## Electronic Communications in Probability

### Non-explosion by Stratonovich noise for ODEs

Mario Maurelli

#### 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
Accepted: 2 September 2020
First available in Project Euclid: 23 September 2020

https://projecteuclid.org/euclid.ecp/1600826536

Digital Object Identifier
doi:10.1214/20-ECP347

Zentralblatt MATH identifier
07252788

#### 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

#### 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.