver for pdfTeX
|
||||
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/base/atveryend-ltx.sty
|
||||
Package: atveryend-ltx 2020/08/19 v1.0a Emulation of the original atveryend pac
|
||||
kage
|
||||
with kernel methods
|
||||
)
|
||||
\Fld@listcount=\count295
|
||||
\c@bookmark@seq@number=\count296
|
||||
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/rerunfilecheck/rerunfilecheck.sty
|
||||
Package: rerunfilecheck 2022-07-10 v1.10 Rerun checks for auxiliary files (HO)
|
||||
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/generic/uniquecounter/uniquecounter.sty
|
||||
Package: uniquecounter 2019/12/15 v1.4 Provide unlimited unique counter (HO)
|
||||
)
|
||||
Package uniquecounter Info: New unique counter `rerunfilecheck' on input line 2
|
||||
85.
|
||||
)
|
||||
\Hy@SectionHShift=\skip72
|
||||
)
|
||||
\c@theorem=\count297
|
||||
\c@notation=\count298
|
||||
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/l3backend/l3backend-pdftex.def
|
||||
File: l3backend-pdftex.def 2024-02-20 L3 backend support: PDF output (pdfTeX)
|
||||
\l__color_backend_stack_int=\count299
|
||||
\l__pdf_internal_box=\box54
|
||||
) (./nota5.aux)
|
||||
\openout1 = `nota5.aux'.
|
||||
|
||||
LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 52.
|
||||
LaTeX Font Info: ... okay on input line 52.
|
||||
LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 52.
|
||||
LaTeX Font Info: ... okay on input line 52.
|
||||
LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 52.
|
||||
LaTeX Font Info: ... okay on input line 52.
|
||||
LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 52.
|
||||
LaTeX Font Info: ... okay on input line 52.
|
||||
LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 52.
|
||||
LaTeX Font Info: ... okay on input line 52.
|
||||
LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 52.
|
||||
LaTeX Font Info: ... okay on input line 52.
|
||||
LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 52.
|
||||
LaTeX Font Info: ... okay on input line 52.
|
||||
LaTeX Font Info: Checking defaults for PD1/pdf/m/n on input line 52.
|
||||
LaTeX Font Info: ... okay on input line 52.
|
||||
LaTeX Font Info: Checking defaults for PU/pdf/m/n on input line 52.
|
||||
LaTeX Font Info: ... okay on input line 52.
|
||||
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/context/base/mkii/supp-pdf.mkii
|
||||
[Loading MPS to PDF converter (version 2006.09.02).]
|
||||
\scratchcounter=\count300
|
||||
\scratchdimen=\dimen182
|
||||
\scratchbox=\box55
|
||||
\nofMPsegments=\count301
|
||||
\nofMParguments=\count302
|
||||
\everyMPshowfont=\toks32
|
||||
\MPscratchCnt=\count303
|
||||
\MPscratchDim=\dimen183
|
||||
\MPnumerator=\count304
|
||||
\makeMPintoPDFobject=\count305
|
||||
\everyMPtoPDFconversion=\toks33
|
||||
)
|
||||
Package caption Info: Begin \AtBeginDocument code.
|
||||
Package caption Info: hyperref package is loaded.
|
||||
Package caption Info: End \AtBeginDocument code.
|
||||
|
||||
*geometry* driver: auto-detecting
|
||||
*geometry* detected driver: pdftex
|
||||
*geometry* verbose mode - [ preamble ] result:
|
||||
* driver: pdftex
|
||||
* paper: a4paper
|
||||
* layout: <same size as paper>
|
||||
* layoutoffset:(h,v)=(0.0pt,0.0pt)
|
||||
* modes:
|
||||
* h-part:(L,W,R)=(56.9055pt, 483.69687pt, 56.9055pt)
|
||||
* v-part:(T,H,B)=(56.9055pt, 731.23584pt, 56.9055pt)
|
||||
* \paperwidth=597.50787pt
|
||||
* \paperheight=845.04684pt
|
||||
* \textwidth=483.69687pt
|
||||
* \textheight=731.23584pt
|
||||
* \oddsidemargin=-15.36449pt
|
||||
* \evensidemargin=-15.36449pt
|
||||
* \topmargin=-52.36449pt
|
||||
* \headheight=12.0pt
|
||||
* \headsep=25.0pt
|
||||
* \topskip=10.0pt
|
||||
* \footskip=30.0pt
|
||||
* \marginparwidth=57.0pt
|
||||
* \marginparsep=11.0pt
|
||||
* \columnsep=10.0pt
|
||||
* \skip\footins=9.0pt plus 4.0pt minus 2.0pt
|
||||
* \hoffset=0.0pt
|
||||
* \voffset=0.0pt
|
||||
* \mag=1000
|
||||
* \@twocolumnfalse
|
||||
* \@twosidefalse
|
||||
* \@mparswitchfalse
|
||||
* \@reversemarginfalse
|
||||
* (1in=72.27pt=25.4mm, 1cm=28.453pt)
|
||||
|
||||
Package hyperref Info: Link coloring OFF on input line 52.
|
||||
(./nota5.out) (./nota5.out)
|
||||
\@outlinefile=\write4
|
||||
\openout4 = `nota5.out'.
|
||||
|
||||
LaTeX Font Info: Trying to load font information for U+msa on input line 54.
|
||||
|
||||
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/amsfonts/umsa.fd
|
||||
File: umsa.fd 2013/01/14 v3.01 AMS symbols A
|
||||
)
|
||||
LaTeX Font Info: Trying to load font information for U+msb on input line 54.
|
||||
|
||||
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/amsfonts/umsb.fd
|
||||
File: umsb.fd 2013/01/14 v3.01 AMS symbols B
|
||||
)
|
||||
LaTeX Font Info: Trying to load font information for U+lasy on input line 54
|
||||
.
|
||||
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/base/ulasy.fd
|
||||
File: ulasy.fd 1998/08/17 v2.2e LaTeX symbol font definitions
|
||||
) [1
|
||||
|
||||
{/nix/store/mhdx3hkmpns8i6czk9ygf2yc7wxlkpy0-texlive-combined-full-2023-final/s
|
||||
hare/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] Excluding 'comment' comment
|
||||
. Excluding 'comment' comment.
|
||||
Overfull \hbox (82.38083pt too wide) detected at line 184
|
||||
[]
|
||||
[]
|
||||
|
||||
[2]
|
||||
LaTeX Font Info: Trying to load font information for U+dsrom on input line 2
|
||||
24.
|
||||
|
||||
(/nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-t
|
||||
exmfdist/tex/latex/doublestroke/Udsrom.fd
|
||||
File: Udsrom.fd 1995/08/01 v0.1 Double stroke roman font definitions
|
||||
)
|
||||
Overfull \hbox (29.63692pt too wide) detected at line 235
|
||||
[]
|
||||
[]
|
||||
|
||||
[3] Excluding 'comment' comment. [4] [5] [6] (./nota5.aux)
|
||||
***********
|
||||
LaTeX2e <2023-11-01> patch level 1
|
||||
L3 programming layer <2024-02-20>
|
||||
***********
|
||||
Package rerunfilecheck Info: File `nota5.out' has not changed.
|
||||
(rerunfilecheck) Checksum: D41D8CD98F00B204E9800998ECF8427E;0.
