# Suitable kernels for Gaussian processes

Consider a stochastic process $$\{X_t \colon t \in T\}$$ indexed by a set $$T$$. We assume for simplicty that $$T \in \mathbb{R}^n$$. We assume that for any choice of indexes $$t_1, \dots, t_n$$, the random variables $$(X_{t_1}, \dots, X_{t_n})$$ are jointly distributed according to a multivariate Gaussian distribution with mean $$\mu = (0, \dots, 0)$$, for a given covariance matrix $$\Sigma$$.

Under this assumptions, the stochastic process is completely determined by the 2nd-order statistics. Hence, if we assume fixed mean at $$0$$, then the stochastic process is fully defined by the covariance matrix. This matrix can be defined in terms of the covariance function $$k(t_i, t_j) = \mbox{cov}(X_{t_i}, X_{t_j}).$$ It is well-known that functions $$k$$ as defined above are admissible Kernels. This fact is often used in probabilistic inference, when performing regression or classification.

Several functions can be suitable kernels, but only a few are used in practise, depending on the application. Given the large amount of related literature, can someone provide an up-to-date list of functions commonly used as kernels in this context?