TY - JOUR
T1 - Invariant measures of stochastic partial differential equations and conditioned diffusions
AU - Reznikoff, Maria G.
AU - Vanden-Eijnden, Eric
N1 - Funding Information:
We thank Gérard Ben Arous, Percy Deift, Weinan E, Bob Kohn, Henry McKean, Chuck Newman, and Esteban Tabak for interesting discussions. This work was partially supported by NSF via grants DMS01-01439, DMS02-09959 and DMS02-39625.
PY - 2005/2/15
Y1 - 2005/2/15
N2 - This work establishes and exploits a connection between the invariant measure of stochastic partial differential equations (SPDEs) and the law of bridge processes. Namely, it is shown that the invariant measure of ut = uxx + f(u) + √2ε η(x,t), where η(x,t) is a space-time white-noise, is identical to the law of the bridge process associated to dU = a(U) dx + √εdW (x), provided that a and f are related by εa″ (u) + 2a′ (u) a(u) = -2f(u), u ∈ ℝ. Some consequences of this connection are investigated, including the existence and properties of the invariant measure for the SPDE on the line, x ∈ ℝ.
AB - This work establishes and exploits a connection between the invariant measure of stochastic partial differential equations (SPDEs) and the law of bridge processes. Namely, it is shown that the invariant measure of ut = uxx + f(u) + √2ε η(x,t), where η(x,t) is a space-time white-noise, is identical to the law of the bridge process associated to dU = a(U) dx + √εdW (x), provided that a and f are related by εa″ (u) + 2a′ (u) a(u) = -2f(u), u ∈ ℝ. Some consequences of this connection are investigated, including the existence and properties of the invariant measure for the SPDE on the line, x ∈ ℝ.
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U2 - 10.1016/j.crma.2004.12.025
DO - 10.1016/j.crma.2004.12.025
M3 - Article
AN - SCOPUS:13844309279
SN - 1631-073X
VL - 340
SP - 305
EP - 308
JO - Comptes Rendus Mathematique
JF - Comptes Rendus Mathematique
IS - 4
ER -