While deep neural networks (DNNs) and Gaussian Processes (GPs) are both popularly utilized to solve problems in reinforcement learning, both approaches feature undesirable drawbacks for challenging problems. DNNs learn complex nonlinear embeddings, but do not naturally quantify uncertainty and are often data-inefficient to train. GPs infer posterior distributions over functions, but popular kernels exhibit limited expressivity on complex and high-dimensional data. Fortunately, recently discovered conjugate and neural tangent kernel functions encode the behavior of overparameterized neural networks in the kernel domain. We demonstrate that these kernels can be efficiently applied to regression and reinforcement learning problems by analyzing a baseline case study. We apply GPs with neural network dual kernels to solve reinforcement learning tasks for the first time. We demonstrate, using the well-understood mountain-car problem, that GPs empowered with dual kernels perform at least as well as those using the conventional radial basis function kernel. We conjecture that by inheriting the probabilistic rigor of GPs and the powerful embedding properties of DNNs, GPs using NN dual kernels will empower future reinforcement learning models on difficult domains.