We establish finite time extinction with probability one for weak solutions of the Cauchy–Dirichlet problem for the 1D stochastic porous medium equation with Stratonovich transport noise and compactly supported smooth initial datum. Heuristically, this is expected to hold because Brownian motion has average spread rate \(\smash {O(t^\frac{1}{2})}\) whereas the support of solutions to the deterministic PME grows only with rate \(\smash {O(t^{\frac{1}{m{+}1}})}\). The rigorous proof relies on a contraction principle up to time-dependent shift for Wong–Zakai type approximations, the transformation to a deterministic PME with two copies of a Brownian path as the lateral boundary, and techniques from the theory of viscosity solutions.

对于具有Stratonovich输运噪声和紧支撑光滑初始基准的一维随机多孔介质方程，我们针对Cauchy-Dirichlet问题的弱解建立了有限时间灭绝。启发式地，由于布朗运动具有平均扩散速率\（\ smash {O（t ^ \ frac {1} {2}）} \），而对确定性PME的解的支持仅随着速率\（ \ smash {O（t ^ {\ frac {1} {m {+} 1}}}} \）。对于Wong-Zakai型近似，严格的证明依赖于直至时间变化的收缩原理，以布朗氏路径的两个副本为横向边界的确定性PME的转换，以及来自粘度解理论的技术。