I'm currently working on a problem where I need to find a probability function, $P(x)$, that remains unchanged after a normal Gaussian convolution. Specifically, the function should satisfy the following integral equation: $$P(x) = \int_{-\infty}^{+\infty}\frac{1}{\sqrt{2\pi}} e^{-\frac{(x-t)^2}{2}}P(t)dt$$ And should be $1\geq P(x) \geq 0$ for any $x$.
I'm not sure how to approach this problem. Could someone guide me?