Electronic Communications in Probability

Non-explosion by Stratonovich noise for ODEs

Mario Maurelli

Full-text: Open access

Abstract

We show that the addition of a suitable Stratonovich noise prevents the explosion for ODEs with drifts of super-linear growth, in dimension $d\ge 2$. We also show the existence of an invariant measure and the geometric ergodicity for the corresponding SDE.

Article information

Source
Electron. Commun. Probab., Volume 25 (2020), paper no. 68, 10 pp.

Dates
Received: 3 February 2020
Accepted: 2 September 2020
First available in Project Euclid: 23 September 2020

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1600826536

Digital Object Identifier
doi:10.1214/20-ECP347

Zentralblatt MATH identifier
07252788

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 37A25: Ergodicity, mixing, rates of mixing

Keywords
stochastic differential equations non-explosion by noise geometric ergodicity

Rights
Creative Commons Attribution 4.0 International License.

Citation

Maurelli, Mario. Non-explosion by Stratonovich noise for ODEs. Electron. Commun. Probab. 25 (2020), paper no. 68, 10 pp. doi:10.1214/20-ECP347. https://projecteuclid.org/euclid.ecp/1600826536


Export citation

References

  • [1] John A. D. Appleby, Xuerong Mao, and Alexandra Rodkina, Stabilization and destabilization of nonlinear differential equations by noise, IEEE Trans. Automat. Control 53 (2008), no. 3, 683–691.
  • [2] D. G. Aronson and James Serrin, Local behavior of solutions of quasilinear parabolic equations, Arch. Rational Mech. Anal. 25 (1967), 81–122.
  • [3] Avanti Athreya, Tiffany Kolba, and Jonathan C. Mattingly, Propagating Lyapunov functions to prove noise-induced stabilization, Electron. J. Probab. 17 (2012), no. 96, 38.
  • [4] Luigi Amedeo Bianchi, Dirk Blömker, and Meihua Yang, Additive noise destroys the random attractor close to bifurcation, Nonlinearity 29 (2016), no. 12, 3934–3960.
  • [5] Luigi Amedeo Bianchi and Franco Flandoli, Stochastic Navier-Stokes equations and related models, Milan J. Math. 88 (2020), no. 1, 225–246.
  • [6] Jeremiah Birrell, David P. Herzog, and Jan Wehr, The transition from ergodic to explosive behavior in a family of stochastic differential equations, Stochastic Process. Appl. 122 (2012), no. 4, 1519–1539.
  • [7] A. P. Carverhill and K. D. Elworthy, Flows of stochastic dynamical systems: the functional analytic approach, Z. Wahrsch. Verw. Gebiete 65 (1983), no. 2, 245–267.
  • [8] R. Catellier and M. Gubinelli, Averaging along irregular curves and regularisation of ODEs, Stochastic Process. Appl. 126 (2016), no. 8, 2323–2366.
  • [9] Pao-Liu Chow and Rafail Khasminskii, Almost sure explosion of solutions to stochastic differential equations, Stochastic Process. Appl. 124 (2014), no. 1, 639–645.
  • [10] Hans Crauel and Franco Flandoli, Additive noise destroys a pitchfork bifurcation, J. Dynam. Differential Equations 10 (1998), no. 2, 259–274.
  • [11] Randal Douc, Gersende Fort, and Arnaud Guillin, Subgeometric rates of convergence of $f$-ergodic strong Markov processes, Stochastic Process. Appl. 119 (2009), no. 3, 897–923.
  • [12] K. D. Elworthy, Stochastic differential equations on manifolds, London Mathematical Society Lecture Note Series, vol. 70, Cambridge University Press, Cambridge-New York, 1982.
  • [13] E. Fedrizzi and F. Flandoli, Pathwise uniqueness and continuous dependence of SDEs with non-regular drift, Stochastics 83 (2011), no. 3, 241–257.
  • [14] F. Flandoli, M. Gubinelli, and E. Priola, Well-posedness of the transport equation by stochastic perturbation, Invent. Math. 180 (2010), no. 1, 1–53.
  • [15] Franco Flandoli, Random perturbation of PDEs and fluid dynamic models, Lecture Notes in Mathematics, vol. 2015, Springer, Heidelberg, 2011, Lectures from the 40th Probability Summer School held in Saint-Flour, 2010.
  • [16] Thomas C. Gard, Introduction to stochastic differential equations, Monographs and Textbooks in Pure and Applied Mathematics, vol. 114, Marcel Dekker, Inc., New York, 1988.