|
||||
)
|
||||
Here is how much of TeX's memory you used:
|
||||
12850 strings out of 474116
|
||||
214715 string characters out of 5717523
|
||||
1952187 words of memory out of 5000000
|
||||
34990 multiletter control sequences out of 15000+600000
|
||||
565113 words of font info for 69 fonts, out of 8000000 for 9000
|
||||
1141 hyphenation exceptions out of 8191
|
||||
63i,16n,67p,849b,509s stack positions out of 10000i,1000n,20000p,200000b,200000s
|
||||
</nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final
|
||||
-texmfdist/fonts/type1/public/amsfonts/cm/cmbx10.pfb></nix/store/bj92vd0x3z4f1y
|
||||
idlv488fis5rx252nk-texlive-combined-full-2023-final-texmfdist/fonts/type1/publi
|
||||
c/amsfonts/cm/cmbx12.pfb></nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-c
|
||||
ombined-full-2023-final-texmfdist/fonts/type1/public/amsfonts/cm/cmex10.pfb></n
|
||||
ix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-texm
|
||||
fdist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></nix/store/bj92vd0x3z4f1yidlv4
|
||||
88fis5rx252nk-texlive-combined-full-2023-final-texmfdist/fonts/type1/public/ams
|
||||
fonts/cm/cmmi5.pfb></nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combine
|
||||
d-full-2023-final-texmfdist/fonts/type1/public/amsfonts/cm/cmmi6.pfb></nix/stor
|
||||
e/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-texmfdist/f
|
||||
onts/type1/public/amsfonts/cm/cmmi7.pfb></nix/store/bj92vd0x3z4f1yidlv488fis5rx
|
||||
252nk-texlive-combined-full-2023-final-texmfdist/fonts/type1/public/amsfonts/cm
|
||||
/cmmi8.pfb></nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2
|
||||
023-final-texmfdist/fonts/type1/public/amsfonts/cm/cmr10.pfb></nix/store/bj92vd
|
||||
0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-texmfdist/fonts/typ
|
||||
e1/public/amsfonts/cm/cmr5.pfb></nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-tex
|
||||
live-combined-full-2023-final-texmfdist/fonts/type1/public/amsfonts/cm/cmr6.pfb
|
||||
></nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-
|
||||
texmfdist/fonts/type1/public/amsfonts/cm/cmr7.pfb></nix/store/bj92vd0x3z4f1yidl
|
||||
v488fis5rx252nk-texlive-combined-full-2023-final-texmfdist/fonts/type1/public/a
|
||||
msfonts/cm/cmr8.pfb></nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combin
|
||||
ed-full-2023-final-texmfdist/fonts/type1/public/amsfonts/cm/cmsy10.pfb></nix/st
|
||||
ore/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-texmfdist
|
||||
/fonts/type1/public/amsfonts/cm/cmsy5.pfb></nix/store/bj92vd0x3z4f1yidlv488fis5
|
||||
rx252nk-texlive-combined-full-2023-final-texmfdist/fonts/type1/public/amsfonts/
|
||||
cm/cmsy6.pfb></nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full
|
||||
-2023-final-texmfdist/fonts/type1/public/amsfonts/cm/cmsy7.pfb></nix/store/bj92
|
||||
vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-texmfdist/fonts/t
|
||||
ype1/public/amsfonts/cm/cmsy8.pfb></nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-
|
||||
texlive-combined-full-2023-final-texmfdist/fonts/type1/public/amsfonts/cm/cmti1
|
||||
0.pfb></nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023-f
|
||||
inal-texmfdist/fonts/type1/public/amsfonts/cm/cmtt10.pfb></nix/store/bj92vd0x3z
|
||||
4f1yidlv488fis5rx252nk-texlive-combined-full-2023-final-texmfdist/fonts/type1/p
|
||||
ublic/doublestroke/dsrom10.pfb></nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-tex
|
||||
live-combined-full-2023-final-texmfdist/fonts/type1/public/amsfonts/symbols/msb
|
||||
m10.pfb></nix/store/bj92vd0x3z4f1yidlv488fis5rx252nk-texlive-combined-full-2023
|
||||
-final-texmfdist/fonts/type1/public/amsfonts/symbols/msbm7.pfb>
|
||||
Output written on nota5.pdf (6 pages, 281177 bytes).
|
||||
PDF statistics:
|
||||
216 PDF objects out of 1000 (max. 8388607)
|
||||
145 compressed objects within 2 object streams
|
||||
31 named destinations out of 1000 (max. 500000)
|
||||
1 words of extra memory for PDF output out of 10000 (max. 10000000)
|
||||
|
||||
BIN
nota5.synctex.gz
Normal file
BIN
nota5.synctex.gz
Normal file
Binary file not shown.
465
nota5.tex
Normal file
465
nota5.tex
Normal file
@ -0,0 +1,465 @@
|
||||
\documentclass[10pt,a4paper]{article}
|
||||
\usepackage[utf8]{inputenc}
|
||||
\usepackage{amsmath}
|
||||
\usepackage{amsfonts}
|
||||
\usepackage{amssymb}
|
||||
\usepackage{dsfont}
|
||||
\usepackage{fancyhdr}
|
||||
\usepackage{indentfirst}
|
||||
\usepackage{graphicx}
|
||||
\usepackage{newlfont}
|
||||
\usepackage{amssymb}
|
||||
\usepackage{amsmath}
|
||||
\usepackage{latexsym}
|
||||
\usepackage{amsthm}
|
||||
\usepackage{mathtools}
|
||||
\usepackage{nicefrac}
|
||||
\usepackage{epstopdf}
|
||||
\usepackage{caption}
|
||||
\usepackage{subcaption}
|
||||
\usepackage{color}
|
||||
\usepackage{dsfont}
|
||||
\usepackage{comment}
|
||||
\usepackage[left=2cm,right=2cm,top=2cm,bottom=2cm]{geometry}
|
||||
\usepackage{hyperref}
|
||||
%\hypersetup{
|
||||
% colorlinks=true,
|
||||
% linkcolor=blue,
|
||||
% filecolor=magenta,
|
||||
% urlcolor=cyan,
|
||||
% pdftitle={Overleaf Example},
|
||||
% pdfpagemode=FullScreen,
|
||||
% }
|
||||
|
||||
|
||||
\newtheorem{theorem}{Theorem}
|
||||
\newtheorem{proposition}[theorem]{Proposition}
|
||||
\newtheorem{corollary}[theorem]{Corollary}
|
||||
\newtheorem{lemma}[theorem]{Lemma}
|
||||
\newtheorem{definition}[theorem]{Definition}
|
||||
\newtheorem{assumption}[theorem]{Assumption}
|
||||
|
||||
|
||||
\newtheorem{remark}[theorem]{Remark}
|
||||
\newtheorem{example}[theorem]{Example}
|
||||
\newtheorem{notation}{Notation}
|
||||
|
||||
\def \R {\mathbb{R}}
|
||||
\def \N {\mathbb{N}}
|
||||
\def \d {\mathrm{d}}
|
||||
\newcommand{\red}[1]{\textcolor{red}{#1}}
|
||||
|
||||
\begin{document}
|
||||
To start off we define the notation for the objects we will work on
|
||||
$$
|
||||
dX_t = B(t,X_t,\mu_{X_t})dt + \Sigma(t,X_t,\mu_{X_t})dW_t,\qquad X_0\sim\mu_0.
|
||||
$$
|
||||
Given the flow of marginals $\mu_t$ we can fix the coefficients and linearize the SDE with the linearized coefficients $B^\mu(t,x)$ and $\Sigma^\mu(t,x)$. Using this, we may define the infinitesimal generator
|
||||
$$
|
||||
\mathcal{A}_t^\mu=\frac{1}{2}\sum_{i,j=1}^N c^\mu_{ij}(t,x)\partial_{x_ix_j}+\sum_{i=1}^NB^\mu_i(t,x)\partial_{x_i}.
|
||||
$$
|
||||
Given this operator, under reasonable assumptions we have the existance of $p(s,x;t,y)$ fundamental solution of
|
||||
\begin{align*}
|
||||
(\partial_s+\mathcal{A}^\mu_s)p^\mu(s,x;t,y)&=0,\\
|
||||
(\partial_t-(\mathcal{A}^\mu_t)^*)p^\mu(s,x;t,y)&=0.
|
||||
\end{align*}
|
||||
Having the transition density $p^\mu$ we may define the forward translation operator
|
||||
$$
|
||||
U_\mu^{t,s}\phi(y)=\int p^\mu(s,x;t,y)\phi(x)dx,
|
||||
$$
|
||||
whose definition may be easily extended to $\mathcal{P}^2(\R^N)$ due to the gaussian estimates on $p$ (which are uniform over the choice of $\mu_t$):
|
||||
$$
|
||||
U_\mu^{t,s}u(y)=\int p^\mu(s,x;t,y)u(dx),\qquad u\in\mathcal{P}^2(\R^N).
|
||||
$$
|
||||
Via this operator we may define
|
||||
$$
|
||||
u^\mu_t(x)=U_\mu^{t,0}\mu_0,
|
||||
$$
|
||||
the density of the solution of the linearized SDE via the marginal flow $(\mu_t)_{t\in[0,T]}$ with initial law $\mu_0$.
