随机偏微分方程
Throughout this section, let $(Omega, calF, calF_t, P)$ be a complete filtered probability space satisfying the usual conditions.
1. Recall the following results:
a) The Doob maximal inequality: if $(N_t)$ is a non-negative $calF_t$-submartingale with $N_0=0$, then for $1<p<infty$, $$ex Esez{sup_{0leq tleq T}sev{N_t}^p} leq sex{frac{p}{p-1}}^p Esez{sev{N_T}^p}. eex$$
b) The set $calS$ of simple processes is dense in the Hilbert space $sex{calH, sen{cdot}_{calH}}$, where $$ex calS:=left{xi_t=sum_{k=0}^n xi_kchi_{[t_k,t_{k+1}]}(t): 0=t_0<t_1<cdots<t_nleq T, ight.\ left.xi_kincalF_{t_k}, sup_ksen{xi_k}_infty<infty ight}, eex$$ and $$ex calH:=left{H: [0,T] imesOmega o bR mbox{ is continuous and } calF_tmbox{-adapted}: ight.\ left. sen{H}_{calH}^2 := Esez{int_0^Tsev{H(s)}^2 d s}<infty ight}. eex$$ Set $$ex calM:=left{ M=(M_t)_{tin [0,T]} mbox{ is continuous } calF_tmbox{-martingales such that } ight.\ left. sen{M}_calM^2 :=sup_{0leq tleq T} Esez{sev{M_t}^2} <+infty ight}. eex$$ Then $(calM,sen{cdot}_calM)$ is a Hilbert space. Let $xi: [0,T] imes Omega o bR$ be the simple process given by $$ex xi_t=sum_{k=0}^n xi_kchi_{[t_k,t_{k+1}]}(t), eex$$ where $0=t_0<t_1<cdots<t_n=T$, and $xi_kin calF_{t_k}$ such that $dps{sup_k sev{xi_k}<infty}$. Define $$ex M_t=int_0^txi_k d W_s :=sum_{k=0}^n xi_ksex{W_{t_{k+1}wedge t-W_{t_kwedge t}}}, eex$$
a) Prove that $M_t$ is a continuous $calF_t$-martingale.
b) Prove the It^o's isometry identity: $$ex Esez{sev{M_t}^2} = Esez{int_0^tsev{xi_s}^2 d s}. eex$$
c) Using the Doob maximal inequality, prove that $$ex Esez{sup_{0leq tleq T} sev{M_t}^2} leq 4 Esez{int_0^T sev{xi_s}^2 d s}. eex$$
d) Given $Hin calH$, let $H_nin calS$ be a sequence such that $sen{H_n-H}_{calH} o 0$ as $n oinfty$. Prove that $dps{M_t^n =int_0^t H_n(s) d W_s}$ is a Cauchy sequence in $sex{calM,sen{cdot}_calM}$. Let $M$ be the limit of $sed{M_n(t); tin [0,T]}$ in $sex{calM,sen{cdot}_calM}$. Prove that this limit does not depend on the choice of the sequence $H_n$ which tends to $H$ in $sex{calH,sen{cdot}_calH}$. Denote by $dps{M_t:=int_0^t H(s) d W_s}$, i.e. $$ex int_0^t H(s) d W_s =lim_{n oinfty} int_0^t H_n(s) d W_s,mbox{ in } sex{calM,sen{cdot}_calM}. eex$$
e) Prove that $dps{M_t=int_0^t H(s) d W_s}$ is a $calF_t$-martingale and satisfies $$ex Esez{sev{M_t}^2} = Esez{int_0^t sev{H(s)}^2 d s}, eex$$ and $$ex Esez{sup_{0leq tleq T}sev{M_t}^2} leq 4 Esez{int_0^T sev{H(s)}^2 d s}. eex$$
f) Using the Borel-Cantelli lemma, prove that $ P$-a.s., $M=(M_t)in C([0,T];bR)$.
