Compact Support Property of Super-Brownian Motion with Irregular Drift

We study the one-dimensional stochastic partial differential equation $$d_t{X}_t(x)=\frac{1}{2}\mathrm{\Delta }{X}_t(x)+b_1\text{unicode}{x1D7D9}_{\{X_t(x)>0\}}+\sqrt{X_t(x)}\dot{W}(t,x),$$ 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.