2022/08/24 by Tom Alberts, Christopher Janjigian, Alberts, Tom +5 · 2 citations
Mathematics · Physics and Astronomy · Computer Science · #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Nonlinear Dynamics and Pattern Formation
paper · pdf · doi:10.48550/arxiv.2208.11255
We build a regular version of the field Zβ(t,x|s,y) which describes the Green's function, or fundamental solution, of the parabolic Anderson model (PAM) with white noise forcing on ℝ1+1: ∂t Zβ(t,x | s,y) = (1)/(2)∂xx Zβ(t,x|s,y) + βZβ(t,x | s,y)W(t,x), Zβ(s,x | s,y) = δ(x-y) for all -∞ < s ≤ t < ∞, all x,y ∈ ℝ, and all β∈ ℝ simultaneously. Through the superposition principle, our construction gives a pointwise coupling of all solutions to the PAM with initial or terminal conditions satisfying sharp growth assumptions, for all initial and terminal times. Using this coupling, we show that the PAM with a (sub-)exponentially growing initial condition admits conserved quantities given by the limits limx→ ±∞ x-1log Zβ(t,x), in addition to proving many new basic properties of solutions to the PAM with general initial conditions. These properties are then connected to the existence, regularity, and continuity of the quenched continuum polymer measures. Through the polymer connection, we also show that the kernel (x,y) ↦ Zβ(t,x | s,y) is strictly totally positive for all t>s and β∈ ℝ.