|
||||
Via this density we are able to construct a new flow of marginals (the one of the solution of the linearized SDE via marginal flow $\mu_t$ and initial law $\mu_0$):
|
||||
$$
|
||||
\mathcal{L}^\mu_t(dy) = u^\mu_t(y)dy = \left(\int p^\mu(0,x;t,y)\mu_0(dx)\right)dy.
|
||||
$$
|
||||
\red{
|
||||
Here we will briefly state what \cite{Kolokoltsov} does to study the contraction properties on $L^1$ norm of $u^\mu$.
|
||||
To start we need the identity\footnote{In my calculations I get the adjoint operator $(\mathcal{A}_s)^*$ but in Kolokoltsov's paper there is the backward one.} (28) of \cite{Kolokoltsov}:
|
||||
\begin{equation}\label{e1}
|
||||
U^{t,0}_\mu - U^{t,0}_\nu = \int_0^t\frac{d}{ds}U^{t,s}_\nu U^{s,0}_\mu ds = \int_0^t U^{t,s}_\nu((\mathcal{A}^{\mu}_s)^*-(\mathcal{A}^{\nu}_s)^*)U^{s,0}_\mu ds.
|
||||
\end{equation}
|
||||
Then we need to observe that
|
||||
\begin{align}
|
||||
||U^{t,s} f||_{L^1} &= \int \left| \int p(s,x;t,y)f(x)dx\right|dy\nonumber\\
|
||||
&\stackrel{Gaussian\ estimates}{\leq}C\int\int |f(x+y)|\Gamma^+(y)dy dx = C||f||_{L^1}.\label{e2}
|
||||
\end{align}
|
||||
Also observe that
|
||||
\begin{align*}
|
||||
||(\mathcal{A}^\mu_s-\mathcal{A}^\nu_s)f||_{L^1}\leq C\sup_{t,x}\left(|c^\mu(t,x)-c^\nu(t,x)|+|B^\mu(t,x)-B^\nu(t,x)|\right)||f||_{W^{1,2}},
|
||||
\end{align*}
|
||||
and by the fact that the operator $U^{s,0}_\mu$ is a bounded operator in $W^{1,2}$, check footnote\footnote{I am unsure of this passage, we don't actually need the $0$-derivative and due to gaussian estimates we should get something similar to $||\partial U\ f||_{L^1}\leq s^{-1/2}||f||_{L^1}$ and $||\partial^2 U\ f||_{L^1}\leq s^{-1}||f||_{L^1}$ where the second derivative is no longer integrable wrt $s$. I think something similar to (24.2.7) in dispense is happening.} we get
|
||||
\begin{equation}\label{e3}
|
||||
||U^{s,0}_\mu f||_{W^{1,2}}\leq C\cdot s^{-1/2}||f||_{W^{1,2}}.
|
||||
\end{equation}
|
||||
By \eqref{e1}, \eqref{e2} and \eqref{e3} we get
|
||||
\begin{align}\nonumber
|
||||
||u^\mu_t-u^\nu_t||_{L^1}=||U^{t,0}_\mu\mu_0-U^{t,0}_\mu\mu_0||_{L^1}&\leq C\int_0^t s^{-1/2}ds ||\mu_0||_{L^1}\sup_{t,x}\left(|c^\mu(t,x)-c^\nu(t,x)|+|B^\mu(t,x)-B^\nu(t,x)|\right)\\
|
||||
&\leq C\sqrt{t} ||\mu_0||_{L^1}\sup_{t,x}\left(|c^\mu(t,x)-c^\nu(t,x)|+|B^\mu(t,x)-B^\nu(t,x)|\right).\label{e4}
|
||||
\end{align}
|
||||
Now that we briefly stated the ideas of \cite{Kolokoltsov} we can begin.}
|
||||
|
||||
To leave as many doors open as possible we first define
|
||||
|
||||
$$
|
||||
I(f) := \int f(x)(u^\mu_t(x)-u^\nu_t(x))dx,
|
||||
$$
|
||||
|
||||
which at the moment may be seen as an indicator of closeness between the two densities, at a later moment we will take the $\sup$ for $f$ in some bounded functional space like the bounded Holder functions or the bounded functions.
|
||||
|
||||
%Now we will change a bit what \cite{Kolokoltsov} did to work in the Wasserstein framework. To start off we apply the duality formula for the Kantorovich–Rubinstein distance (Remark 6.5 of \cite{Villani}):
|
||||
%\begin{align*}
|
||||
%W^{(1)}(\mathcal{L}^\mu_t,\mathcal{L}^\nu_t) = \sup_{||f||_{Lip}\leq1}\left(\int f(x)(u^\mu_t(x)-u^\nu_t(x))dx\right)=:\sup_{||f||_{Lip}\leq1}I(f).
|
||||
%\end{align*}
|
||||
Now we apply this useful trick: if we write explicitly the definition of $u^\mu$ as the evaluation of $p^\mu$ on the distribution $\mu_0$ in $I(f)$ we can change the order of integration to evaluate $p^\mu$ on the regular distribution $f(x)dx$, this is useful because it switches the operators in Kolokoltsov's formula \eqref{e1} from being forward to being backwards while changing only marginally everything else. If we define the backward propagator operator
|
||||
\begin{equation}\label{e4-1}
|
||||
V^{s,t}_\mu g(y):=\int p^\mu(s,y;t,x)g(x)dx,
|
||||
\end{equation}
|
||||
we can expand $I(f)$ this way
|
||||
\begin{align*}
|
||||
I(f) &= \int\int f(x)(p^\mu-p^\nu)(0,y;t,x)\mu_0(dy)dx = \mu_0\left(\int f(x)(p^\mu-p^\nu)(0,\cdot;t,x)dx\right)\\
|
||||
&=\mu_0\left(\int_0^t\frac{d}{ds}\left(\int\int p^\mu(0,\cdot;s,z)p^\nu(s,z;t,x)f(x)dxdz\right)ds\right)\\
|
||||
&=\mu_0\left(\int_0^t \int\int\partial_{t_2}p^\mu(0,\cdot;s,z)p^\nu(s,z;t,x)f(x)dxdz + \int\int p^\mu(0,\cdot;s,z)\partial_{t_1}p^\nu(s,z;t,x)f(x)dxdz ds\right)
|
||||
\intertext{by the fact that $p$ is the fundamental solution for both the forward and backward PDEs}
|
||||
&=\mu_0\left(\int_0^t \int\int(\mathcal{A}_s^\mu)^* p^\mu(0,\cdot;s,z)p^\nu(s,z;t,x)f(x)dxdz - \int\int p^\mu(0,\cdot;s,z)\mathcal{A}_s^\nu p^\nu(s,z;t,x)f(x)dxdz ds\right)\\
|
||||
&=\mu_0\left(\int_0^t \int(\mathcal{A}_s^\mu)^* p^\mu(0,\cdot;s,z)V_\nu^{s,t}f(z)dz - \int (\mathcal{A}_s^\nu)^*p^\mu(0,\cdot;s,z) V_\nu^{s,t}f(z)dz ds\right)\\
|
||||
&=\mu_0\left(\int_0^t \int p^\mu(0,\cdot;s,z)\left(\mathcal{A}_s^\mu - \mathcal{A}_s^\nu\right)V_\nu^{s,t}f(z)dz ds\right)\\
|
||||
&=\mu_0\left(\int_0^t V_\mu^{0,s}\left(\mathcal{A}_s^\mu - \mathcal{A}_s^\nu\right)V_\nu^{s,t}f ds\right)=\int_0^t \mu_0\left(V_\mu^{0,s}\left(\mathcal{A}_s^\mu - \mathcal{A}_s^\nu\right)V_\nu^{s,t}f \right) ds.
|
||||
\end{align*}
|
||||
Using similar arguments it is possible to obtain also Kolokoltsov's formula \eqref{e1}:
|
||||
\begin{equation*}
|
||||
I(f) = \int_0^t f\left(U_\nu^{t,s}\left((\mathcal{A}_s^\mu)^* - (\mathcal{A}_s^\nu)^*\right)U_\mu^{s,0}\mu_0 \right) ds.
