Is an ITO diffusion time slice always Normally distributed?

As the title says, if we take a time slice on any Ito diffusion - are we guaranteed that the data is always Normally distributed?

• This seems like a useful property for computer generalization and when the distribution of the forward propagation is hard to solve.

Note: My question does not include fractal diffusion.

• This seems trivially false since Ito diffusions are closed under "nice" functions (like the conditions of Ito's lemma, say). So if $X_t$ is a Brownian motion (therefore an Ito process), then $Y_t = e^{X_t}$ is also an Ito process, and has lognormal distribution at a fixed time. Was there a particular reason you expected this to be always normal? – Chris Haug Jun 1 '18 at 12:10
• @ChrisHaug Thank you - that makes sense. The reason I ask is because I am writing a web-app and instead of transferring all the path data from one point to another - I wanted to return an array with mean and sd for each time index - which would be much smaller in size. – Edv Beq Jun 1 '18 at 12:59
• @ChrisHaug A numerical schema for the forward kolmogorov would be helpful too if one exists. – Edv Beq Jun 1 '18 at 13:09