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Stochastic Hamiltonian flows with singular coefficients

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  • ReceivedFeb 7, 2017
  • AcceptedJun 16, 2017
  • PublishedMar 23, 2018

Abstract

In this paper, we study the following stochastic Hamiltonian system in ${\mathbb~R}^{2d}$ (a second order stochastic differential equation): where $b(x,\mathrm{v}):{\mathbb~R}^{2d}\to{\mathbb~R}^d$ and $\sigma(x,\mathrm{v}):{\mathbb~R}^{2d}\to{\mathbb~R}^d\otimes{\mathbb~R}^d$ are two Borel measurable functions.We show that if $\sigma$ is bounded and uniformly non-degenerate, and $b\in~H^{2/3,0}_p$ and $\nabla\sigma\in~L^p$ for some $p>2(2d+1)$,where $H^{\alpha,\beta}_p$ is the Bessel potential space with differentiability indices $\alpha$ in $x$ and $\beta$ in $\mathrm{v}$,then the above stochastic equation admits a unique strong solution so that $(x,\mathrm{v})\mapsto~Z_t(x,\mathrm{v}):=(X_t,\dot~X_t)(x,\mathrm{v})$forms a stochastic homeomorphism flow, and $(x,\mathrm{v})\mapsto~Z_t(x,\mathrm{v})$ is weakly differentiable withess.$\sup_{x,\mathrm{v}}{\rm~E}(\sup_{t\in[0,T]}|\nabla~Z_t(x,\mathrm{v})|^q~)<\infty$ for all $q\geqslant~1$ and $T\geqslant~0$. Moreover, we also show the uniqueness ofprobability measure-valued solutions for kinetic Fokker-Planck equations with rough coefficientsby showing the well-posedness of the associated martingale problem and using the superposition principle established by Figalli (2008) and Trevisan (2016).


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Appendix

The following stochastic Gronwall's type lemma is probably well-known. Since we cannot find it in the literature, a proof is provided here for the reader's convenience.

Lemma 9. For a given $T>0$, let $(\xi_t)_{t\in~[0,~T]}$ and $(\beta_t)_{t\in~[0,~T]}$ $($resp. $(\alpha_t)_{t\in~[0,~T]})$ be two real-valued $($resp. ${\mathbb~R}^d$-valued$)$ measurable ${\mathscr~F}_t$-adapted processes. Let $\zeta_t$ be an Itô process with the form $$ \zeta_t=\zeta_0+\int^t_0\zeta^{(1)}_s{{{d}}} s+\int^t_0\zeta^{(2)}_s{{{d}}} W_s. $$ Suppose that for any $\gamma>0$, \begin{align}\kappa_\gamma:={\rm E}\exp\bigg\{\gamma\int^T_0 (|\beta_s|+|\alpha_s|^2 ){{{d}}} s\bigg\}<\infty, \tag{103} \end{align} and \begin{align} 0\leqslant\xi_t\leqslant\zeta_t+\int^t_0\xi_s\beta_s{{{d}}} s+\int^t_0\xi_s\alpha_s{{{d}}} W_s. \tag{104} \end{align} Then for any $q_0\in[1,\infty)$ and $q_1,q_2,q_3>q_0$, there is a constant $C>0$ only depending on $q_i,\kappa_{\gamma},~i=0,1,2,3$ such that \begin{align}\|\xi^*_T\|_{q_0}\leqslant C\bigg(\|\zeta_0\|_{q_1}+\bigg\|\int^T_0|\zeta_s^{(1)}|{{{d}}} s\bigg\|_{q_2} +\bigg\|\int^T_0|\zeta_s^{(2)}|^2{{{d}}} s\bigg\|^{1/2}_{q_3/2}\bigg), \tag{105} \end{align} where $\xi^*_T:=\sup_{t\in[0,T]}\xi_t$ and $\|\cdot\|_{q_i}$ denotes the norm in $L^{q_i}(\Omega)$.

Proof. Write \begin{align*}\eta_t&:=\zeta_t+\int^t_0\xi_s\beta_s{{{d}}} s+\int^t_0\xi_s\alpha_s{{{d}}} W_s=\zeta_t+\int^t_0\eta_s\bar\beta_s{{{d}}} s+\int^t_0\eta_s\bar\alpha_s{{{d}}} W_s, \end{align*} where $\bar\beta_s:=\xi_s\beta_s/\eta_s$ and $\bar\alpha_s:=\xi_s\alpha_s/\eta_s$. Here, we use the convention $\frac{0}{0}:=0$.

