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

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  • Kirchner, Kristin
  • Willems, Joshua

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.

Suggested Citation

  • Kirchner, Kristin & Willems, Joshua, 2025. "Multiple and weak Markov properties in Hilbert spaces with applications to fractional stochastic evolution equations," Stochastic Processes and their Applications, Elsevier, vol. 186(C).
    • Handle: RePEc:eee:spapps:v:186:y:2025:i:c:s0304414925000808
    DOI: 10.1016/j.spa.2025.104639
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