Let $f$ be a function on $\mathbb{R}_+\times\mathbb{R}^d$. Let $L$ be some differential operator, like $L=\frac{\partial}{\partial t}+\frac{\partial^2}{\partial x^2}$. Consider for some function $g$ the PDE
$$ L[f]=g$$
equipped with the initial condition that $f(0,x)=h(x)$. If we are told the fundamental solution ( i.e the solution when $f(0,x)=\delta(x_0)$ ) how do I from this get the solution for when $f(0,x)=h(x)$ ?
As an example I am thinking of eq (12.27) the Langevin equation from the book Elements of Nonequilibrium Statistical Mechanics by V. Balakrishnan.
Im confused when comparing it to the wikipedia https://en.wikipedia.org/wiki/Fundamental_solution#Proof_that_the_convolution_is_a_solution because one is relating fundamental solutions about the RHS of the PDE and the other is relating them to initial conditions.