On the convergence of solutions of stochastic ordinary differential equations as stochastic flows of diffeomorphisms
Let \(\psi_ t^{\epsilon}\) be a solution of the ordinary stochastic differential equation \(dx/dt=\epsilon F(\epsilon^ 2t,x,t,\omega)+\epsilon^ 2G(\epsilon^ 2t,x,t,\omega)\) where \(F(\tau\),x,t,\(\omega)\) and \(G(\tau\),x,t,\(\omega)\) be random vector fields on \(R^ d\). Put \(\phi_ t^{\epsilon}\equiv\psi^{\epsilon}_{t/\epsilon^ 2}\), \(F_{\epsilon}(t,x)=(1/\epsilon)F(t,x,t/\epsilon^ 2)+G(t,x,t/\epsilon^ 2)\), \(X_ t^{\epsilon}(x)=\int_{[0,t]} F_{\epsilon}(s,x)ds.\) It is shown that the pair \((\phi^{\epsilon},X^{\epsilon})\) converges weakly to (\(\phi\),X) as \(\epsilon\to 0\), where \(\phi\) is a diffusion process with local characteristics determined from the random vector fields F and G in a suitable way, X is a Brownian motion with values in the space V of vector fields, \(\phi\) and X are connected by the stochastic differential equation \(d\phi_ t=dX_ t(\phi_ t)+c(t,\phi_ t)dt\) with a certain ''correction'' term \(c_ t(x)\). The main result is formulated as a limit theorem on the convergence of measures on the space C([0,T]\(\times D)\times C([0,T]\times V)\) where D is the diffeomorphism group on \(R^ d\).
scientific article; zbMATH DE number 3876307