  • [17] Paul Gassiat and Benjamin Gess, Regularization by noise for stochastic Hamilton-Jacobi equations, Probab. Theory Related Fields 173 (2019), no. 3-4, 1063–1098.
  • [18] Benjamin Gess and Scott Smith, Stochastic continuity equations with conservative noise, J. Math. Pures Appl. (9) 128 (2019), 225–263.
  • [19] Martin Hairer and Jonathan C. Mattingly, Yet another look at Harris’ ergodic theorem for Markov chains, Seminar on Stochastic Analysis, Random Fields and Applications VI, Progr. Probab., vol. 63, Birkhäuser/Springer Basel AG, Basel, 2011, pp. 109–117.
  • [20] R. Z. Has’minskii, Ergodic properties of recurrent diffusion processes and stabilization of the solution of the Cauchy problem for parabolic equations, Teor. Verojatnost. i Primenen. 5 (1960), 196–214.
  • [21] David P. Herzog and Jonathan C. Mattingly, Noise-induced stabilization of planar flows I, Electron. J. Probab. 20 (2015), no. 111, 43.
  • [22] David P. Herzog and Jonathan C. Mattingly, Noise-induced stabilization of planar flows II, Electron. J. Probab. 20 (2015), no. 113, 37.
  • [23] Ioannis Karatzas and Steven E. Shreve, Brownian motion and stochastic calculus, second ed., Graduate Texts in Mathematics, vol. 113, Springer-Verlag, New York, 1991.
  • [24] Rafail Khasminskii, Stochastic stability of differential equations, second ed., Stochastic Modelling and Applied Probability, vol. 66, Springer, Heidelberg, 2012, With contributions by G. N. Milstein and M. B. Nevelson.
  • [25] Tiffany Kolba, Anthony Coniglio, Sarah Sparks, and Daniel Weithers, Noise-induced stabilization of perturbed Hamiltonian systems, Amer. Math. Monthly 126 (2019), no. 6, 505–518.
  • [26] Matti Leimbach and Michael Scheutzow, Blow-up of a stable stochastic differential equation, J. Dynam. Differential Equations 29 (2017), no. 2, 345–353.
  • [27] Xue-Mei Li and Michael Scheutzow, Lack of strong completeness for stochastic flows, Ann. Probab. 39 (2011), no. 4, 1407–1421.
  • [28] Lei Liu and Yi Shen, Noise suppresses explosive solutions of differential systems with coefficients satisfying the polynomial growth condition, Automatica J. IFAC 48 (2012), no. 4, 619–624.
  • [29] Xuerong Mao, Glenn Marion, and Eric Renshaw, Environmental Brownian noise suppresses explosions in population dynamics, Stochastic Process. Appl. 97 (2002), no. 1, 95–110.
  • [30] H. P. McKean, Jr., Stochastic integrals, Probability and Mathematical Statistics, No. 5, Academic Press, New York-London, 1969.
  • [31] Sean Meyn and Richard L. Tweedie, Markov chains and stochastic stability, second ed., Cambridge University Press, Cambridge, 2009, With a prologue by Peter W. Glynn.
  • [32] Salah-Eldin A. Mohammed, Torstein K. Nilssen, and Frank N. Proske, Sobolev differentiable stochastic flows for SDEs with singular coefficients: applications to the transport equation, Ann. Probab. 43 (2015), no. 3, 1535–1576.
  • [33] Daniel Revuz and Marc Yor, Continuous martingales and Brownian motion, third ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, Springer-Verlag, Berlin, 1999.
  • [34] M. Scheutzow, Stabilization and destabilization by noise in the plane, Stochastic Anal. Appl. 11 (1993), no. 1, 97–113.
  • [35] Michael Scheutzow, An integral inequality and its application to a problem of stabilization by noise, J. Math. Anal. Appl. 193 (1995), no. 1, 200–208.
  • [36] Daniel W. Stroock and S. R. Srinivasa Varadhan, Multidimensional diffusion processes, Classics in Mathematics, Springer-Verlag, Berlin, 2006, Reprint of the 1997 edition.
  • [37] A. Ju. Veretennikov, Strong solutions and explicit formulas for solutions of stochastic integral equations, Mat. Sb. (N.S.) 111(153) (1980), no. 3, 434–452, 480.
  • [38] A. Yu. Veretennikov, On polynomial mixing bounds for stochastic differential equations, Stochastic Process. Appl. 70 (1997), no. 1, 115–127.
  • [39] A. Yu. Veretennikov, On polynomial mixing and the rate of convergence for stochastic differential and difference equations, Teor. Veroyatnost. i Primenen. 44 (1999), no. 2, 312–327.
  • [40] Fuke Wu and Shigeng Hu, Suppression and stabilisation of noise, Internat. J. Control 82 (2009), no. 11, 2150–2157.