I would like to collect a list of known explicit expression of the form $$h(x,t) = \Bbb E [ f(Z\sqrt{t} +x)],$$ where $f:\Bbb R \to \Bbb [0,\infty )$ is a measurable, non-negative function and $Z \sim \mathcal{N}(0,1)$, this means $Z$ is normally distributed with mean $0$ and variance $1$. Alternatively, the quantity above can be written as $$\int_{-\infty}^\infty f(u \sqrt{t} + x) (2\pi)^{-1/2} e^{-u^2 /2} du = \int_{-\infty}^\infty f(u \sqrt{t} + x) \phi (u) du,$$ where $\phi(u) = (2\pi)^{-1/2} e^{-u^2 /2}$. Further define $\Phi (z) = \int_{-\infty}^z \phi(u)du$.
By "explicitely" I mean that the expression $h(x,t)=0$ has a chance to be solved for the variable $x$ in dependence of $t$ in an explicit way.
There are of course some trivial/known examples (normal distribution german wikipedia):
- $f(u) = 1$
- $f(u) = u^{n}$, then $h(x,t) = \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n}{2k}(2k-1)!!t^k x^{n-2k}$
- $f(u) = e^{\lambda u}$, then $h(x,t) = e^{\lambda x + \frac{t\lambda^2}{2}}$
Of special interest for me are functions $f$, which are symmetric, i.e. $f(u)=f(-u)$,
- $f(u) = \vert u\vert$, then $h(x,t) = \sqrt{t \frac{2}{\pi}} e^{-x/2t} + x(1 - 2\Phi (-x /\sqrt{t}))$ (folded normal distribution)
- $f(u) = \phi(u) $, then $h(x,t)= \frac{1}{\sqrt{2\pi}}\sqrt{\frac{t}{t+1}}e^{\frac{x^2}{2t(t+1)} - \frac{x^2}{2t}}$ (Example 6, Hans Rudolf Lerche, Boundary Crossing of Brownian Motion)
or functions, which vanish for $u < 0$, i.e. $f(u) = 0$ for $u<0$,
- $f(u) = 1_{[0,\infty )}(u)$, then $h(x,t) = 1 - \Phi (x/\sqrt{t})$