Stochastic hyperbolic systems, small perturbations and pathwise approximation (Q2215992)

The author addresses several problems concerning systems of linear hyperbolic equations driven by a one-dimensional Brownian motion \(w\). Let \(s\in\mathbb R\) and let \(d,d^\prime\) be positive integers. First, the existence and uniqueness of solutions of the equation \[ \frac{\partial u(t,x)}{\partial t}=a_t(x,D)u(t,x)\circ\frac{dw(t)}{dt}+b_t(x,D)u(t,x)+f(t,x)\circ\frac{dw(t)}{dt}+g(t,x),\quad u(0,\cdot)=u_0(\cdot) \] on \(\mathbb R^d\) that are bounded in \(H^s\) and \(\gamma\)-Hölder continuous in \(H^{s-2}\) for every \(\gamma<\frac{1}{4}\) are proved. Here \(u_0\in(H^s)^{d^\prime}\), \(f\) and \(g\) have paths in \(C^0([0,T],(H^s)^{d^\prime})\), paths of \(f\) are bounded in \(L^4(\Omega,H^{s+1})\), paths of \(g+\langle f,w\rangle\) are bounded in \(L^4(\Omega,H^s)\), \(a_t(x,D),b_t(x,D)\) are a bounded family of pseudo-differential operators satisfying \[ |D^\alpha_\xi D^\beta_xc(x,\xi)|\le C(\alpha,\beta)(1+|\xi|))^{1-|\alpha|} \] for all multiindices \(\alpha,\beta\) and every \(c\in\{a_t,b_t:\,t\in[0,T]\}\), \(t\mapsto a_t,b_t\) are continuous in \(C^\infty(\mathbb R^{2d})\) and \[ |D^\alpha_\xi D^\beta_xc(x,\xi)|\le C(\alpha,\beta)(1+|\xi|))^{-|\alpha|} \] holds for all multiindices \(\alpha,\beta\) and every \(c\in\{A_t,B_t,L_t:\,t\in[0,T]\}\), where \[ A_t=a_t+a^*t,\quad B_t=b_t+b^*_t,\quad L_t=A_ta_t+a^*_tA_t. \] Second, it is shown that solutions of \[ du^\varepsilon=\sqrt\varepsilon a_t(x,D)u^\varepsilon\circ dw+b_t(x,D)u^\varepsilon\,dt,\quad u^\varepsilon(0)=u_0\in H^s \] converge (in probability) uniformly in \(H^{s-2}\), as \(\varepsilon\to 0+\), to the solution of \[ du=b_t(x,D)u,\quad u=u_0 \] and the laws of \(u^\varepsilon\) satisfy a large deviation principle in \(C_{u_0}([0,T],H^{s-2})\) with the good rate function \[ I_u(\phi)=\inf\left\{\frac{1}{2}\int_0^T|\dot h(t)|\,dt:\Psi(h)=\phi\right\}, \] where \(\Phi:C_0([0,T],\mathbb R)\to C_{u_0}([0,T],H^{s-2})\) is given by \[ \Phi(h)(t)=u_0+\int_0^ta(x,D)\Phi(h)(\tau)\dot h(\tau)\,d\tau+\int_0^tb(x,D)\Phi(h)(\tau)\,d\tau. \] Third, if one considers solutions of the equations \[ \frac{\partial u^n}{\partial t}=a_t(x,D)u^n(t)\dot w^n(t)+b_t(x,D)u^n+f(t,x)\dot w^n(t)+g(t,x),\quad u^n(0,\cdot)=u_0(\cdot), \] where \(w^n\) is the polygonal approximation of \(w\) on the partition \(\{iT/n:0\le i\le n\}\) then \[ \lim_{n\to\infty}\mathbb E\,\sup_{t\in[0,T]}|u^n(t)-u(t)|^2_{H^{s-2}}=0 \] and the topological support of the law of \(u\) in \(C^0([0,T];(H^{s-2})^{d^\prime})\) coincides with the closures of the sets \[ \{v(t,\phi):\,\phi\in H^\infty\},\quad\{v(t,\phi):\,\phi\in H_p^\infty\} \] in \(C^0([0,T];(H^{s-2})^{d^\prime})\), where \(H^\infty\) and \(H^\infty_p\) denote the sets of real infinitely and piecewise infinitely differentiable functions, respectively, starting from the origin and \(v\) solves the equation \[ dv(t)=a_t(x,D)v(t)\dot\phi(t)\,dt+b(x,D)v(t)\,dt,\quad v(0)=u_0. \] The author subsequently applies the above Wong-Zakai approximation result to a problem of existence of a stochastic flow of diffeomorphisms for a class of backward stochastic hyperbolic equations. Finally, the scalar equation \[ du=a_t(x,D)u(t)\circ dw(t)+b_t(x,D)u(t)\,dt,\quad u(0)=u_0\in H^s \] is considered where, in addition, the principal symbols of \(a_t\) and \(b_t\) are imaginary, and \(s>d/2\). Then \(u\) is defined for every \((t,x)\), \(u(t)\in\mathbb D^{1,2}(H^{s-2})\), \(D_\theta u(t)=0\) for \(\theta>t\) and \[ D_\theta u(t)=a_\theta(x,D)u(\theta)+\int_\theta^ta_\theta(x,D)D_\theta u(\tau)\circ dw(\tau)+\int_\theta^tb_\theta(x,D)D_\theta u(\tau)\,d\tau. \] Here \(D\) stands for the Malliavin derivative.