Persistence Exponent for the Simple Diffusion Equation: The Exact Solution for any Integer Dimension

The persistence exponent ${\theta }_o$ for the simple diffusion equation ${\varphi }_t(x,t)=\mathrm{△}\varphi (x,t)$ , with random Gaussian initial condition {\color{red},} has been calculated exactly using a method known as selective averaging. The probability that the value of the field $\varphi$ at a specified spatial coordinate remains positive throughout for a certain time $t$ behaves as $t^{-{\theta }_o}$ for asymptotically large time $t$. The value of ${\theta }_o$, calculated here for any integer dimension $d$, is $\theta_o = \frac{d}{4}$for$d\leq 4$and$1$ otherwise. This exact theoretical result is being reported possibly for the first time and is not in agreement with the accepted values ${\theta }_o=0.12,0.18,0.23$ for $d=1,2,3$ respectively.