more precise hypotheses
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Alessio Rondelli
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PR_1.aux
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PR_1.aux
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\citation{Pascu_PB2}
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\citation{Pascu_PB2}
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\bibcite{YAOZHONG}{1}
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\bibcite{Kolokoltsov}{2}
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\bibcite{LucePagliaPascu}{3}
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PR_1.log
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PR_1.tex
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PR_1.tex
@ -87,9 +87,16 @@ $$
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$$
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a sufficient condition for this to occur under assupmtion \ref{a1} is if $\sigma$ is a uniformly positive definite matrix.
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\end{assumption}
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\begin{assumption}[Regularity]\label{a3}
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Assume assumption \ref{a1}. The coefficients $b$ and $\sigma$ are functions in $L^\infty([0,T],bC^\alpha)$. Explicitly
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\begin{gather*}
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\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,\\
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\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).
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\end{gather*}
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\end{assumption}
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\section{Results}
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\begin{theorem}[Non-degenerate case]\label{t1}
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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.
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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.
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\end{theorem}
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\section{Proofs}
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\begin{proof}of Theorem \ref{t1}.
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@ -124,8 +131,11 @@ and observe that the distribution of $X^\mu_t$ is absolutely continuous with den
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$$
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u^\mu_t(y)=U^{t,0}_\mu\mu_0.
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$$
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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.
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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 $\mu_0$.
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\begin{remark}
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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 $\mu_0$.
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%Thus we image of $\mathcal{L}$ is actually contained inside $$ \left\lbrace (\nu_t)_{t\in[0,T]}\ |\ \nu_0=\mu_0 \right\rbrace.$$
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\end{remark}
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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
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$$
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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,
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