Concentration estimates for SPDEs driven by fractional Brownian motion
Abstract: The main goal of this work is to provide sample-path estimates for the solution of slowly time-dependent SPDEs perturbed by a cylindrical fractional Brownian motion. Our strategy is similar to the approach by Berglund and Nader for space-time white noise. However, the setting of fractional Brownian motion does not allow us to use any martingale methods. Using instead optimal estimates for the probability that the supremum of a Gaussian process exceeds a certain level, we derive concentration estimates for the solution of the SPDE, provided that the Hurst index $H$ of the fractional Brownian motion satisfies $H>\frac14$. As a by-product, we also obtain concentration estimates for one-dimensional fractional SDEs valid for any $H\in(0,1)$.
- N. Berglund and B. Gentz. Pathwise description of dynamic pitchfork bifurcations with additive noise. Probability Theory Related Fields. 122: 341–388, 2002.
- N. Berglund and R. Nader. Stochastic resonance in stochastic PDEs. SPDEs: Analysis and Computation. 11(1): 348–387, 2023.
- N. Berglund and R. Nader. Concentration estimates for slowly time-dependent singular SPDEs on the two-dimensional torus. Electronic Journal of Probability. 29: 1–35 (2024).
- A. Blessing (Neamţu) and D. Blömker. Finite-time Lyapunov exponents for SPDEs with fractional noise. arXiv:2309.12189, 2023.
- D. Blömker and A. Neamţu. Amplitude equations for SPDEs driven by fractional additive noise with small Hurst parameter. Stoch. Dyn., 22(03):2240013, 2022.
- Hitting times for Gaussian processes. Ann. Probab., 319–330, 2008.
- T.E. Duncan, B. Pasik-Duncan and B. Maslowski. Fractional Brownian motion and stochastic equations in Hilbert spaces. Stoch. Dyn., 2(2):225–250, 2002.
- T.E. Duncan, B. Pasik-Duncan and B. Maslowski. Linear stochastic equations in a Hilbert space with a fractional Brownian motion. Book chapter (pp. 201–221) in Stochastic Processes, Optimization, and Control Theory: Applications in Financial Engineering, Queueing Networks, and Manufacturing Systems, 2006.
- T.E. Duncan, B. Pasik-Duncan and B. Maslowski. Semilinear stochastic equations in a Hilbert space with a fractional Brownian motion. SIAM. J. Math. Anal., 40(6):2286–2315, 2009.
- K. Eichinger, C. Kuehn and A. Neamţu. Sample Paths Estimates for Stochastic Fast-Slow Systems driven by Fractional Brownian Motion. J. Stat. Phys, 179(5):1222–1266, 2020.
- M. Hairer and X.-M. Li. Averaging dynamics driven by fractional Brownian motion. Ann. Probab., 48(4):1826–1860, 2020.
- M. Hairer and A. Ohasi. Ergodic theory for SDEs with extrinsic memory. Ann. Probab., 35(5):1950–1977, 2007.
- K. Kubilius, Y. Mishura and K. Ralchenko. Parameter Estimation in Fractional Diffusion Models, Springer 2017.
- C. Kuehn, K. Lux and A. Neamţu. Warning Signs for Non-Markovian Bifurcations: Color Blindness and Scaling Laws. Proc. Royal Soc. A, 478:20210740, 2022.
- F.W.J. Olver. Asymptotics and Special Functions. Academic Press, 1974.
- A. Pazy. Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer Applied Mathematical Series. Springer–Verlag, Berlin, 1983.
- V.I. Piterbarg. Asymptotic Methods in the Theory of Gaussian Processes and Fields. American Mathematical Society, 1996.
- L.C.G. Rogers. Arbitrage with Fractional Brownian Motion. Wiley, 2002.
- D. Slepian. The one-side barrier problem for Gaussian noise. Bell System Tech., 41(2):436– 501, 1962.
- S. Tindel, C.A. Tudor and F. Viens. Stochastic evolution equations with fractional Brownian motion. Probab. Theory. Rel. Fields, 127:186–204, 2003.
- M. Weiss. Single-particle tracking data reveal anticorrelated fractional Brownian motion in crowded fluids. Phys. Rev. E, 88(1):010101, 2013.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.