2. Consider the following SDE on $bR^m$: $$ex d X_t= d W_t- V(X_t) d t,quad X_0=x, eex$$ where $Vin C_b^2(bR^m)$. Fix $T>0$. Suppose that $u(t,x)in C_b^{1,2}([0,T] imesbR^m,bR)$ is a solution of the heat equation $$ex left{a{ll} frac{p u}{p t}(t,x) =frac{1}{2}lap u(t,x) -sef{ V(x), u(t,x)},&mbox{in }[0,T) imes bR^m,\ u(0,x)=f(x),&xin bR^m, ea ight. eex$$ where $fin C_b(bR^m)$. Applying It^o's formula to $u(T-t,X_t)$, prove that $$ex u(t,x)= E_xsez{f(X_t)},quad forall tgeq 0, xin bR^m. eex$$
3. Consider the following SPDE on $[0,T] imes S^1$: $$eelabel{1} frac{p}{p t}u(t,x) =lap u+dot W(t,x), eee$$ where $tin [0,infty)$ and $xin S^1=[0,2pi]$, $dps{lap=frac{p^2}{p x^2}}$ is the Laplace operator on $S^1$, and $W(t,x)$ is the space-time white noise on $[0,infty) imes S^1$. Recall that $lap$ is a compact operator on $L^2(S^1, d x)$ and the spectral of $lap$ is given by $$ex mbox{Sp}(lap)=sed{-n^2; nin bN}. eex$$ Indeed, let $$ex e_{2n}(x)=frac{1}{sqrt{pi}}cos(nx),quad e_{2n+1}(x)=frac{1}{sqrt{pi}} sin (nx),quad ninbN, xin S^1. eex$$ Then $$ex lap e_{2n}=-n^2 e_{2n},quad lap e_{2n+1}=-n^2e_{2n+1},quad forall ninbN. eex$$ The set $sed{e_n}$ consists of a complete orthonormal basis of $L^2(S^1, d x)$. Write $$ex W(t,x)=sum_{n=1}^infty W_n(t)e_n(x), eex$$ where $W_n(t)$ are i.i.d Brownian motion on $bR^1$.
(a) Let $$ex X_t(cdot) =u(t,cdot)in L^2(S^1, d x). eex$$ Prove that $X_t$ satisfies the Ornstein-Uhlenbeck SDE on $L^2(S^1, d x)$: $$ex d X_t=lap X_t+ d W_t, eex$$ and $dps{W_t=sum_{n=0}^infty W_n(t)e_n}$ is the cylinder Brownian motion on $L^2(S^1, d x)$.
(b) Let $dps{u(t,x)=sum_{nin bN} u_n(t)e_n(x)}$ be the orthogonal decomposition of $u(t,cdot)$ in $L^2(S^1, d x)$. Prove that $u_n(t)$ satisfies the Langevin SDE on $bR$: $$ex d u_n(t)=-n^2 u_n(t) d t+ d W_n(t), eex$$ and solve this Langevin SDE with initial condition $u_n(0)=u_nin bR$.
© Find the mild solution to the SPDE eqref{1} with initial condition $dps{u(0,x)=sum_{n=0}^infty u_ne_n(x)}$ for $dps{sum_{n=0}^infty sev{u_n}^2<+infty}$.
(d) Recall that the domain of $lap$ is given by $$ex H_0=left{u=sum_{n=1}^infty u_ne_nin L^2(S^1, d x); u_n=sef{u,e_n}, ight.\ left.mbox{ and } sum_{n=1}^infty n^2 sev{u_n}^2<infty ight} . eex$$ Let $$ex d mu(u)=prod_{n=1}^infty frac{n}{sqrt{2pi}} mbox{exp}sez{-frac{n^2sev{u_n}^2}{2}} d u_n. eex$$ Prove that $mu$ is a Gaussian measure on $(H,calB(H))$ with mean zero and with covariance matrix $Q=sex{q_{ij}}_{bN imesbN}$ with $$ex q_{ij}=frac{1}{i^2}delta_{ij}, eex$$ i.e., $mu=calN(0,Q)$. Formally we write $$ex Q=sex{-lap}^{-1},quad mu=calN(0,sex{-lap}^{-1}). eex$$
(e) Prove that $mu$ is an invariant measure for the Ornstein-Uhlenbeck processs $X_t$ on $L^2(S^1, d x)$.
