The parametrix method approach to diffusions in a turbulent Gaussian environment
Let \((\Omega , \mathcal F,\mathbf P)\) be a probability space, \((T_{t,x}; (t,x)\in \mathbb R\times \mathbb R^{n})\) a group of measurable transformations of \(\Omega \), \(Z:\Omega \to \mathbb R^{n}\) a random vector. Define \(V(t,x,\omega) = Z(T_{t,x}(\omega))\) and assume that the stationary random field \(V\) is Gaussian, zero-mean, and has paths continuous in \(t\) and twice continuously differentiable in \(x\) almost surely. Let the field \(V\) be incompressible, i.e. \(\operatorname {div} V(t,\cdot) =0\), and denote by \(R_{ij}(t,x) = \mathbf E V_{i}(t,x)V_{j}(0,0)\), \(1\leq i,j\leq n\), the covariance matrix. Let \((\Theta ,\mathcal G,\mathbf Q)\) be another probability space carrying a standard \(n\)-dimensional Brownian motion \(B\); a stochastic process \(X^{s,x}_{\varepsilon}(t)\) on \(\Omega \times \Theta \) solving the equation \[ X^{s,x}_{\varepsilon}(t) = x + \frac 1{\varepsilon} \int ^{t}_{s} V\left ( \frac {r}{\varepsilon ^{2}},\frac {X^{s,x}_{\varepsilon}(r)} {\varepsilon ^{\alpha}}\right)dr + \sqrt {2} B_{t-s}, \quad t\geq s, \tag{1} \] for some fixed \(\alpha \in \left [0,1\right [\), is considered. (Equation (1) is a rescaled version of an equation modelling a particle in a turbulent fluid.) Under a suitable assumption on the coefficients of \(\alpha \)-mixing of the random field \(V\) the family of processes \(X_\varepsilon (t)\) is shown to converge weakly over the space \(C(\left [0,\infty \right [;\mathbb R^{n})\) as \(\varepsilon \to 0+\) to an \(n\)-dimensional Wiener process whose covariance matrix \(A\) is given by \(A_{ij} = \int ^{\infty}_{-\infty} R_{ij}(t,0) dt + 2\delta_{ij}\), \(1\leq i,j\leq n\). The proof relies on careful analysis of the fundamental solution to the (random) Kolmogorov equation corresponding to (1).
weak convergence of stochastic processes
scientific article; zbMATH DE number 1364463
scientific article