When is the Gaussian smoothed version of a function the same as the original function?

Question

To be more precise:

If $f(x) = \mathbb{E}_{\delta \sim \mathcal{N}(0,\,\mathbb{I}\sigma^{2})} u(x+\delta)$, what sort of function can $u(x)$ be s.t $f(x) = u(x)$ ?

I'm not sure what tags to attach, any suggestions for them are welcome

Answer 1

This is just a conjecture: Harmonic functions likely have this property.

We say a function $u:\mathbb R^d\rightarrow \mathbb R$ is harmonic if $$\Delta u = \sum_{i=1}^d \partial_{x_i}^2 u=0.$$

It is well know that the infinitesimal generator of a Brownian Motion is given by the operator $Lf(x)=\frac{1}{2}\Delta f(x)$. This means that if $B_t$ is Brownian Motion with $B_t\sim N(0,t\text{I})$ then for small $t$ we have $$\mathbb E[f(B_t+x)]=\mathbb E[f(B_t)|B_0=x]\approx f(x)+t\cdot\frac{1}{2}\Delta f(x)=f(x)$$ if $f$ is harmonic.

This can also be seen as harmonic functions satisfy the mean value property.

If your function is supposed to be one-dimensional, meaning $f:\mathbb R\rightarrow\mathbb R$ then linear functions are the only harmonic functions.