Urbana Campus Research Calendar (OVCRI)

dv_t = 1/2 ∆v_tdt+α_ε(∇v_t)dW_t^ε,   t≥0,x∈R

and the 1-D stochastic heat equation with multiplicative conservative noise:

du_t = 1/2 ∆u_tdt+α_ε ∇(u_tdW_t^ε),   t≥0,x∈R.

Assume that W_t^ε = W_t∗ h_ε is the spatial convolution between the cylindrical Wiener process and a smooth compactly supported mollifier h_ε:=ε⁻¹h(ε⁻¹·). We show that when ε ↓ 0, the solutions (v_t^ε)_t≥0, and respectively (u_t^ε)_t≥0, converge in distribution to the cumulative Arratia flow, and respectively, the massive Arratia flow when the noise strength α_ε is properly tuned. This is based on ongoing joint work with Xiaolong Zhang.