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We provide a framework for probabilistic reasoning in Vadalog-based Knowledge Graphs (KGs), satisfying the requirements of ontological reasoning: full recursion, powerful existential quantification, expression of inductive definitions. Vadalog is a Knowledge Representation and Reasoning (KRR) language based on Warded Datalog+/-, a logical core language of existential rules, with a good balance between computational complexity and expressive power. Handling uncertainty is essential for reasoning with KGs. Yet Vadalog and Warded Datalog+/- are not covered by the existing probabilistic logic programming and statistical relational learning approaches for several reasons, including insufficient support for recursion with existential quantification, and the impossibility to express inductive definitions. In this work, we introduce Soft Vadalog, a probabilistic extension to Vadalog, satisfying these desiderata. A Soft Vadalog program induces what we call a Probabilistic Knowledge Graph (PKG), which consists of a probability distribution on a network of chase instances, structures obtained by grounding the rules over a database using the chase procedure. We exploit PKGs for probabilistic marginal inference. We discuss the theory and present MCMC-chase, a Monte Carlo method to use Soft Vadalog in practice. We apply our framework to solve data management and industrial problems, and experimentally evaluate it in the Vadalog system. Under consideration in Theory and Practice of Logic Programming (TPLP).