Multiple and weak Markov properties in Hilbert spaces with applications to fractional stochastic evolution equations

Kristin Kirchner and Joshua Willems

Stochastic Processes and their Applications, 2025, vol. 186, issue C

Abstract: We define a number of higher-order Markov properties for stochastic processes (X(t))t∈T, indexed by an interval T⊆R and taking values in a real and separable Hilbert space U. We furthermore investigate the relations between them. In particular, for solutions to the stochastic evolution equation LX=Ẇ, where L is a linear operator acting on functions mapping from T to U and (Ẇ(t))t∈T is the formal derivative of a U-valued cylindrical Wiener process, we prove necessary and sufficient conditions for the weakest Markov property via locality of the precision operator L∗L.

Keywords: Higher-order Markov property; Infinite-dimensional fractional Wiener process; Matérn covariance; Spatiotemporal Gaussian process (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1016/j.spa.2025.104639

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