diff --git a/PR_1.aux b/PR_1.aux index 5dc8d4c..c59fbdc 100644 --- a/PR_1.aux +++ b/PR_1.aux @@ -7,14 +7,15 @@ \@writefile{toc}{\contentsline {section}{\numberline {2}Preliminaries}{1}{section.2}\protected@file@percent } \newlabel{a1}{{4}{1}{Structural Assumption}{theorem.4}{}} \newlabel{a2}{{5}{1}{Non degeneracy}{theorem.5}{}} +\newlabel{a3}{{6}{1}{Regularity}{theorem.6}{}} \@writefile{toc}{\contentsline {section}{\numberline {3}Results}{1}{section.3}\protected@file@percent } -\newlabel{t1}{{6}{1}{Non-degenerate case}{theorem.6}{}} +\newlabel{t1}{{7}{1}{Non-degenerate case}{theorem.7}{}} \@writefile{toc}{\contentsline {section}{\numberline {4}Proofs}{1}{section.4}\protected@file@percent } \@writefile{toc}{\contentsline {section}{\numberline {5}Appendix}{3}{section.5}\protected@file@percent } -\newlabel{ap1}{{7}{3}{}{theorem.7}{}} +\newlabel{ap1}{{9}{3}{}{theorem.9}{}} \citation{Pascu_PB2} +\newlabel{ap2}{{10}{4}{}{theorem.10}{}} \citation{Pascu_PB2} -\newlabel{ap2}{{8}{4}{}{theorem.8}{}} \bibcite{YAOZHONG}{1} \bibcite{Kolokoltsov}{2} \bibcite{LucePagliaPascu}{3} diff --git a/PR_1.log b/PR_1.log index d70e85c..38cc7ed 100644 --- a/PR_1.log +++ b/PR_1.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.25 (TeX Live 2023/nixos.org) (preloaded format=pdflatex 1980.1.1) 21 NOV 2024 12:58 +This is pdfTeX, Version 3.141592653-2.6-1.40.25 (TeX Live 2023/nixos.org) (preloaded format=pdflatex 1980.1.1) 21 NOV 2024 19:23 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -607,7 +607,7 @@ File: ulasy.fd 1998/08/17 v2.2e LaTeX symbol font definitions {/nix/store/mhdx3hkmpns8i6czk9ygf2yc7wxlkpy0-texlive-combined-full-2023-final/s hare/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] [2] [3] LaTeX Font Info: Trying to load font information for U+dsrom on input line 2 -13. +23. 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PDF statistics: - 169 PDF objects out of 1000 (max. 8388607) - 126 compressed objects within 2 object streams - 28 named destinations out of 1000 (max. 500000) + 173 PDF objects out of 1000 (max. 8388607) + 130 compressed objects within 2 object streams + 30 named destinations out of 1000 (max. 500000) 41 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/PR_1.pdf b/PR_1.pdf index b9e4e37..86d3d28 100644 Binary files a/PR_1.pdf and b/PR_1.pdf differ diff --git a/PR_1.synctex.gz b/PR_1.synctex.gz index 130cdfd..fc34f2e 100644 Binary files a/PR_1.synctex.gz and b/PR_1.synctex.gz differ diff --git a/PR_1.tex b/PR_1.tex index 9fedbcc..55d1f4a 100644 --- a/PR_1.tex +++ b/PR_1.tex @@ -87,9 +87,16 @@ $$ $$ a sufficient condition for this to occur under assupmtion \ref{a1} is if $\sigma$ is a uniformly positive definite matrix. \end{assumption} +\begin{assumption}[Regularity]\label{a3} +Assume assumption \ref{a1}. The coefficients $b$ and $\sigma$ are functions in $L^\infty([0,T],bC^\alpha)$. Explicitly +\begin{gather*} +\exists C>0,\ s.t.\ \forall (x_1,x_2,y_1,y_2)\in \R^{4N},\ t-\mathrm{a.s.},\qquad |b(t,x,y)|+|\sigma(t,x,y)|\leq C,\\ +\mathrm{and}\ s.t.\ |b(t,x_1,y_1)-b(t,x_2,y_2)| + |\sigma(t,x_1,y_1)-\sigma(t,x_2,y_2)|\leq C\left(|x_1-x_2|^\alpha+|y_1-y_2|^\alpha\right). +\end{gather*} +\end{assumption} \section{Results} \begin{theorem}[Non-degenerate case]\label{t1} -Consider a MKV SDE under assumptions \ref{a1} and \ref{a2} with $\alpha$-Holder bounded coefficients in $(x,y)$ uniformly in $t$ and initial distribution $\mu_0\in\mathcal{P}(\R^N)$; then we have weak existance and uniqueness of the solution of the MKV SDE. +Consider a MKV SDE under assumptions \ref{a1}, \ref{a2} and \ref{a3} and initial distribution $\mu_0\in\mathcal{P}(\R^N)$; then we have weak existance and uniqueness of the solution of the MKV SDE. \end{theorem} \section{Proofs} \begin{proof}of Theorem \ref{t1}. @@ -124,8 +131,11 @@ and observe that the distribution of $X^\mu_t$ is absolutely continuous with den $$ u^\mu_t(y)=U^{t,0}_\mu\mu_0. $$ -Thus we may define the application $\mathcal{L}:C([0,T], \mathcal{P}(\R^N))\rightarrow C([0,T], \mathcal{P}(\R^N))$ such that $\mathcal{L}((\mu_t)_{t\in[0,T]})=(\mathcal{L}^\mu_t)_{t\in[0,T]}=([X^\mu_t])_{t\in[0,T]}$. Less abstractly this is the application that given a flow of marginals returns the flow of marginals of the solution of the "linearized" SDE with the first flow of marginals. - +Thus we may define the application $\mathcal{L}:C([0,T], \mathcal{P}(\R^N))\rightarrow C([0,T], \mathcal{P}(\R^N))$ such that $\mathcal{L}((\mu_t)_{t\in[0,T]})=(\mathcal{L}^\mu_t)_{t\in[0,T]}=([X^\mu_t])_{t\in[0,T]}$. Less abstractly this is the application that given a flow of marginals returns the flow of marginals of the solution of the "linearized" SDE with the first flow of marginals and initial datum $\mu_0$. +\begin{remark} +Since $\mathcal{L}^\mu_0=[X^\mu_0]=\mu_0$ for any flow of marginals the image through $\mathcal{L}$ has initial law equal to $\mu_0$. +%Thus we image of $\mathcal{L}$ is actually contained inside $$ \left\lbrace (\nu_t)_{t\in[0,T]}\ |\ \nu_0=\mu_0 \right\rbrace.$$ +\end{remark} Now we need to observe carefully the definition of $d_{bC^\alpha}([X^\mu_t],[X^\nu_t])$: if we remove the $\sup$ we get $$ I(f)=\int f(x)\left( [X^\mu_t](dx)-[X^\nu_t](dx) \right)=\int f(x)\left( u^\mu_t(x)-u^\nu_t(x) \right)dx,