|
||||
\end{equation*}
|
||||
|
||||
Now we can try to estimate $I(f)$:
|
||||
\begin{align}\nonumber
|
||||
I(f)&\leq \int_0^t \left|\mu_0\left( V^{0,s}_\mu\left( \mathcal{A}^\mu_s - \mathcal{A}^\nu_s \right)V^{s,t}_\nu f\right)\right| ds,
|
||||
\intertext{since $\mu_0$ is a probability measure we can bound $\mu_0(g)$ with the uniform bound of $g$: $\mu_0(g)\leq |g|_\infty$:}
|
||||
&\leq \int_0^t \sup_x\left(V^{0,s}_\mu\left( \mathcal{A}^\mu_s - \mathcal{A}^\nu_s \right)V^{s,t}_\nu f(x)\right) ds.\label{f1}
|
||||
\end{align}
|
||||
We observe that due to Holder's inequality and the fact that $p^\mu(0,x;s,y)dy$ is a probability measure for any fixed $x$ we have uniformly in $x$
|
||||
$$
|
||||
|V^{0,s}_\mu g(x)|=\left| \int p^\mu(0,x;s,y)g(y)dy \right|\leq ||g||_{L^\infty}.
|
||||
$$
|
||||
Thus continuing from \eqref{f1} we have
|
||||
\begin{align*}
|
||||
I(f)&\leq \int_0^t ||\left( \mathcal{A}^\mu_s - \mathcal{A}^\nu_s \right)V^{s,t}_\nu f(x)||_{L^\infty} ds\leq \\
|
||||
&\leq \sup_{s,x}\left(|B^\mu(s,x)-B^\nu(s,x)|+|c^\mu(s,x)-c^\nu(s,x)|\right)\int_0^t ||\bigtriangledown V^{s,t}_\nu f||_{L^\infty} + ||Hess\ V^{s,t}_\nu f||_{L^\infty} ds.
|
||||
\end{align*}
|
||||
|
||||
\begin{theorem}\label{a1}
|
||||
\begin{comment}
|
||||
If $f$ is a Lipschitz function with $[f]_{Lip}\leq 1$ then
|
||||
\begin{enumerate}
|
||||
\item $||Hess\ V^{s,t}_\nu f||_{L^\infty}\leq \frac{C}{\sqrt{t-s}}$.
|
||||
\item $||\bigtriangledown V^{s,t}_\nu f||_{L^\infty}\leq C$.
|
||||
\end{enumerate}
|
||||
\end{comment}
|
||||
If $f$ is a $C^\alpha_B$ function with $||f||_{C^\alpha_B}\leq 1$ then
|
||||
\begin{enumerate}
|
||||
\item $||Hess\ V^{s,t}_\nu f||_{L^\infty}\leq \frac{C}{|t-s|^{1-\frac{\alpha}{2}}}$.
|
||||
\end{enumerate}
|
||||
\end{theorem}
|
||||
\begin{proof}
|
||||
\begin{comment}
|
||||
Let's first tackle the case with $f$ Lipschitz. Let $x\in\R^N$. We have
|
||||
\begin{align*}
|
||||
\left| \partial_{x_ix_j}V^{s,t}_\nu f (x) \right|&=\left| \int\partial_{x_i x_j}p^\nu(s,x;t,y)f(y)dy \right|\leq \left| \int\partial_{x_i x_j}p^\nu(s,x;t,y)(f(y)-f(x))dy \right|+ \left| \int\partial_{x_i x_j}p^\nu(s,x;t,y)dyf(x) \right|\\
|
||||
&\leq \int |\partial_{x_ix_j}p^\nu(s,x;t,y)||x-y|dy + \left| \partial_{x_ix_j}\underbrace{\int p^\nu(s,x;t,y) dy}_{=1} \right||f(x)|\\
|
||||
&\leq \frac{C}{|t-s|}\int \Gamma^+(t-s,x-y)|x-y|dy + 0\leq \frac{C}{\sqrt{t-s}}.
|
||||
\end{align*}
|
||||
The second inequality for the Lipschitz case is done in a completely analogous manner. Let's consider the Holder case
|
||||
\end{comment}
|
||||
\begin{align*}
|
||||
\left| \partial_{x_ix_j}V^{s,t}_\nu f (x) \right|&=\left| \int\partial_{x_i x_j}p^\nu(s,x;t,y)f(y)dy \right|\leq \left| \int\partial_{x_i x_j}p^\nu(s,x;t,y)(f(y)-f(e^{(t-s)B}x))dy \right|+ \left| \int\partial_{x_i x_j}p^\nu(s,x;t,y)dyf(e^{(t-s)B}x) \right|\\
|
||||
&\leq \int |\partial_{x_ix_j}p^\nu(s,x;t,y)||e^{(t-s)B}x-y|^\alpha_B dy + \left| \partial_{x_ix_j}\underbrace{\int p^\nu(s,x;t,y) dy}_{=1} \right||f(e^{(t-s)B}x)|\\
|
||||
&\leq \frac{C_{B,\alpha}}{|t-s|}\int \Gamma^+(t-s,x-y)|x-e^{-(t-s)B}y|^\alpha_B dy + 0\leq \frac{C_{B,\alpha}}{|t-s|^{1-\frac{\alpha}{2}}}.
|
||||
\end{align*}
|
||||
by Lemma (A.5) of \cite{LucePagliaPascu}.
|
||||
\end{proof}
|
||||
|
||||
We will now define
|
||||
$$
|
||||
d_{C^{\alpha}_B}(\mu,\nu)=\sup_{||f||_{C^{\alpha}_B}\leq 1}\left|\int f(x)\left(\mu(dx)-\nu(dx)\right) \right|
|
||||
$$
|
||||
the bounded anisotropic $\alpha$-Holder distance.
|
||||
\begin{theorem}
|
||||
The bounded anisotropic $\alpha$-Holder distance metrizes weak convergence of measures. More precisely given $(\mu_n)_{n\in\N}$ and $\mu$ probability measures
|
||||
$$
|
||||
d_{C^{\alpha}_B}(\mu_n,\mu)\rightarrow 0 \Leftrightarrow \mu_n\stackrel{d}{\rightarrow}\mu.
|
||||
$$
|
||||
\end{theorem}
|
||||
\begin{proof}
|
||||
The proof will be divided in two steps and is mostly taken from \url{https://sites.stat.washington.edu/jaw/COURSES/520s/522/HO.522.20/ch11c.pdf}
|
||||
|
||||
1) First we prove that
|
||||
$$
|
||||
\mu_n\stackrel{d}{\rightarrow}\mu \Leftrightarrow \int f d\mu_n\rightarrow\int f d\mu,\ \forall f\in C^{\alpha}_B.
|
||||
$$
|
||||
If $\mu_n\stackrel{d}{\rightarrow}\mu$ then equivalently $\int f\mu_n\rightarrow\int f\mu$ for any function $f\in bC$ which in particular means that it is true for any $f\in C^{\alpha}_B$. The converse is true because if $\int f\mu_n\rightarrow\int f\mu$ for any function $f\in C^{\alpha}_B$ then in particular it is true for any $f\in bLip$ which by Portmanteau's theorem implies weak convergence.
|
||||
|
||||
2) We will now prove that
|
||||
$$
|
||||
\int f d\mu_n\rightarrow\int fd\mu \ \forall f\in C^{\alpha}_B \Leftrightarrow d_{C^{\alpha}_B}(\mu_n,\mu)\rightarrow0.
|
||||
$$
|
||||
The easy implication is the right-to-left one: indeed by comparison theorem
|
||||
$$
|
||||
\lim_n \int f(x) \left( \mu_n(dx)-\mu(dx) \right)\leq \lim_n \sup_{||f||_{C^{\alpha}_B}\leq 1}\left|\int f(x) \left( \mu_n(dx)-\mu(dx) \right)\right| = \lim_n d_{C^{\alpha}_B}(\mu_n,\mu)\rightarrow0.
