The Linear Self-Attracting Diffusion Driven by Weighted-Fractional Brownian Motion I: Large Time Behaviors

Abstract

Let $B^{a,b}$ be a weighted-fractional Brownian motion with Hurst indexes a and b such that $a>-1$ and $0<b<1\wedge (1+a)$. In this paper, we consider the linear self-attracting diffusion

$ dX^{a,b}_t=dB^{a,b}_t-\theta \left( \int _0^t(X^{a,b}_t-X^{a,b}_s)ds-\nu \right) dt $

with $X^{a,b}_0=0$, where $\theta >0$, $\nu \in {{\mathbb {R}}}$ are two real parameters. The model is an analogue of the linear self-attracting diffusion (Cranston and Le Jan in Math Ann 303:87–93, 1995). We study large time behavior of the model. For this seemingly trivial generalization, we find that the large time behavior is much more complex than that of fractional Brownian motion, which can not be observed in the model driven by fractional Brownian motion.

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Keywords

Weighted-fractional Brownian motion / Self-attracting diffusion / The law of large numbers / Asymptotic distribution / 60G22 / 60H07 / 60F05 / 62M09

References

[1] AlósE, MazetO, NualartD. Stochastic calculus with respect to Gaussian processes. Ann. Prob., 2001, 29: 766-801

[2] AlsenafiA, Al-ForaihM, Es-SebaiyK. Least squares estimation for non-ergodic weighted fractional Ornstein–Uhlenbeck process of general parameters. AIMS Math., 2021, 6: 12780-12794

[3] BenaïmM, CiotirI, GauthierC-E. Self-repelling diffusions via an infinite dimensional approach. Stoch PDE Anal. Comp., 2015, 3: 506-530

[4] BenaïmM, LedouxM, RaimondO. Self-interacting diffusions. Probab. Theory Relat. Fields, 2002, 122: 1-41

[5] BertoinJ. Sur une intégrale pour les processus á $\alpha $-variation borné. Ann. Probab., 1989, 17: 1521-1535

[6] BojdeckiT, GorostizaLG, TalarczykA. Some extensions of fractional Brownian motion and sub-fractional Brownian motion related to particle systems. Elect. Commun. Probab., 2007, 12: 161-172

[7] BojdeckiT, GorostizaLG, TalarczykA. Occupation time limits of inhomogeneous Poisson systems of independent particles. Stoch. Proc. Appl., 2008, 118: 28-52

[8] BojdeckiT, GorostizaLG, TalarczykA. Self-similar stable processes arising from high density limits of occupation times of particle systems. Potent. Anal., 2008, 28: 71-103

[9] ChakravartiN, SebastianKL. Fractional Brownian motion model for polymers. Chem. Phys. Lett., 1997, 267: 9-13

[10] ChenQ, ShenG, WangQ. The local time of the linear self-attracting diffusion driven by weighted fractional Brownian motion. Bull. Korean Math. Soc., 2020, 57: 547-568

[11] CherayilJ, BiswasP. Path integral description of polymers using fractional Brownian walks. J. Chem. Phys., 1993, 11: 9230-9236

[12] ChowY, TeicherHProbability Theory, 19782New York. Springer.

[13] CranstonM, Le JanY. Self-attracting diffusion: two case studies. Math. Ann., 1995, 303: 87-93

[14] CranstonM, MountfordTS. The strong law of large numbers for a Brownian polymer. Ann. Probab., 1996, 24: 1300-1323

[15] DurrettR, RogersLCG. Asymptotic behavior of Brownian polymer. Probab. Theory Relat. Fields, 1992, 92: 337-349

[16] FöIllmer, H.: Calcul d’Itô sans probabilités. In: Azéme and Yor (eds.) Séinaire de Probabilités XV, Lecture Notes in Mathematics, pp. 143-145, No. 50. Springer (1980)

[17] GauthierC-E. Self attracting diffusions on a sphere and application to a periodic case. Electron. Commun. Probab., 2016, 21(53): 1-12

[18] HerrmannS, RoynetteB. Boundedness and convergence of some self-attracting diffusions. Math. Ann., 2003, 325: 81-96

[19] HerrmannS, ScheutzowM. Rate of convergence of some self-attracting diffusions. Stoch. Process. Appl., 2004, 111: 41-55

[20] KeddiA, MadaniF, BouchentoufAA. Nonparametric estimation of trend function for stochastic differential equations driven by a weighted fractional Brownian motion. Appl. Appl. Math., 2020, 15: 708-724

[21] MahmoudiF, TahmasebiM. The convergence of exponential Euler method for weighted fractional stochastic equations. Comput. Methods Differ. Equ., 2022, 10: 538-548

[22] MountfordT, TarrésP. An asymptotic result for Brownian polymers. Ann. Inst. H. Poincaré Probab. Stat., 2008, 44: 29-46

[23] NualartDMalliavin Calculus and Related Topics, 20062New York. Springer.

[24] PemantleR. Phase transition in reinforced random walk and RWRE on trees. Ann. Probab., 1988, 16: 1229-1241

[25] PipirasV, TaqquMS. Integration questions related to fractional Brownian motion. Probab. Theory Relat. Field., 2000, 118: 121-291

[26] SerflingRJApproximation Theorems of Mathematical Statistics, 2002, New York. Wiley.

[27] SunX, YanL, ZhangQ. The quadratic covariation for a weighted-fractional Brownian motion. Stoch. Dyn., 2017, 171750029

[28] SunX, YanL. Central limit theorems and parameter estimation associated with weighted-fractional Brownian motion. J. Stat. Plann. Inference, 2018, 192: 45-64

[29] SunX, YanL. A convergence on the linear self-interacting diffusion driven by $\alpha $-stable motion. Stochastics, 2021, 93: 1186-1208

[30] SunX, YanL. The laws of large numbers associated with the linear self-attracting diffusion driven by fractional Brownian motion and applications. J. Theor. Prob., 2022, 35: 1423-1478

[31] YanL, SunY, LuY. On the linear fractional self-attracting diffusion. J. Theor. Probab., 2008, 21: 502-516

[32] YanL, WangZ, JingH. Some path properties of weighted fractional Brownian motion. Stochastics, 2014, 86: 721-758

Funding

National Natural Science Foundation of China(11971101)

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature