Although social and biomedical scientists have long been interested in the process through which ideas and behaviors diffuse, the identification of causal diffusion effects, also known as peer and contagion effects, remains challenging. Many scholars consider the commonly used assumption of no omitted confounders to be untenable due to contextual confounding and homophily bias. To address this long-standing problem, we examine the causal identification under a new assumption of structural stationarity, which formalizes the underlying diffusion process with a class of dynamic causal directed acyclic graphs. First, we develop a statistical test that can detect a wide range of biases, including the two types mentioned above. We then propose a difference-in-differences style estimator that can directly correct biases under an additional parametric assumption. Leveraging the proposed methods, we study the spatial diffusion of hate crimes against refugees in Germany. After correcting large upward bias in existing studies, we find hate crimes diffuse only to areas that have a high proportion of school dropouts.