Onsager Machlup Functional for Stochastic Evolution Equations in a Class of Norms

Transition pathways for a class of high dimensional stochastic dynamical systems with Lévy noise

“…Equipped with this uniform norm, D x 0 ([0, 1]) is a Banach space. Now, we define the Onsager-Machlup action functional [5] associated with the stochastic differential equation (2.1).…”

“…The Karhunen-Loève expansion [5] method is applied to investigate the Onsager-Machlup action functional by converting stochastic integrals to independent random variables, which is invalid for Lévy-driven process due to the nonequivalence of irrelevance and independence. It is hard to analyze the limiting behaviors.…”

Onsager-Machlup action functional for stochastic partial differential equations with Levy noise

“…Now, we define the Onsager-Machlup action functional [4] associated with the stochastic partial differential equation (2.6). Our goal is to study the limiting behaviour of ratios of the form…”

“…The first result has been obtained by Dembo and Zeitouni [14] for trace class elliptic stochastic partial differential equations on a bounded domain of R d , and then by Mayer-Wolf and Zeitouni [41] in the non-trace case. Bardina, Rovira and Tindel [4] derived the Onsager-Machlup action functional for stochastic evolution equations in Hilbert space. But for infinite dimensional systems with Lévy process, as far as we know, the Onsager-Machlup action functional is not yet derived.…”

“…Moreover, the Lévy process is much more complex (Brownian motion is a special case of Lévy process) which is hard to handle. The methods in [4,14,41] encouter some new difficulties here, because the random variables represented by Karhunen-Loève expansion are not independent. Inspired by the works [10,18], we are able to overcome these difficulties by a path representation and derive the Onsager-Machlup action functional for stochastic partial differential equations with Lévy process as well as Brownian motion.…”