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Hinojosa-Calleja A. Exact uniform modulus of continuity for q-isotropic Gaussian random fields. Stat Probab Lett 2023. [DOI: 10.1016/j.spl.2023.109813] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 03/02/2023]
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Strong Local Nondeterminism and Exact Modulus of Continuity for Isotropic Gaussian Random Fields on Compact Two-Point Homogeneous Spaces. J THEOR PROBAB 2023. [DOI: 10.1007/s10959-022-01231-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/05/2023]
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Wang Y, Wu F, Yin G, Zhu C. Stochastic functional differential equations with infinite delay under non-Lipschitz coefficients: Existence and uniqueness, Markov property, ergodicity, and asymptotic log-Harnack inequality. Stoch Process Their Appl 2022. [DOI: 10.1016/j.spa.2022.03.008] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
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Exact Uniform Modulus of Continuity and Chung’s LIL for the Generalized Fractional Brownian Motion. J THEOR PROBAB 2022. [DOI: 10.1007/s10959-021-01148-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
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Temporal Moduli of Non-Differentiability for Linearized Kuramoto–Sivashinsky SPDEs and Their Gradient. Symmetry (Basel) 2021. [DOI: 10.3390/sym13071306] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/16/2022] Open
Abstract
Let U=U(t,x) for (t,x)∈R+×Rd and ∂xU=∂xU(t,x) for (t,x)∈R+×R be the solution and gradient solution of the fourth order linearized Kuramoto–Sivashinsky (L-KS) SPDE driven by the space-time white noise in one-to-three dimensional spaces, respectively. We use the underlying explicit kernels and symmetry analysis, yielding exact, dimension-dependent, and temporal moduli of non-differentiability for U(·,x) and ∂xU(·,x). It has been confirmed that almost all sample paths of U(·,x) and ∂xU(·,x), in time, are nowhere differentiable.
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Spatial Moduli of Non-Differentiability for Linearized Kuramoto–Sivashinsky SPDEs and Their Gradient. Symmetry (Basel) 2021. [DOI: 10.3390/sym13071251] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/17/2022] Open
Abstract
We investigate spatial moduli of non-differentiability for the fourth-order linearized Kuramoto–Sivashinsky (L-KS) SPDEs and their gradient, driven by the space-time white noise in one-to-three dimensional spaces. We use the underlying explicit kernels and symmetry analysis, yielding spatial moduli of non-differentiability for L-KS SPDEs and their gradient. This work builds on the recent works on delicate analysis of regularities of general Gaussian processes and stochastic heat equation driven by space-time white noise. Moreover, it builds on and complements Allouba and Xiao’s earlier works on spatial uniform and local moduli of continuity of L-KS SPDEs and their gradient.
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Spatial Moduli of Non-Differentiability for Time-Fractional SPIDEs and Their Gradient. Symmetry (Basel) 2021. [DOI: 10.3390/sym13030380] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/17/2022] Open
Abstract
High order and fractional PDEs have become prominent in theory and in modeling many phenomena. In this paper, we study spatial moduli of non-differentiability for the fourth order time fractional stochastic partial integro-differential equations (SPIDEs) and their gradient, driven by space-time white noise. We use the underlying explicit kernels and spectral/harmonic analysis, yielding spatial moduli of non-differentiability for time fractional SPIDEs and their gradient. On one hand, this work builds on the recent works on delicate analysis of regularities of general Gaussian processes and stochastic heat equation driven by space-time white noise. On the other hand, it builds on and complements Allouba and Xiao’s earlier works on spatial uniform and local moduli of continuity of time fractional SPIDEs and their gradient.
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Wang W, Su Z, Xiao Y. The moduli of non-differentiability for Gaussian random fields with stationary increments. BERNOULLI 2020. [DOI: 10.3150/19-bej1162] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Shen Y, Wang Y. Operator-scaling Gaussian random fields via aggregation. BERNOULLI 2020. [DOI: 10.3150/19-bej1133] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Lee CY, Xiao Y. Local nondeterminism and the exact modulus of continuity for stochastic wave equation. ELECTRONIC COMMUNICATIONS IN PROBABILITY 2019. [DOI: 10.1214/19-ecp264] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Tudor CA, Xiao Y. Sample paths of the solution to the fractional-colored stochastic heat equation. STOCH DYNAM 2016. [DOI: 10.1142/s0219493717500046] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
Let [Formula: see text] be the solution to the linear stochastic heat equation driven by a fractional noise in time with correlated spatial structure. We study various path properties of the process [Formula: see text] with respect to the time and to the space variable, respectively. In particular, we derive exact uniform moduli of continuity and Chung-type laws of iterated logarithm.
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Affiliation(s)
- Ciprian A. Tudor
- Laboratoire Paul Painlevé, Université de Lille 1, F-59655 Villeneuve d’Ascq, France
- Academy of Economical Studies, Piata Romana nr.6, Sector 1, Bucharest, Romania
| | - Yimin Xiao
- Department of Statistics and Probability, Michigan State University, East Lansing, MI 48824, USA
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Stationary Increments Harmonizable Stable Fields: Upper Estimates on Path Behaviour. J THEOR PROBAB 2016. [DOI: 10.1007/s10959-016-0698-0] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/21/2022]
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Li Y, Wang W, Xiao Y. Exact moduli of continuity for operator-scaling Gaussian random fields. BERNOULLI 2015. [DOI: 10.3150/13-bej593] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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