can someone help me with this Question? $\\ $ Let $d \geq 2 $ and let $ f: \mathbb{R}^d \rightarrow \mathbb{R} $ be a twice differentiable function. Let $ p(t,x)= \frac{1}{(2\pi t)^{\frac{d}{2}}} e^{\frac{- \vert \vert x \vert \vert ^2}{2t}} $ for $ (t,x) \in \mathbb{R}^d \times (0, \infty) $. Then $$ \int p(t,x) \frac{1}{2} \Delta f(x) dx = \int f(x) \frac{dp(t,x)}{dt} dx $$
I tried just to calculate the derivative and then put it into the Right side but this dosent work very well. I know p(t,x) is just the multivariate normal Density for a $ N(0,t)$ Random variable but i dont know how to use it. Thanks for helping!!