In the context of generalised Black-Scholes models, $$\frac{\partial V}{\partial t}+\frac{1}{2} \sigma^2(S, t) S^2 \frac{\partial^2 V}{\partial S^2}+(r(t)-D(t)) S \frac{\partial V}{\partial S}-r(t) V=0,$$or
$$\frac{\partial V}{\partial t}+\frac{1}{2} \sigma^2(S, t) S^2 \frac{\partial^2 V}{\partial S^2}+(r(S,t)-D(S,t)) S \frac{\partial V}{\partial S}-r(S,t) V=0,$$
the price-asymptotic behaviour of call options can be mathematically described by, $$V(S,t) \approx S\,e^{-\int_0^t D(s)\,ds}-Ee^{-\int_0^t r(s)\,ds} ~~\text{as}~S \to \infty,$$where, $V(S,t)$ is the value of call option which depends on the asset price $S$ and time $t$ to the expiry $T$, $D(t)$ is the dividend yield, $E$ is the strike price, and $r(t)$ is the interest rate. This makes sense as in the long price run, there is a high likelihood that the underlying asset's price will eventually equal or exceed the strike price, and thereby, the option's intrinsic value approaches the difference between the underlying asset's price and the strike price.
However, while thinking about the case $T \to \infty,$ Of course, the distribution of option value tends to become more tightly centred around the intrinsic value $\max\{S-E,0\},$ and it is observable that option premium is higher than the terminal condition $\max\{S-E,0\},$ in some neighbourhood of expiry and it diffuses to $\max\{S-E,0\},$ near the strike. But I feel that the diffusion process is not much explicit (as in the case of heat distribution on a finite rod) in the distribution of option value although the underlying dynamics agrees with diffusion-dominated Brownian motion. Why it is so?
Thanks in advance