QuoteOriginally posted by: AlanThe normative Ito sde is dX = b dt + a dW, where W(t) is a Brownian motion.However, you believe that there exists an alternative sde, the Fermion sde, dX = b dt + a dZ, where Z(t) is some non-Gaussianprocess. You also believe this alternative sde leads to the standard Fokker-Planck equation: p_t = 0.5 (a^2 p)_xx - (b p)_x So, you are looking for a 'gap' in the standard theory that allows your alternative theory to exist. No. You clearly haven't understood what I have written since your post of Jan 14. I completely accept the idea that the process I am discussing is discrete and has the moment behavior (for the first two moments) that you described there. That makes complete sense to me. If you are fixated on believing what you just wrote, then it is no wonder that you keep trying to talk past me.QuoteAll you can find are "pre deltaT->0" versions where Z(t) is non-Gaussian before the continuum limit.As I said, I pretty much accepted that many posts ago -- except for one minor detail which is if we are only interested in stuff that doesn't depend on an sde (e.g. moments, FPE), does delta Z necessarily become Gaussian in the continuum limit? I don't have an opinion on that, just guessing that if it doesn't, then the distribution may not be identical for all t. And if I am wrong, then what does happen to delta Z in the limit deltaT->0 if it is incompatible with a Gaussian (except in the steady state or the limit t->infinity).QuoteAt this point, you are looking for gaps in Levy's characterization of a Brownian motion and higher moments conditions that allows for your theory to exist.Yes -- once we are quite clear that I am only speculating on an idea and do not confuse it with your view of what "my theory" is.QuoteFine, maybe some other board member will explore those issues with you. Hopefully so.QuoteIn the end, you have to make a belief choice.Belief #1 is that you have discovered some new thing, but are not mathematically equipped to quite 'prove it'.Belief #2 is that the standard Ito theory is really all there is -- given that the standard FPE is the target.Not really. I am more interested in extending my knowledge beyond the Ito/Gaussian box in a way that is relevant to my project. As it happens, thanks to responses from you, polter and frenchX in other threads, none of this thread is critical for my work so I am not forced to make a choice. I am just curious in a way that you seem to find too much of a stretch.
Last edited by Fermion
on January 15th, 2012, 11:00 pm, edited 1 time in total.