We prove absolute continuity of the law of the solution, evaluated at fixed points in time and space, to a parabolic dissipative stochastic PDE on L²(G), where G is an open bounded domain in \(\mathbb {R}^{d}\) with smooth boundary. The equation is driven by a multiplicative Wiener noise and the nonlinear drift term is the superposition operator associated to a real function that is assumed to be monotone, locally Lipschitz continuous, and growing not faster than a polynomial. The proof, which uses arguments of the Malliavin calculus, crucially relies on the well-posedness theory in the mild sense for stochastic evolution equations in Banach spaces.

我们证明了在L ²（G）上到抛物线型耗散随机PDE的解的定律的绝对连续性，该定律在时间和空间上是固定的，其中G是\（\ mathbb {R} ^ { d} \），边界平滑。该方程由乘法维纳噪声驱动，非线性漂移项是与实函数相关的叠加算子，该实函数被认为是单调的，局部为Lipschitz连续的，并且增长速度不快于多项式。该证明使用Malliavin微积分的论点，在很大程度上依赖于Banach空间中随机演化方程的适度适度性理论。