Rescaled interacting diffusions converge to super Brownian motion (Q1413677)

The process \((X_t)_{t\geq 0}\) is considered that takes values in a suitable subspace \(\mathcal{X}\) of \([0,\infty)^{\mathbb{Z}^d}\) and that is the unique strong solution of the stochastic differential equation \[ dX_t=\mathcal{A}X_tdt+\sqrt{g(X_t)}dB_t, \] where \(\mathcal{A}\) is the \(q\)-matrix of a random walk on \(\mathbb{Z}^d\), \(g(x)=\kappa x(1-x/b)^+, x\geq 0\), and \(\big\{\big(B_t(i)\big)_{t\geq 0}\), \(i\in \mathbb{Z}^d\big\}\) is an independent family of standard Brownian motions. Let \[ X_t^N=\frac{1}{N}\sum_{i\in\mathbb{Z}^d}X_{tN}(i)\delta_{i/\sqrt{n}},\quad N\in \mathcal{N}, t\geq 0, \] be the rescaled process. It is shown that if the initial measure \(X_0^N\) converges as \(N\to\infty\) to a finite measure, then \(X^N\) converges to super Brownian motion with branching rate \(\gamma\) calculated explicitly. Other choices of \(g\) are considered as well.