dvt = 1/2 ∆vtdt+αε(∇vt)dWtε, t≥0,x∈R
and the 1-D stochastic heat equation with multiplicative conservative noise:
dut = 1/2 ∆utdt+αε ∇(utdWtε), t≥0,x∈R.
Assume that Wtε = Wt∗ hε is the spatial convolution between the cylindrical Wiener process and a smooth compactly supported mollifier hε:=ε−1h(ε−1·). We show that when ε ↓ 0, the solutions (vtε)t≥0, and respectively (utε)t≥0, converge in distribution to the cumulative Arratia flow, and respectively, the massive Arratia flow when the noise strength αε is properly tuned. This is based on ongoing joint work with Xiaolong Zhang.