more precise hypotheses

PR_1.aux

@@ -7,14 +7,15 @@
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 \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 }
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+\newlabel{t1}{{7}{1}{Non-degenerate case}{theorem.7}{}}
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 \@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}

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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,