|
||||
$$
|
||||
The other way is more challenging, first by continuity from below of probability measures for any fixed $\epsilon>0$ there exists $K$ a compact set such that $\mu(K)>1-\epsilon$. Let $\mathcal{H}=\left\lbrace f\in C^{\alpha}_B\ |\ ||f||_{C^{\alpha}_B}\leq 1 \right\rbrace$, if we restrict each of these functions on $K$ we have that $\mathcal{H}\vert_K$ is totally bounded with respect to the $||\cdot||_\infty$ norm by Ascoli-Arzelà's theorem, in particular $\exists k$ finite and $f_1,\cdots f_k\in \mathcal{H}\vert_K$ such that for any $f\in\mathcal{H}$ $\exists j$ such that $\sup_K|f-f_j|\leq\epsilon$.
|
||||
|
||||
Now if we consider $d_B(x,y)=|x-y|_B$ and $K^\epsilon=\left\lbrace x\in\R^N\ |\ d_B(x,K)\leq\epsilon \right\rbrace$ and $f,f_j$ as before we have
|
||||
$$
|
||||
\sup_{x\in K^\epsilon}|f(x)-f_j(x)|\leq \sup_{x\in K^\epsilon}\left( |f(x)-f(y_x)| + |f(y_x)-f_j(y_x)| + |f_j(y_x)-f_j(x)|\right)\leq \sup_{x\in K^\epsilon}\left( 2\epsilon^\alpha + \epsilon \right)\leq C_\alpha\epsilon^\alpha.
|
||||
$$
|
||||
where $y_x$ is a point in $K$ such that $|x-y|_B<\epsilon$. $C_\alpha$ may be taken uniformly of $\epsilon$ as long as $\epsilon\leq 1$.
|
||||
|
||||
Let $g(x)=\max\left( 0, 1-\frac{d_B(x,K)}{\epsilon} \right)$, evidently $g\in bLip\subseteq C^{\alpha}_B$ and $\mathds{1}_K\leq g\leq \mathds{1}_{K^\epsilon}$. Thus by taking $n$ big enough we have by convergence against $C^{\alpha}_B$ functions that
|
||||
$$
|
||||
\mu_n(K^\epsilon)\geq\int g(x) \mu_n(dx) > 1 - 2\epsilon.
|
||||
$$
|
||||
Thus by taking $f\in\mathcal{H}$ and the associated $f_j$ we have
|
||||
\begin{align*}
|
||||
\left| \int f(x)\left(\mu_n(dx)-\mu(dx)\right) \right|&= \left| \int (f(x)-f_j(x))\left(\mu_n(dx)-\mu(dx)\right) \right| + \left| \int f_j(x)\left(\mu_n(dx)-\mu(dx)\right) \right|\\
|
||||
&\leq \left| \int (f(x)-f_j(x))\mu_n(dx) \right| + \left| \int (f(x)-f_j(x))\mu(dx) \right| + \left| \int f_j(x)\left(\mu_n(dx)-\mu(dx)\right) \right|\\
|
||||
&\leq \left| \int_{K^\epsilon} (f(x)-f_j(x))\mu_n(dx) \right| + \left| \int_{(K^\epsilon)^c} (f(x)-f_j(x))\mu_n(dx) \right| + \left| \int_{K^\epsilon} (f(x)-f_j(x))\mu(dx) \right| + \\
|
||||
&\qquad + \left| \int_{(K^\epsilon)^c} (f(x)-f_j(x))\mu(dx) \right| + \left| \int f_j(x)\left(\mu_n(dx)-\mu(dx)\right) \right|\\
|
||||
&\leq C_\alpha\epsilon^\alpha + 4\epsilon + C_\alpha\epsilon^\alpha+2\epsilon+\epsilon\leq C_\alpha\epsilon^\alpha,
|
||||
\end{align*}
|
||||
where the last term gets bounded by taking $n$ big enough and by using convergence against $C^{\alpha}_B$ functions, this gives us the final result.
|
||||
\end{proof}
|
||||
\begin{theorem}
|
||||
For small values of $T$; if the coefficients of the SDE are $C^\alpha_B$ functions of $y$ uniformly in $(t,x)$ (the $C^\alpha_B$ norm is uniformly bounded in $(t,x)$) we have that the application $\mathcal{L}:C([0,T],\mathcal{P}(\R^N))\rightarrow C([0,T],\mathcal{P}(\R^N))$ that $\mathcal{L}((\mu_t)_{t\in[0,T]})=(\mathcal{L}^\mu_t)_{t\in[0,T]}$ is a contraction wrt the distance
|
||||
$$
|
||||
d((\mu_t)_{t\in[0,T]},(\nu_t)_{t\in[0,T]})=\sup_{t\in[0,T]}d_{C^\alpha_B}(\mu_t,\nu_t).
|
||||
$$
|
||||
\end{theorem}
|
||||
\begin{proof}
|
||||
We have
|
||||
\begin{align*}
|
||||
d_{C^{\alpha}_B}(\mathcal{L}^\mu_t,\mathcal{L}^\nu_t)&=\sup_{||f||_{C^{\alpha}_B}\leq 1}|I(f)|\leq\\
|
||||
&\stackrel{Th.\ \ref{a1}}{\leq} C\sup_{s,x} \left(|B^\mu(s,x)-B^\nu(s,x)|+|c^\mu(s,x)-c^\nu(s,x)|\right)\int_0^t \left(\frac{1}{|t-s|^{1-\frac{\alpha}{2}}} + \frac{1}{\sqrt{t-s}}\right)ds\\
|
||||
&\leq C_T|t|^\frac{\alpha}{2}\sup_{s,x} \left(|B^\mu(s,x)-B^\nu(s,x)|+|c^\mu(s,x)-c^\nu(s,x)|\right).
|
||||
\end{align*}
|
||||
Now we observe that since the coefficients are uniformly $C^\alpha_B$ we have
|
||||
$$
|
||||
|B^\mu(s,x)-B^\nu(s,x)|=\left| \int b(s,x,y)\mu_s(dy) - \int b(s,x,y)\nu_s(dy) \right|\leq C d_{C^{\alpha}_B}(\mu_s,\nu_s),
|
||||
$$
|
||||
where $C=||b||_{C^{\alpha}_B}$, a priori it depends on $(s,x)$ but since $b$ uniformly $C^\alpha_B$ it can be taken uniformly in $(s,x)$. It is also possible to prove that
|
||||
$$
|
||||
|c^\mu(s,x)-c^\nu(s,x)|\leq C ||\sigma||_{\infty}d_{C^{\alpha}_B}(\mu_s,\nu_s).
|
||||
$$
|
||||
with these we can conclude that
|
||||
$$
|
||||
d_{C^{\alpha}_B}(\mathcal{L}^\mu_t,\mathcal{L}^\nu_t)\leq C|t|^\frac{\alpha}{2}\sup_{s\in[0,t]} d_{C^{\alpha}_B}(\mu_s,\nu_s),
|
||||
$$
|
||||
and thus
|
||||
$$
|
||||
\sup_{t\in[0,T]}d_{C^{\alpha}_B}(\mathcal{L}^\mu_t,\mathcal{L}^\nu_t)\leq C|T|^\frac{\alpha}{2}\sup_{t\in[0,T]}d_{C^{\alpha}_B}(\mu_t,\nu_t).
|
||||
$$
|
||||
which proves contraction for small valus of $T$.
|
||||
\end{proof}
|
||||
\begin{comment}
|
||||
\red{
|
||||
We will now define\footnote{to be precise the Wasserstein metric is defined differently, still by the duality formula for the Wasserstein $1$-distance (Remark 6.5 of \cite{Villani}) the two definitions are equivalent}
|
||||
\begin{align*}
|
||||
d_{bL}(\mu,\nu)&=\sup_{||f||_{bLip}\leq 1} \left|\int f(x)\left(\mu(dx)-\nu(dx)\right)\right|,\\
|
||||
W^{(1)}(\mu,\nu)&=\sup_{||f||_{Lip}\leq 1} \left|\int f(x)\left(\mu(dx)-\nu(dx)\right)\right|,
|
||||
\end{align*}
|
||||
respectively the bounded Lipschitz distance and the Wasserstein $1$-distance. It is possible to prove that convergence in these distances implies weak convergence of measures.
