Invariant measures for SPDEs with reflection

J. Math. Sci. Univ. Tokyo
Vol. 11 (2004), No. 4, Page 425--446.

Otobe, Yoshiki
Invariant measures for SPDEs with reflection
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Abstract:
We investigate the stationary distribution for a time evolution of continuous random fields on $\R$ driven by Langevin equation taking nonnegative values only. The dynamics have a reflecting wall at zero. It is known that a stationary distribution for the dynamics without reflection is expressed by locally transforming a shift-invariant Gaussian measure on $C(\R,\R)$ in a proper way. The purpose of this paper is to establish a similar relationship for the dynamics with reflection. It will be shown that a Gibbs measure with hard-wall external potential, which is expressed by using 3-dimensional Bessel bridge, is a reversible (and therefore stationary) measure for such dynamics. When the potential is strictly convex, the stationary distribution and the Gibbs measure are both unique in a class of tempered distributions and therefore coincide with each other.

Keywords: SPDE with reflection, Gibbs measure with hard-wall external potential

Mathematics Subject Classification (2000): 60H15, 82C21.
Mathematical Reviews Number: MR2110922