In this article, we close a gap in the literature by proving existence of invariant measures for reflected stochastic partial differential equations with only one reflecting barrier. This is done by arguing that the sequence $$(u(t,\cdot ))_{t \ge 0}$$ ( u ( t , · ) ) t ≥ 0 is tight in the space of probability measures on continuous functions and invoking the Krylov–Bogolyubov theorem. As we no longer have an a priori bound on our solution as in the two-barrier case, a key aspect of the proof is the derivation of a suitable $$L^p$$ L p bound which is uniform in time.

中文翻译：

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反射随机偏微分方程不变测度的存在性

在本文中，我们通过证明只有一个反射屏障的反射随机偏微分方程存在不变测度，从而弥补了文献中的空白。这是通过论证序列 $$(u(t,\cdot ))_{t \ge 0}$$ ( u ( t , · ) ) t ≥ 0 在连续函数的概率测度空间中是紧的并调用 Krylov-Bogolyubov 定理。由于我们不再像在两个障碍的情况下那样对我们的解决方案有先验界限，因此证明的一个关键方面是推导合适的 $$L^p$$L p 界限，该界限在时间上是一致的。