|
||||
First let's concentrate on the bounded Lipschitz case, consider the coefficients of the SDE to be bounded and globally Lipschitz in the $y$ variable ($b(t,x,y)$), then we have
|
||||
\begin{align*}
|
||||
d_{bL}(\mathcal{L}^\mu_t,\mathcal{L}^\nu_t)&= \sup_{||f||_{bLip}\leq 1}|I(f)|\leq\\
|
||||
&\stackrel{Assumption\ 1}{\leq} C\sup_{s,x} \left(|B^\mu(s,x)-B^\nu(s,x)|+|c^\mu(s,x)-c^\nu(s,x)|\right)\int_0^t \frac{1}{\sqrt{t-s}}\left(||f||_{L^\infty}+||f||_{Lip}\right)ds\\
|
||||
\intertext{but the sum of these norms of $f$ is equal to $||f||_{bLip}$ which is less than $1$,}
|
||||
&\leq C\sqrt{t}\sup_{s,x} \left(|B^\mu(s,x)-B^\nu(s,x)|+|c^\mu(s,x)-c^\nu(s,x)|\right).
|
||||
\end{align*}
|
||||
Now we most importantly observe that since the coefficients are bounded and Lipschitz we have
|
||||
$$
|
||||
|B^\mu(s,x)-B^\nu(s,x)|=\left| \int b(s,x,y)\mu_s(dy) - \int b(s,x,y)\nu_s(dy) \right|\leq C d_{bL}(\mu_s,\nu_s),
|
||||
$$
|
||||
where $C=||b||_{bLip}$, a priori it depends on $(s,x)$ but since $b$ is globally Lipschitz and bounded it can be taken uniformly in $(s,x)$. It is also possible to prove that
|
||||
$$
|
||||
|c^\mu(s,x)-c^\nu(s,x)|\leq C ||\sigma||_{\infty}d_{bL}(\mu_s,\nu_s).
|
||||
$$
|
||||
with these we can conclude that
|
||||
$$
|
||||
d_{bL}(\mathcal{L}^\mu_t,\mathcal{L}^\nu_t)\leq C\sqrt{t}\sup_{s\in[0,t]} d_{bL}(\mu_s,\nu_s),
|
||||
$$
|
||||
and thus
|
||||
$$
|
||||
\sup_{t\in[0,T]}d_{bL}(\mathcal{L}^\mu_t,\mathcal{L}^\nu_t)\leq C\sqrt{T}\sup_{t\in[0,T]}d_{bL}(\mu_t,\nu_t).
|
||||
$$
|
||||
which proves contraction for small valus of $T$.
|
||||
If we consider the case with coefficients globally Lipschitz in $y$ but with $b$ possibly unbounded we have
|
||||
\begin{align*}
|
||||
W^{(1)}(\mathcal{L}^\mu_t,\mathcal{L}^\nu_t)&=\sup_{||f||_{Lip}\leq 1}|I(f)|\leq\\
|
||||
&\stackrel{Assumption\ 1}{\leq} C\sqrt{t} \sup_{s,x} \left(|B^\mu(s,x)-B^\nu(s,x)|+|c^\mu(s,x)-c^\nu(s,x)|\right)
|
||||
\intertext{using the estimates on $B$ and $c$ as above but with the Wasserstein distance since now $b$ is not bounded}
|
||||
&\leq C\sqrt{t}\sup_{s\in[0,t]}W^{(1)}(\mu_s,\nu_s)
|
||||
\end{align*}
|
||||
and thus
|
||||
$$
|
||||
\sup_{t\in[0,T]}W^{(1)}(\mathcal{L}^\mu_t,\mathcal{L}^\nu_t)\leq C\sqrt{T}\sup_{t\in[0,T]}W^{(1)}(\mu_t,\nu_t).
|
||||
$$
|
||||
which proves contraction for small valus of $T$.}
|
||||
\end{comment}
|
||||
\begin{remark}
|
||||
This approach of having the $\sup$ in the distance over the space of functions of the same regularity of the coefficients of the SDE seems quite natural ($b$ and $f$ in the same bounded space). It doesn't seem impossible to use these types of techniques for even broader classes of coefficients as long as there are gaussian estimates.
|
||||
\end{remark}
|
||||
%At this point we can try to bound $I(f)$:
|
||||
%\begin{equation}\label{e6}
|
||||
%\mu_0(g)=\int g(y)\mu_0(dy)\leq ||g||_{L^1(\mu_0)}.
|
||||
%\end{equation}
|
||||
%By Gaussian estimates we can observe that the backward propagation operator $V^{0,s}$ is a bounded operator\footnote{Here there is an error in the last equality since $\mu_0$ is not translation invariant. We could try to work with the $W^{(2)}$ distance or make this last inequality work.} in $L^1(\mu_0)$:
|
||||
%\begin{align}\label{e7}
|
||||
%||V^{0,s}_\mu g||_{L^1(\mu_0)}\leq \int\left|\int p^\mu(0,x;s,y)g(y)dy\right|\mu_0(dx)\leq C\int\int |g(z+x)|\Gamma^+(|s|,z)dz\mu_0(dx)=C||g||_{L^1(\mu_0)}.
|
||||
%\end{align}
|
||||
%Similarly to \cite{Kolokoltsov}, but without using the adjoint operator so without the need to ask for regularity of the coefficients, we can show the following bound:
|
||||
%\begin{align}\nonumber
|
||||
%||(\mathcal{A}^\mu_s-\mathcal{A}^\nu_s)g||_{L^1(\mu_0)}&=||(B^\mu_s-B^\nu_s)\bigtriangledown g +\frac{1}{2}\left\langle (c^\mu_s-c^\nu_s)\bigtriangledown, \bigtriangledown\right\rangle g||_{L^1(\mu_0)}\\ \label{e8}
|
||||
%&\leq \sup_{s,x}\left( |B^\mu_s(x)-B^\nu_s(x)| + |c^\mu_s(x)-c^\nu_s(x)| \right)||g||_{W^{2,1}(\mu_0)}.
|
||||
%\end{align}
|
||||
%The following inequality still needs to be properly proved but is stated as (13) in \cite{Kolokoltsov} and seems reasonable
|
||||
%\begin{equation}\label{e9}
|
||||
%||V_\nu^{s,t}f||_{W^{2,1}(\mu_0)}\leq Cs^{-\frac{1}{2}}||f||_{W^{1,1}(\mu_0)}.
|
||||
%\end{equation}
|
||||
%Now we employ a couple of observations. Firstly $I(f)$ is invariant with respect to translations: $I(f+c)=I(f),\ \forall c\in\R$; for this reason without loss of generality $f(0)=0$ and thus
|
||||
%\begin{align*}
|
||||
%||f||_{L^1(\mu_0)}&=\int|f(x)|\mu_0(dx)\leq \int|f(x)-f(0)|\mu_0(dx) + \int|f(0)|\mu_0(dx)\leq ||f||_{Lip}\int |x|\mu_0(dx) = C,\\
|
||||
%||\bigtriangledown f||_{L^1(\mu_0)}&\leq \int ||f||_{Lip}\mu_0(dx) = 1.
|
||||
%\end{align*}
|
||||
%Thus putting everything together in \eqref{e6}-\eqref{e9} and noticing that the estimate is uniform for any $f$ with Lipschitz constant bounded by $1$ we obtain
|
||||
%$$
|
||||
%W^{1}(\mathcal{L}^\mu_t,\mathcal{L}^\nu_t)\leq \int_0^t C\sup\left( |B^\mu-B^\nu| + |a^\mu-a^\nu| \right)s^{-\frac{1}{2}}ds = Ct^{\frac{1}{2}}\sup\left( |B^\mu-B^\nu| + |c^\mu-c^\nu| \right).
