Verify that the probability density (f_{t}(x)) of (B(t)), [f_{t}(x)=frac{1}{sqrt{2 pi t} sigma} e^{-x^{2} /left(2 sigma^{2} t ight)},

Verify that the probability density \(f_{t}(x)\) of \(B(t)\),

\[f_{t}(x)=\frac{1}{\sqrt{2 \pi t} \sigma} e^{-x^{2} /\left(2 \sigma^{2} t\right)}, \quad t>0\]

satisfies with a positive constant \(c\) the thermal conduction equation

\[\frac{\partial f_{t}(x)}{\partial t}=c \frac{\partial^{2} f_{t}(x)}{\partial x^{2}}\]