Multilevel bilinear systems of stochastic differential equations
The multilevel bilinear systems investigated in the paper are stochastic differential equations with a mean field type drift and a linear diffusion term of the form \[ dX^{N}_{i}(t) = \sum ^{L}_{k=1}\frac {c_{k}}{N^{k-1}} \left (\widetilde X^{N}_{i,k}(t)-X^{N}_{i}(t)\right ) dt + \frac {c_{L+1}}{N^{L}}\left (\theta -X^{N}_{i}(t)\right ) dt +\gamma X^{N}_{i} dW_{i}(t), \tag{1} \] \(i = (i_{1},\ldots ,i_{L})\), \(0\leq i_{j}\leq N-1\), where \(c_{1},\ldots ,c_{L+1},\theta ,\gamma \) are nonnegative constants, \(W=(W_{i})\) is an \(N^{L}\)-dimensional Wiener process, the initial condition \(X^{N}(0)\) is nonnegative and independent of \(W\), and \(\widetilde X^{N}_{i,k}(t)\) stands for the process \(N^{-k}\sum ^{N-1}_{i_{1},\ldots ,i_{k}=0}X^{N}_{i}(t)\). The long time behaviour of solution to (1) depends on the rate of spatial mixing (given by \(c_{j}\)'s) and on the noise intensity \(\gamma \). In the subcritical case \(\sum ^{L}_{k=1}(2c_{k}) ^{-1}<\gamma ^{-2}\), when the noise intensity is small compared to the rate of spatial mixing, the author proves that measures \(P_{N}\), induced on the space \({\mathcal C}(\mathbb R_{+},\mathbb R_{+})\) by processes \((\widetilde X^{N}_{i,L}(N^{L}t),\;t\geq 0)\), converge weakly as \(N\to \infty \) to the law of a solution to a one-dimensional stochastic bilinear differential equation whose coefficients depend only on \(c_{j}\)'s, \(\theta \) and \(\gamma \). Moreover, particular results concerning the critical case are established.
scientific article; zbMATH DE number 1134824