|
||||
%$$
|
||||
%Now, if we are able to prove an inequality in the form $\sup\left( |B^\mu-B^\nu| + |c^\mu-c^\nu|\right)\leq \sup_t W^{(1)}(\mu_t,\nu_t)$ we have a contraction. If the coefficients are Lipschitz in the third variable this is mostly trivial
|
||||
%\begin{gather*}
|
||||
%\sup\left( |B^\mu-B^\nu|\right)\leq \sup_{t,x}\left(\left|\int b(t,x,y_1)\mu_t(dy_1)-\int b(t,x,y_2)\nu_t(dy_2)\right| \right)\leq L\sup_{t,x}\left( \int|y_1-y_2|\gamma(dy_1,dy_2) \right)\\
|
||||
%\stackrel{\inf}{\Rightarrow} \sup\left( |B^\mu-B^\nu|\right)\leq L\sup_{t,x}W^{(1)}(\mu_t,\nu_t),
|
||||
%\end{gather*}
|
||||
%this should also work for $c$ but the matrix form is a bit more challenging. If everything goes according to plan we should achieve that for small $T$ we have the contraction
|
||||
%$$
|
||||
%\sup_tW^{(1)}(\mathcal{L}^\mu_t,\mathcal{L}^\nu_t)\leq C\sqrt{T} \sup_tW^{(1)}(\mu_t,\nu_t).
|
||||
%$$
|
||||
|
||||
|
||||
|
||||
%\begin{small}
|
||||
%\red{
|
||||
%Let's concentrate for a moment on $B$:
|
||||
%\begin{align*}
|
||||
%|B^\mu(t,x)-B^\nu(t,x)|&=|B(t,x,\mu_t)-B(t,x,\nu_t)|=\left|\int b(t,x,y)\mu_t(dy)-\int b(t,x,z)\nu_t(dz)\right|
|
||||
%\intertext{Given $\gamma$ a law of marginals $\mu_t$ and $\nu_t$}
|
||||
%&\leq \left|\int b(t,x,y)-b(t,x,z)\gamma(dy,dz)\right|
|
||||
%\stackrel{\alpha-Hold}{\leq} ||b||_{C^{0,\alpha}}\int |y-z|^\alpha\gamma(dy,dz)
|
||||
%\intertext{This is uniform over $\gamma$, by passing to the inf we arrive to the Wasserstein distance}
|
||||
%|B^\mu(t,x)-B^\nu(t,x)|&\leq ||b||_{C^{0,\alpha}} W^{(\alpha)}(\mu_t,\nu_t)^\alpha\stackrel{Jensen}{\leq} ||b||_{C^{0,\alpha}}W^{(2)}(\mu_t,\nu_t)^\alpha.
|
||||
%\end{align*}
|
||||
%Probably something similar may be proven for $c$, effectively $c=\Sigma\Sigma^*$ and $\Sigma$ is $\alpha$-Holderian wrt the Wasserstein distance.\\
|
||||
%This should almost prove what we need, we just need an inequality to bound\footnote{to accomplish such a feat the "$\inf$" part of the Wasserstein is fundamental since if for simplicity we consider the product of $Unif_{[0,1]}$ with itself it is very far from $0$ while the Wasserstein distance between a law and itself is $0$. More heuristically with the $\inf$ the Wasserstein considers heavily correlated random variables pair which drastically lower the value of the distance.} $W^{(2)}(\mathcal{L}^\mu_t,\mathcal{L}^\nu_t)$ with $||u^\mu_t-u^\nu_t||_{L^1}$. Alternatively we could bound $\sup_tW^{(2)}(\mu_t,\nu_t)$ with the $L^1$ norm of the difference of the densities associated with $\mu_t$ and $\nu_t$. Depending on the case the contraction is applied on $C([0,T]\times\mathcal{P}^2(\R^n))$ or in $C([0,T]\times L^1(\R^N))$.\\
|
||||
%A little observation with the $W^{(1)}$ distance: due to Kantarovich's duality theorem we have (\cite{Villani} remark 6.5)
|
||||
%\begin{align*}
|
||||
%W^{(1)}(\mu,\nu)&=\sup_{||f||_{Lip}\leq1}\left(\int f\mu-\int f\nu\right)\stackrel{AC}{=}\sup_{||f||_{Lip}\leq1}\left(\int f(x)(\gamma_\mu(x)-\gamma_\nu(x))dx\right)\\
|
||||
%&\stackrel{Lipschitz}{\leq}\int|x-x_0|\cdot|\gamma_\mu(x)-\gamma_\nu(x)|dx
|
||||
%\end{align*}
|
||||
%In some cases the inequality seems an equality. For simplicity let's consider the $1$-d case and define $\beta(dt)=|\gamma_\mu(t)-\gamma_\nu(t)|dt$:
|
||||
%\begin{align*}
|
||||
%W^{(1)}(\mu,\nu)&\leq\int_{-\infty}^{+\infty}|t|\beta(dt)=\int_0^{+\infty}t\beta(dt)-\int_{-\infty}^0t\beta(dt)\\
|
||||
%&=\int_0^{+\infty}\int_0^tdx\beta(dt)+\int_{-\infty}^0\int_t^0dx\beta(dt)=\int_0^{+\infty}\beta([x,+\infty[)dx+\int_{-\infty}^0\beta(]-\infty,x])dx\\
|
||||
%&=\int_0^{+\infty}\beta(]-\infty,-x]\cup[x,+\infty[)dx=\int_0^{+\infty}\int_{\R\setminus[-x,x]}|\gamma_\mu(t)-\gamma_\nu(t)|dtdx.
|
||||
%\end{align*}
|
||||
%Unfortunately here there is no space for bounding the Wasserstein distance with the $L^1$ distance: if we consider the densities uniform in $[n,n+1]$ and $[n+1/2,n+3/2]$ we get that the constant $C$ in $W^{(1)}(\mu,\nu)\leq C||\gamma_\mu-\gamma_\nu||_{L^1}$ needs to be arbitrarily big. The following theorem should shelter us from this type of problems.}
|
||||
%\end{small}
|
||||
|
||||
There is the property of tightness for the family of measures that are solution of an SDE with $\alpha$-Holder coefficients and with initial law with finite $p$-moment:
|
||||
\begin{theorem}\label{t1}
|
||||
Let $\mu_0\in\mathcal{P}^p(\R^N)$. Let $p(s,x;t,y)$ be a fundamental solution of a forward Kolmogorov equation with $\alpha$-Holderian coefficients so that Gaussian estimates exist. Then for any $\epsilon>0$ there exists $K>0$ such that
|
||||
$$
|
||||
\int_{B_K^c}u_t(x)dx<\epsilon,\qquad \int_{B_K^c}|x|^pu_t(x)dx<\epsilon.
|
||||
$$
|
||||
Where $u_t(x)=\int p(0,y;t,x)\mu_0(dy)$.
|
||||
\end{theorem}
|
||||
\begin{proof}
|
||||
The proof is a little variation of (3.2) in \cite{Kolokoltsov}; indeed the first inequality is proved there. Fix $\epsilon>0$. Let $\tilde{\epsilon}>0$ that we will fix later. Since $\mu_0$ is a measure with finite $p$-moment there exists $K>0$ such that
|
||||
$$
|
||||
\mu_0(B_K^c)<\tilde{\epsilon},\qquad \int_{B_K^c}|x|^p\mu_0(dx)<\tilde{\epsilon}.
|
||||
$$
|
||||
Let $\tilde{K}>0$ that we will fix later.
|
||||
\begin{align*}
|
||||
\int_{|x|\geq K+\tilde{K}}|x|^pu_t(x)dx&\stackrel{Gaussian\ estimates}{\leq} C\int_{|x|\geq K+\tilde{K}}|x|^p\int\Gamma^+(x-\xi,t)\mu_0(d\xi)dx\\
|
||||
&\leq C\int_{|\xi|\geq K,y\in\R^N}|y+\xi|^p\Gamma^+(y,t)\mu_0(d\xi)dy + C\int_{|\xi|\leq K,y\geq\tilde{K}}|y+\xi|^p\Gamma^+(y,t)\mu_0(d\xi)dy\\
|
||||
&\leq C_p\mu_0(B_K^c)\int |y|^p\Gamma^+(y,t)dy+C_p\int_{B_K^c}|\xi|^p\mu_0(d\xi)\\
|
||||
&\qquad+C_p\mu_0(B_k)\int_{B_{\tilde{K}}^c}|y|^p\Gamma^+(y,t)dy + C_p\int|\xi|^p\mu_0(d\xi)\int_{B_{\tilde{K}}^c}\Gamma^+(y,t)dy
|
||||
\intertext{The first two terms get bounded by the preliminary inequalities and the fact that the Gaussian has finite $p$ moment. The last two terms get bounded by a constant $\tilde{C}_{p,\tilde{K},T}$ that goes to $0$ as $\tilde{K}$ goes to $+\infty$.}
|
||||
&\leq C_{T,p}\tilde{\epsilon} + C_p\tilde{\epsilon} + C_p\tilde{C}_{p,\tilde{K},T} + C_{p,\mu_0}\tilde{C}_{p,\tilde{K},T}.
