On the rate of convergence of solutions of the first order partial differential equations (Q2784981)

Let \(\eta(t)\), \(t\geq 0,\) be a stationary, in the restricted sense random process with zero mean satisfying the condition NEWLINE\[NEWLINEE\sup_{B\in F_{t+\tau}^{+\infty}}\left|P(B/F_0^{t})-P(B) \right|=\beta(\tau)\to 0,\quad \tau\to\infty,NEWLINE\]NEWLINE where \(F_0^{t}=\sigma\{\eta(s), s\in [0,t]\}\), \(F_{t+\tau}^{+\infty}=\sigma\{\eta(s), s\in [t+\tau,+\infty), \tau\geq 0\}\). The author proves an existence and uniqueness theorem for the solution of the Cauchy problem NEWLINE\[NEWLINE{\partial X^{\varepsilon}\over\partial t}+a(t,x){\partial X^{\varepsilon}\over\partial x}=\sigma(t,x)f(t,x,X^{\varepsilon}(t,x))+\sigma(t,x)\eta(t/\varepsilon),NEWLINE\]NEWLINE NEWLINE\[NEWLINEX^{\varepsilon}(t,x)|_{t=0}=\varphi(x),\quad \lim_{x\to-\infty}X^{\varepsilon}(t,x)=\lim_{x\to+\infty} X^{\varepsilon}(t,x),\quad t\in[0,T],NEWLINE\]NEWLINE where \(\varepsilon>0\) is a small parameter. Also the author studies the rate of convergence of solutions \(X^{\varepsilon}(t,x)\) of this Cauchy problem to solutions of the following Cauchy problem NEWLINE\[NEWLINE\partial\xi^{\varepsilon}+a(t,x){\partial \xi^{\varepsilon}\over\partial x} dt=\sigma(t,x)f(t,x,\xi^{\varepsilon}(t,x)) dt+ \sqrt{\varepsilon}\sigma(t,x) d\mu_{\varepsilon}(t),NEWLINE\]NEWLINE NEWLINE\[NEWLINE\xi^{\varepsilon}(t,x)|_{t=0}=\varphi(x),\quad \lim_{x\to-\infty}\xi^{\varepsilon}(t,x)=\lim_{x\to+\infty} \xi^{\varepsilon}(t,x),NEWLINE\]NEWLINE where \(\mu_{\varepsilon}(t)\) is a martingale.