Weak Existence and Uniqueness for Super-Brownian Motion with Irregular Drift

TL;DR

This work proves weak existence and weak uniqueness for a one-dimensional SPDE with square-root noise and irregular drift. By extending the duality between measure-valued branching processes and log-Laplace equations to include jump-noise perturbations and infinite jumps, the authors accommodate drifts that are non-Lipschitz or discontinuous at zero, including Hölder-at-zero drifts and piecewise-constant drifts. The analysis hinges on an initial-trace framework for the log-Laplace equation and a carefully constructed dual process, enabling control of exponential moments and moments of the dual jump count. They derive not only well-posedness but also finite cozero-set measure for zero-drift at zero, and conditions for survival with positive probability via rescaling and comparison to KPP-type equations with branching noise. The results substantially broaden the class of admissible drifts in SBM-type SPDEs and offer duality-based tools that may extend to other non-Lipschitz SPDEs.

Abstract

We establish weak existence and uniqueness for random field solutions of the one-dimensional SPDE \[ d_tX_t = \frac{1}{2}ΔX_t +h(X_t)+ \sqrt{X_t}\dot{W}, \quad t\geq 0,\] where $\dot{W}$ is space-time white noise and $h$ is a bounded drift with $h(0)\geq 0$. The proof relies on an extension of the duality relation of the super-Brownian motion, which allows us to treat a broad class of admissible drifts, including functions that are non-Lipschitz or discontinuous at zero. In particular, well-posedness is derived for certain drifts that are Hölder continuous at zero with exponent $α\in(0,1)$. We also allow discontinuous drifts of the form $h(x) = b_0\unicode{x1D7D9}_{x = 0} + b_1\unicode{x1D7D9}_{x>0},$ where $b_0 \geq 0$, $b_1 \in \mathbb{R}$. Additionally, if $h(0)=0$ and the initial condition is continuous and compactly supported, we show that the Lebesgue measure of the non-zero set of $X$ is finite. The proofs are based on duality. We use a log-Laplace equation, which is perturbed by jump noise as the equation for the dual process, and the jumps of the dual process are allowed to take infinite values. We believe that the results for the dual process are also of independent interest. Under suitable assumptions on $h$ we also prove survival of $X$ with positive probability, using rescaling and comparison to the KPP-equation with branching noise.

Theorems & Definitions (67)

• Theorem 1.1
• Theorem 1.2
• Conjecture 1.3
• Theorem 1.4
• Proposition 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Definition 3.5
• Proposition 3.6