Multiple and weak Markov properties in Hilbert spaces with applications to fractional stochastic evolution equations
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=W...
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| Published in: | Stochastic processes and their applications Vol. 186; p. 104639 |
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| Language: | English |
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01.08.2025
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| ISSN: | 0304-4149, 1879-209X |
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| 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.
As an application, we consider the space–time fractional parabolic operator L=(∂t+A)γ of order γ∈(1/2,∞), where −A is a linear operator generating a C0-semigroup on U. We prove that the resulting solution process satisfies an Nth order Markov property if γ=N∈N and show that a necessary condition for the weakest Markov property is generally not satisfied if γ∉N. The relevance of this class of processes is twofold: Firstly, it can be seen as a spatiotemporal generalization of Whittle–Matérn Gaussian random fields if U=L2(D) for a spatial domain D⊆Rd. Secondly, we show that a U-valued analog to the fractional Brownian motion with Hurst parameter H∈(0,1) can be obtained as the limiting case of L=(∂t+ɛIdU)H+12 for ɛ↓0. |
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| AbstractList | 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. As an application, we consider the space–time fractional parabolic operator L=(∂t+A)γ of order γ∈(1/2,∞), where −A is a linear operator generating a C0-semigroup on U. We prove that the resulting solution process satisfies an Nth order Markov property if γ=N∈N and show that a necessary condition for the weakest Markov property is generally not satisfied if γ∉N. The relevance of this class of processes is twofold: Firstly, it can be seen as a spatiotemporal generalization of Whittle–Matérn Gaussian random fields if U=L2(D) for a spatial domain D⊆Rd. Secondly, we show that a U-valued analog to the fractional Brownian motion with Hurst parameter H∈(0,1) can be obtained as the limiting case of [Formula presented] for ɛ↓0. 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. As an application, we consider the space–time fractional parabolic operator L=(∂t+A)γ of order γ∈(1/2,∞), where −A is a linear operator generating a C0-semigroup on U. We prove that the resulting solution process satisfies an Nth order Markov property if γ=N∈N and show that a necessary condition for the weakest Markov property is generally not satisfied if γ∉N. The relevance of this class of processes is twofold: Firstly, it can be seen as a spatiotemporal generalization of Whittle–Matérn Gaussian random fields if U=L2(D) for a spatial domain D⊆Rd. Secondly, we show that a U-valued analog to the fractional Brownian motion with Hurst parameter H∈(0,1) can be obtained as the limiting case of L=(∂t+ɛIdU)H+12 for ɛ↓0. |
| ArticleNumber | 104639 |
| Author | Kirchner, Kristin Willems, Joshua |
| Author_xml | – sequence: 1 givenname: Kristin orcidid: 0000-0002-3609-9431 surname: Kirchner fullname: Kirchner, Kristin email: k.kirchner@tudelft.nl organization: Delft Institute of Applied Mathematics, Delft University of Technology, P.O. Box 5031, 2600 GA, Delft, The Netherlands – sequence: 2 givenname: Joshua orcidid: 0009-0000-2085-1935 surname: Willems fullname: Willems, Joshua email: j.willems@tudelft.nl organization: Delft Institute of Applied Mathematics, Delft University of Technology, P.O. Box 5031, 2600 GA, Delft, The Netherlands |
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| Cites_doi | 10.1142/S0219493702000340 10.1090/tran/6329 10.1080/01621459.2019.1611582 10.1007/s00211-020-01151-x 10.1016/j.spasta.2022.100599 10.1080/10618600.2019.1665537 10.1007/s10584-016-1809-8 10.1111/j.1467-9868.2011.00777.x 10.1214/21-AOS2138 10.1002/wics.1315 10.1137/S0040585X97986540 10.1093/imanum/dry091 10.1214/aoms/1177731234 10.1137/21M144788X 10.1137/1010093 10.1142/S0218202520500050 10.1007/s10543-018-0719-8 |
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| Keywords | Higher-order Markov property 60G22 35R60 Infinite-dimensional fractional Wiener process Matérn covariance 60J25 Spatiotemporal Gaussian process 60G15 |
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| Title | Multiple and weak Markov properties in Hilbert spaces with applications to fractional stochastic evolution equations |
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