Invariant measures of stochastic partial differential equations and conditioned diffusions

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 u t = u x x + f ( u ) + 2 ɛ η ( x , t ) , where η ( x , t ) is a space–time white-noise, is identical to the law of the bridge process associated to d U = a ( U ) d x + ɛ d W ( x ) , provided that a and f are related by ɛ a ″ ( u ) + 2 a ′ ( u ) a ( u ) = − 2 f ( 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 .