(f) (Not required) Prove that $mu$ is the unique invariant measure for the Ornstein-Uhlenbeck process $X_t$ on $L^2(S^1, d x)$.
可压 Navier-Stokes 方程
1. Consider the compressible fluid flow with damping: $$ex left{a{ll} p_t ho+Div( hobu)=0,\ p_t( hobu)+Divsex{ hobuotimesbu} + p=- hobu. ea ight. eex$$ Can this system satisfy Kawashima's condition?
2. Follow the similar analysis for Lemma 2.1 to prove (2.18) in Proposition 2.2.
3. Give the details of the proof of Lemma 3.1.
4. Give the details of the proof of Theorem 5.3.
5. Give the complete proof of Lemmas 6.4 and 6.5.
几何分析 [参考答案链接]
1.(15') 设 $R(X,Y): calX(M) o calX(M)$ 为曲率, 求证:
(1) $R(X,Y)(fZ_1+gZ_2) =fR(X,Y)Z_1+gR(X,Y)Z_2$, $forall X,Y,Z_1,Z_2in calX(M), f,hin C^infty (M)$;
(2) $R(X,Y)Z+R(Y,Z)X+R(Z,X)Y=0$, $forall X,Y,Zin calX(M)$.
2.(10') 设 $V(t), J(t)$ 是沿最短测地线 $gamma(t), tin [0,1]$ 的向量场, 它们满足 $$ex V(t)perp dotgamma(t),quad J(t)perp dotgamma(t),quad V(0)=J(0),quad V(1)=J(1), eex$$ 且 $J(t)$ 是 Jacobi 场, 求证: $$ex I(J,J)leq I(V,V), eex$$ 其中 $I$ 为 $gamma$ 上的指标形式.
3.(10') 设 $gamma(t): (-infty,+infty) o M$ 为一条测地直线, 相应地记 $$ex gamma_+=gamma|_{[0,+infty)},quad gamma_-=gamma|_{(-infty,0]} eex$$ 及两 Busemann 函数 $$ex B_{gamma_+}(x)=lim_{t o+infty}sez{d(x,gamma(t))-t}; eex$$ $$ex B_{gamma_-}(x)=lim_{t o-infty}sez{d(x,gamma(t))+t}. eex$$ 求证: $$ex B_{gamma_+}+B_{gamma_-}=0,quadmbox{在 } gamma mbox{ 上}; eex$$ $$ex B_{gamma_+}+B_{gamma_-}geq 0,quadmbox{在 } M mbox{ 上}. eex$$
4.(15') 设 $M$ 为紧流形, 再设 $g_{ij}(t)$ 满足 Ricci 流, 且 $f(t), au(t)$ 满足 $$ex frac{p }{p t}f=-lap f+sev{ f}^2-R+frac{n}{2 au},quad frac{p }{p t} au =-1. eex$$ 求证:
(1) $$ex frac{ d}{ d t}int_M sex{4pi au}^{-frac{n}{2}} e^{-f}\, d vol_{g_{ij}}=0; eex$$
(2) $$ex & &frac{ d }{ d x}int_M sez{ au sex{R+sev{ f}^2} +f-n}(4pi^ au)^{-frac{n}{2}} e^{-f}\, d vol_{g_{ij}}\ & &=int_M 2 au sev{R_{ij}+ _i _j f-frac{1}{2 au}g_{ij}}^2 (4pi au)^{-frac{n}{2}} e^{-f}\, d vol_{g_{ij}}. eex$$