I have to compute the characteristic function of a random variable with density $$ f(y) = \int_0^{+\infty} \frac{1}{\sqrt{2\pi t}} \exp{\left( -\frac{|x-y|^2}{2t} \right)} p e^{-pt} \, \text{d} t $$ so I get the following integral: $$ \varphi_{p,x}(\xi) = \int_{\mathbb{R}} e^{i\xi y} \,\text{d} y \left[ \int_0^{+\infty} \frac{1}{\sqrt{2\pi t}} \exp{\left( -\frac{|x-y|^2}{2t} \right)} p e^{-pt} \, \text{d} t \right] \overbrace{=}^{\text{Fubini}} \int_0^{+\infty} p e^{-pt} \left[ \int_{\mathbb{R}} e^{i\xi y} \frac{\exp{\left( -\frac{|x-y|^2}{2t} \right)}}{\sqrt{2\pi t}} \, \text{d} y \right] \, \text{d} t, $$
but I have no idea how to continue this exercise.
There is a common strategy to break this kind of integrals?