The Green's function of the parabolic Anderson model and the continuum directed polymer

TL;DR

This work constructs a regular Green's function for the parabolic Anderson model driven by space-time white noise and uses a superposition principle to couple all PAM solutions across general initial data. It proves strong growth, Hölder regularity, and a conservation law for growing initial conditions, and develops a comprehensive continuum directed polymer framework with existence, regularity, strong Feller properties, and a Karlin–McGregor–based strict total positivity result. The PAM/KPZ connection is made precise via the Hopf–Cole transform, enabling a unified treatment of Green's functions, KPZ solutions, and quenched continuum polymers. Together these results provide a robust, parallel framework for PAM, KPZ, and polymer measures, with implications for synchronization and ergodicity analyses in related work.

Abstract

We build a regular version of the field $Z_β(t,x|s,y)$ which describes the Green's function, or fundamental solution, of the parabolic Anderson model (PAM) with white noise forcing on $\mathbb{R}^{1+1}$: $\partial_t Z_β(t,x | s,y) =$ $\frac{1}{2}\partial_{xx} Z_β(t,x|s,y) + βZ_β(t,x | s,y)W(t,x)$, $Z_β(s,x | s,y) = δ(x-y)$ for all $-\infty < s \leq t < \infty$, all $x,y \in \mathbb{R}$, and all $β\in \mathbb{R}$ 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 $\displaystyle \lim_{x\to \pm\infty} x^{-1}\log Z_β(t,x)$, 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 $(x,y) \mapsto Z_β(t,x | s,y)$ is strictly totally positive for all $t>s$ and $β\in \mathbb{R}$.

Figures (3)

• Figure 5.1: The path $X^i$ starts at $y_i$ at time $s$ and reaches $B_i$ (dashed, at level $r$) at time $r$, while remaining inside the tube connecting $y_i$ and $B_i$ (drawn in solid lines) between times $s$ and $r$. Before time $v$, the tube is a symmetric cylinder with diameter $4\epsilon$ around the straight line segment connecting $y_i$ and $c_i$. The tube expands at a time $v$ (which is slightly smaller than $r$) in order to allow the path enough space to reach any point in $B_i$, while still not allowing for intersections with any other path.
• Figure 5.2: On $E_i$, the $i^{th}$ path is required to lie within $\epsilon$ of $\ell_{i,u_{m-1}}(u_j)$ at the times $\{u_j : j \in [m-1]\}$. Between these times, the path is required to remain inside the larger cylinder of radius $2\epsilon$ (the union of the light and dark grey regions) around $\ell_{i,v}(\cdot)$ (thick). The points $\ell_{i,u_{m-1}}(u_j)$ and some admissible values of $X_{u_j}^i$ are marked as bullets. An inadmissible path between these values (due to exiting the cylinder between $u_{m-2}$ and $u_{m-1}$) is drawn as a zigzag. In order to exit in this way, the path between $X_{u_{j-1}}^i$ and $X_{u_{j}}^i$ deviates from the straight line between those points by more than $\epsilon$.
• Figure 5.3: By Lemma \ref{['lem:rndbd']}, $X_{u_{m-1}}^i \in (c_i - \epsilon, c_i + \epsilon)$ (solid at level $u_{m-1}$) and $X_r^i \in B_i$ (dashed, at level $r$) with positive probability. If $\epsilon$ is such that $(c_i - \epsilon, c_i + \epsilon) \subset [a_i,b_i]$, then in order to exit the interval $[a_i-\delta/4,b_i+\delta/4]$, the path must deviate by more than $\delta/4$ from the straight line connecting $X_{u_{m-1}}^i$ and $X_r^i$ on the time interval $[u_{m-1},r]$.

Theorems & Definitions (89)

• Example 2.1
• Theorem 2.2
• Proposition 2.3
• Corollary 2.4
• Remark 2.5
• Theorem 2.6
• Remark 2.7
• Remark 2.8
• Theorem 2.9
• Corollary 2.10