|
||||
\end{align*}
|
||||
if we choose $\tilde{\epsilon}$ small enough and $\tilde{K}$ big enough the final result will be smaller than $\epsilon$. We must also notice that all the estimates and the constant do not depend directly on $u_t(x)$ but on the Gaussian estimates and so they hold uniformly for the whole family of solutions.
|
||||
\end{proof}
|
||||
We may notice that the family of the marginals is bounded in $\mathcal{P}^p$ with the Wasserstein metric.
|
||||
\begin{theorem}
|
||||
Let $(\mu_i)_{i\in\mathcal{I}}$ be the family of the marginals of solutions to SDEs with the same initial datum and $\alpha$-Holderian coefficients. (Written like this is not very rigorous but for example given $(\mu_t)_{t\in[0,T]}$ the flow of marginals of a solution to an SDE as in the hypothesis we have that $\mu_t$ is an element of the family for every $t\in[0,T]$).
|
||||
|
||||
Then the family is bounded as a subset of $\mathcal{P}^p(\R^N)$ equipped with the Wasserstein metric .
|
||||
\end{theorem}
|
||||
\begin{proof}
|
||||
Let $\mu_1$ and $\mu_2$ be elements of the family. Fix $\epsilon>0$. By theorem \ref{t1} we know that exists $K>0$ such that
|
||||
$$
|
||||
\int_{B^c_K}\mu_i(dx)<\epsilon,\qquad \int_{B^c_K}|x|^p\mu_i(dx)<\epsilon,\qquad i=1,2.
|
||||
$$
|
||||
In particular given a measure $\gamma$ on $\R^{2N}$ with marginals $\mu_1$ and $\mu_2$ we have that exists $\tilde{K}$ (uniformly in $\gamma$) such that
|
||||
\begin{equation*}
|
||||
\int_{B^c_{\tilde{K}}}\gamma(dy,dz)\leq \int\int_{B^c_{K}\times B^c_{K}}\gamma(dy,dz)\leq \int\int_{B^c_{K}\times \R^N}\gamma(dy,dz)=\int_{B^c_K}\mu_1(dy)<\epsilon.
|
||||
\end{equation*}
|
||||
This also works for the $p$-moment and we get
|
||||
\begin{equation*}
|
||||
\int_{B^c_{\tilde{K}}}|y|^p\gamma(dy,dz)\leq \int\int_{B^c_{K}\times B^c_{K}}|y|^p\gamma(dy,dz)\leq \int\int_{B^c_{K}\times \R^N}|y|^p\gamma(dy,dz)=\int_{B^c_K}|y|^p\mu_1(dy)<\epsilon.
|
||||
\end{equation*}
|
||||
This means that we can bound the Wasserstein distance between the two in the following way:
|
||||
\begin{align*}
|
||||
W^{(p)}(\mu_1,\mu_2)^p&=\inf_{\gamma}\int\int(y-z)^p\gamma(dy,dz)\leq\inf_\gamma \int_{B^c_{\tilde{K}}}(y-z)^p\gamma(dy,dz) + \int_{B_{\tilde{K}}}(y-z)^p\gamma(dy,dz)\\
|
||||
&\leq C_p\left(\int_{B^c_{\tilde{K}}}|y|^p\gamma(dy,dz) + \int_{B^c_{\tilde{K}}}|z|^p\gamma(dy,dz)\right) + \int_{B_{\tilde{K}}}diam(B_{\tilde{K}})^p\gamma(dy,dz)\\
|
||||
&\leq 2C_p\epsilon + diam(B_{\tilde{K}})^p.
|
||||
\end{align*}
|
||||
\end{proof}
|
||||
The preceding theorem in particular proves that the $p$-moments are uniformly bounded, for this reason the following theorem is valid in our case.
|
||||
\begin{theorem}
|
||||
Let $(\mu_i)_{i\in\mathcal{I}}\subset\mathcal{P}^p(\R^N)$ be a family of probability measures with tightness property as of Theorem \ref{t1} and such that the $p$-moments are uniformly bounded. Then for any sequence $(\mu_n)_{n\in\N}$ there exists a subsequence $(\mu_{n_m})_{m\in\N}$ and a measure $\mu\in\mathcal{P}^p(\R^N)$ such that $$\mu_{n_m}\stackrel{Wasserstein}{\rightarrow}\mu.$$
|
||||
\end{theorem}
|
||||
\begin{proof}
|
||||
By theorem (6.9) of \cite{Villani} we have that Wasserstein convergence in $\mathcal{P}^p$ is equivalent to weak convergence and convergence of the $p$-moment. We know that
|
||||
$$
|
||||
\mu_i(B_K^c)<\epsilon,\qquad \int_{B_K^c}|x|^p\mu_i(dx)<\epsilon.
|
||||
$$
|
||||
Thus if we define $P_i(dx)=|x|^p\mu_i(dx)$ we have that $(P_i)_{i\in\mathcal{I}}$ is a tight family of uniformly finite measures, in particular we may use a generalization of Prokhorov's theorem (Theorem 8.6.2 of \cite{Bogachev}) and get that for any sequence there exists a subsequence $P_{n_m}$ that converges weakly to a finite measure $P$.
|
||||
\begin{equation}\label{e10}
|
||||
P_{n_m}(\phi)\rightarrow P(\phi),\qquad\forall \phi\in C_0^{\infty}(\R^N).
|
||||
\end{equation}
|
||||
By using the same theorem on $\mu_{n_m}$ we can find a new subsequence (that will still be witten as $\mu_{n_m}$) such that $\mu_{n_m}$ converges weakly to $\mu$.
|
||||
\begin{equation}\label{e11}
|
||||
\mu_{n_m}(\phi)\rightarrow \mu(\phi),\qquad\forall \phi\in C_0^{\infty}(\R^N).
|
||||
\end{equation}
|
||||
Consider now $\phi\in C_0^{\infty}(\R^N)$.
|
||||
\begin{align*}
|
||||
&\int |x|^p\phi(x)\mu_{n_m}(dx)\stackrel{\eqref{e11}}{\rightarrow} \int |x|^p\phi(x)\mu(dx)\\
|
||||
&=P_{n_m}(\phi)\stackrel{\eqref{e10}}{\rightarrow} P(\phi).
|
||||
\end{align*}
|
||||
By uniqueness of the limit we have for any $\phi\in C_0^{\infty}$ that $P(\phi)=\int|x|^p\phi(x)\mu(dx)$. This proves convergence of the $p$-moment and thus with weak convergence we have Wasserstein convergence.
|
||||
\end{proof}
|
||||
%Most probably some of these $L^p$ spaces will need to be changed, for example it could be helpful to only work with $W^{(1)}$ and $L^1$ to follow more closely Kolokoltsov.
|
||||
\begin{thebibliography}{90}
|
||||
\bibitem{Bogachev} Vladimir I. Bogachev - Measure Theory (2007)
|
||||
\bibitem{Kolokoltsov} Vassili N. Kolokoltsov - Nonlinear Diffusions and Stable-Like Processes
|
||||
with Coefficients Depending on the Median or VaR (2013)
|
||||
\bibitem{LucePagliaPascu} G. Lucertini, A. Pagliarani, A. Pascucci - Optimal regularity for degenerate Kolmogorov equations in non-divergence form with rough-in-time coefficients (2024), https://doi.org/10.1007/s00028-023-00916-9
|
||||
\bibitem{YAOZHONG} Yaozhong Hu, Michael A. Kouritzin, Jiayu Zheng - Nonlinear McKean-Vlasov diffusions under the weak Hormander condition with quantile-dependent coefficients (2021), https://arxiv.org/abs/2101.04080
|
||||
\bibitem{Villani} Cédric Villani - Optimal Transport Old and New (2009)
|
||||
\end{thebibliography}
|
||||
\end{document}
|
||||
Loading…
Reference in New Issue
Block a user