Define $$ M_t:=\exp\bigg\{\int^t_0\bar\alpha_s{{{d}}} W_s+\int^t_0(\bar\beta_s-\tfrac{1}{2}|\bar\alpha_s|^2){{{d}}} s\bigg\}. $$ By Itô's formula, we have $$ M_t=1+\int^t_0M_s\bar\beta_s{{{d}}} s+\int^t_0M_s\bar\alpha_s{{{d}}} W_s $$ and \begin{align*}\eta_t=M_t\bigg[\zeta_0+\int^t_0 M^{-1}_s(\zeta^{(1)}_s-\langle\bar\alpha_s,\zeta_s^{(2)}\rangle){{{d}}} s+\int^t_0M^{-1}_s \zeta^{(2)}_s{{{d}}} W_s\bigg]. \end{align*} Hence, \begin{align*}\|\eta^*_T\|_{q_0}&\leqslant\|M^*_T\zeta_0\|_{q_0}+\bigg\|M^*_T(M^{-1})^*_T\int^T_0|\zeta^{(1)}_s|{{{d}}} s\bigg\|_{q_0} \\ & +\bigg\|M^*_T(M^{-1})^*_T\int^T_0|\bar\alpha_s|\cdot|\zeta_s^{(2)}|{{{d}}} s\bigg\|_{q_0} \\ & +\bigg\|M^*_T\sup_{t\in[0,T]}\bigg|\int^t_0M^{-1}_s\zeta_s^{(2)}{{{d}}} W_s\bigg|\bigg\|_{q_0} \\ &=:I_1+I_2+I_3+I_4. \end{align*} Noticing that by 104, \begin{align} |\bar\beta_s|\leqslant|\beta_s|, |\bar\alpha_s|\leqslant|\alpha_s|, \tag{106} \end{align} for any $p\in{\mathbb~R}$, by 103, Hölder's inequality and Doob's maximal inequality, we have \begin{align} {\rm E}\Big(\sup_{t\in[0,T]}M_t^p\Big)<\infty. \tag{107} \end{align} Thus, by Hölder's inequality and 107, we have $$ I_1\leqslant C\|\zeta_0\|_{q_1}, I_2\leqslant C\bigg\|\int^T_0|\zeta^{(1)}_s|{{{d}}} s\bigg\|_{q_2}, $$ and by 106, \begin{align*}I_3&\leqslant\bigg\|M^*_T(M^{-1})^*_T\bigg(\int^T_0|\alpha_s|^2{{{d}}} s\bigg)^{1/2}\bigg(\int^T_0|\zeta_s^{(2)}|^2{{{d}}} s\bigg)^{1/2}\bigg\|_{q_0} \\ &\leqslant C\bigg\|\bigg(\int^T_0|\zeta_s^{(2)}|^2{{{d}}} s\bigg)^{1/2}\bigg\|_{q_3}=C\bigg\|\int^T_0|\zeta_s^{(2)}|^2{{{d}}} s\bigg\|^{1/2}_{q_3/2}. \end{align*} Similarly, by Hölder and Burkholder's inequalities, we also have $$ I_4\leqslant C \bigg\|\int^T_0|\zeta_s^{(2)}|^2{{{d}}} s \bigg\|^{1/2}_{q_3/2}. $$ Combining the above estimates, we obtain 105.

Let $f$ be a locally integrable function on ${\mathbb~R}^d$. The Hardy-Littlewood maximal function is defined by $$ {\mathcal M} f(x):=\sup_{0<r<\infty}\frac{1}{|B_r|}\int_{B_r}f(x+y){{{d}}} y, $$ where $B_r:=\{x\in{\mathbb~R}^{d}:~|x|<r\}$. The following result can be found in [-1].

Lemma 10. rm(i) There exists a constant $C_d>0$ such that for all $f\in~C^\infty({\mathbb~R}^{d})$ and $x,y\in~{\mathbb~R}^{d}$, \begin{align}|f(x)-f(y)|\leqslant C_d |x-y|({\mathcal M}|\nabla f|(x)+{\mathcal M}|\nabla f|(y)). \tag{108} \end{align} rm(ii) For any $p>1$, there exists a constant $C_{d,p}$ such that for all $f\in~L^p({\mathbb~R}^d)$, \begin{align}\|{\mathcal M} f\|_p\leqslant C_{d,p}\|f\|_p. \tag{109} \end{align}


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