Problem: Let $K:\mathbb{R} \times (0,\infty)\to \mathbb{R}$ be a function such that the solution of the initial value problem \[ \frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}, ~u(x,0)=f(x), ~x\in \mathbb{R}, ~t>0, \] is given by \[ u(x,t)=\int_{\mathbb{R}}K(x-y,t)f(y){\mathrm{d} }y \] for all bounded continuous functions $f$. Then find the value of \[ \int_{\mathbb{R}}K(x,t){\mathrm{d}}x. \]