Friends, I have come up with my postulated theory (which can be wrong. Not tested yet) of hermite polynomials in relation with normal dnesity, general hermite series densities and diffusions.

We know that a standard normal density can be expanded in terms of hermite series as derivatives of normal density can be described in terms of (after multiplication with density) hermite polynomial terms.

A random process is basically a random particle that is moving randomly to and fro along standard normal axis. This random particle is divided into several states and its properties have to be described as a sum/composition of properties of its different states. These states of random particle are called hermite polynomial states and these states vary in relative magnitude as the particle moves to and fro on standard normal axis. Towards the mid of the density smaller hermite polynomial states are dominant and towards the extremes of the density, larger hermite polynomial states are dominant. I suppose that each point on standard normal axis, the relative magnitude of the hermite states can be determined by relative contribution to standardized density at that point by different hermite polynomials.

The concept of hermite states is important since we can determine their relative magnitude in density at any point along standard normal axis as the particle moves to and fro on the normal axis. We have not done this with hermite polynomials earlier and this is where this goes beyond simple hermite polynomial analytics.

Each hermite state has its own variance and variance of the particle at a certain point on standard normal axis is combined weighted sum of the variances in each of the hermite states.

We can define other processes from standard normal density simply by assigning different coefficients to hermite states that magnify or diminish the variances of each hermite state.

Correlations are between hermite states of the same order between two stochastic processes.

When we develop the theory of transition probabilities of a stochastic process across time, these hermite states are very helpful. Our marginal density in state two in future has coefficients that are expected values. However transition density has higher variance than marginal density since there are many transitions that have zero expected value but they have significant variances.

On top of expected values of coefficients of the hermite series of marginal density in future (these expected values are associated with zeroth hermite transition. They are non-zero expectations of coefficients), there are transitions from 1st hermite state at time one to all the 1st, 2nd, 3rd up to fifth hermite(in our set up where fifth hermite is largest) at 2nd time in future. Similalry there are transitions from 2nd hermite state at time one to all the first to fifth hermite states at time two. And so on from 3rd, fourth and fifth hermite at time one to 1st to fifth hermite states at time two. All these hermite state(at time one) to hermite state(at time two) transitions are pure variances and have zero expected values. These pure variances do not contribute to marginal density at time two which comes solely from zeroth expectation hermite.

In our set up, all terms with single hermite polynomials contribute to marginal density coefficients. All terms with products of hermite polynomials are pure variances with zero expected values and they do not contribute to marginal density. They are purely related to transition portion of transition densities.

However, I believe that we can actually take all the realizations of stochastic process at time one and divide each realization to hermite states from knowledge of associated value of standard normal (this is different from only knowing polynomial value there and requires knowledge of relative magnitude of hermite state in density at that point). Similalry we can take associated realization of stochastic process at time two and divide it into hermite states again from knowledge of its associated standard normal. We can then study relative transitions from various hermite states at time one to various other hermite states at time two and calculate their transtioin coefficients numerically from pairs of historical data.

I am trying to brainstorm and think of other ideas and hope to share them with friends as I get to know something.

Please know that I have not tested anything in above theory so I may be totally wrong.