Making sense of covariance of space-time white noise as a product of delta distributions

Question

The covariance of space-time white noise $\dot{W}(x,t)$ is given by $\mathbb{E}\dot{W}(x,t)\dot{W}(y,s) = \delta(t-s)\delta(x-y)$, where the $\delta$-distribution satisfies $\delta(x) = 0$ if $x\neq 0$ and $\delta(x) = \infty$ if $x = 0$. What I don't get is, isn't the product of distributions undefined first of all? And also, what if $t=s$ but $y\neq x$ above? Then you'd get $0\times\infty$. What to do then?

Answer 1

One can multiply distributions when they contain different sets of variables. This operation is called the tensor product of distributions. One can also invertible do a linear change of variables inside distributions.