Itô’s Stochastic Calculus and Probability Theory pp 251-259 | Cite as

# A Martin boundary connected with the ∞-volume limit of the focussing cubic Schrödinger equation

## Summary

The existence of a change of phase in the micro-canonical ensemble for the focussing cubic Schrödinger system was suggested by Lebowitz-Rose-Speer [1989]. Chorin [1994] disputes their numerical evidence; his own is based on a more sophisticated approximation to the micro-canonical distribution and leads him to the opposite conclusion. Perhaps the source of this contradictory testimony is the fact, proved here, that *the ∞-volume limit does not exist at any temperature* 0 < T < ∞ *or density* 0 < D < ∞. This does not preclude a more or less dramatic change in the ensemble, from high to low temperatures, but it does guarantee the existence of several distinct ∞-volume Gibbs states. These are related to a sort of “boundary” of the type introduced by Martin [1941] for the description of classical harmonic functions in general 3-dimensional regions, as will be explained below.

## Keywords

Schrodinger Equation Gibbs State Formal Density Martin Boundary Positive Harmonic Function## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Bourgain, J.: The Fourier transform restriction phenomena for certain lattice subsets and applications to non-linear evolution equations. IAS preprint (1994), to appear 1995.Google Scholar
- 2.Chorin, A.: Univ. CA, Berkeley preprint (1994).Google Scholar
- 3.Krylov, N. V.,
*Non-Linear Elliptic and Parabolic Equations of Second Order*, Riedel, Boston, Tokyo, 1987.Google Scholar - 4.Lebowitz,
**J**H. Rose, and**E**Speer: Statistical mechanics of the non-linear Schrödinger equation (2).*J. Stat. Phys***54**(1989), 17–56.Google Scholar - 5.
- 6.McKean, H. P.: Brownian motion with restoring drift: the micro-canonical ensemble.
*Comm. Math. Phys***160**(1994), 615–630.MathSciNetMATHCrossRefGoogle Scholar - 7.McKean, H. P. and K. Vaninsky: Statistical mechanics of non-linear wave equations: the petit and micro-canonical ensembles.
*Trends and Perspective Appl. Math.*, ed. L. Sirovich, Springer-Verlag, New York, 1994.Google Scholar - 8.Statistical mechanics of non-linear wave equations (4): cubic Schrödinger.
*Comm. Math. Phys.***168**(1995), 479–491.Google Scholar - 9.Williams,
**D**.:*Diffusions*,*Markov Processes*,*and Martingales*.**J**. Wiley & Sons, New York, 1979.Google Scholar