# Convolution of gaussian process

Let $$\text{GP}(\mu(x),k(x,x^{\prime})$$ be a gaussian process. Here, $$\mu$$ is the mean function. Typically, $$\mu$$ equals to $$0$$. $$k$$ is the kernel function.

Can you define a convolution of the gaussian process, $$\int_{x} \text{GP}\left(\mu(x), k(x,x^{\prime})\right)f(x)dx$$?

Here, $$f$$ is the $$L_2$$ function.

Intuitively, this is a limitation of gaussian random variables, $$\int_{x} \text{GP}\left(\mu(x), k(x,x^{\prime})\right)f(x)dx \approx \sum_{i=0}^{N} f(x_i)X_{x_i}\approx N\left(\int_{x} \mu(x) f(x)dx, \int_{x,x^\prime}k(x,x^\prime)f(x)f(x^\prime)dxdx^\prime\right)$$

This intuition is true? And, how to formulate it? Maybe, this is a stochastic integral. What field in math should I study?