Compact Support Property of Super-Brownian Motion with Irregular Drift

Abstract

We study the one-dimensional stochastic partial differential equation \[ d_t X_t(x)=\frac{1}{2}ΔX_t(x) +b_1\unicode{x1D7D9}_{\{X_t(x)>0\}} +\sqrt{X_t(x)}\dot W(t,x), \] where $b_1>0$, $\dot W$ is space-time white noise, and the initial condition is a nonnegative, compactly supported continuous function. We prove that its unique weak solution has the compact support property. The drift term lies outside the usual regularity assumptions for the Dawson--Girsanov theorem. Instead, the proof is based on a layer decomposition in which the solution is constructed as the monotone limit of sums of super-Brownian motions with random immigration rates determined recursively by the positivity sets of the preceding layers. By comparison, the compact support property extends to a broader class of bounded nonnegative drifts vanishing at the origin. Finally, support-radius estimates yield a comparison of the total mass of $X$ with a squared Bessel process, which allows us to show that $X$ has a positive extinction probability.

Disclosure

“the authors was supported in part by the ISF grant No. 1985/22. Statement on AI use The mathematical results and core arguments of this paper were devel- oped by the authors independently. At a late stage of preparation, the authors used OpenAI’s Chat- GPT [5.6 Pro; accessed August 2026] for critical reading and language editing of the manuscript, including identifying passages and proof steps requiring further clarification. All resulting revi- sions were independently evaluated,”

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2026-08-23T05:17:25.323140Z

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