On regularization by a small noise of multidimensional ODEs with non-Lipschitz coefficients

A. Pilipenko; A. Kulik

doi:10.37863/umzh.v72i9.6292

A. Pilipenko Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv,and Nat. Techn. Univ. Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”
A. Kulik Wroclaw Univ. Sci. and Technology, Poland

DOI: https://doi.org/10.37863/umzh.v72i9.6292

Abstract

UDC 519.21

In this paper we solve a selection problem for multidimensional SDE
$d X^{\epsilon}(t)=a(X^{\epsilon}(t))\, d t + \epsilon\sigma(X^{\epsilon}(t))\, d W(t),$
where the drift and diffusion are locally Lipschitz continuous outside of a fixed hyperplane $H.$
It is assumed that $X^{\epsilon}(0)=x^0\in H,$ the drift $a(x)$ has a Hoelder asymptotics as $x$ approaches $H,$ and the limit ODE $d X(t)=a(X(t))\, d t$ does not have a unique solution.
We show that if the drift pushes the solution away from $H,$ then the limit process with certain probabilities selects some extremal solutions to the limit ODE.
If the drift attracts the solution to $H,$ then the limit process satisfies an ODE with some averaged coefficients.
To prove the last result we formulate an averaging principle, which is quite general and new.

References

F. Attanasio, F. Flandoli, Zero-noise solutions of linear transport equations without uniqueness: an example, C. R. Acad. Sci. Paris, Ser. I, 347, no. 13-14, 753 – 756 (2009), https://doi.org/10.1016/j.crma.2009.04.027 DOI: https://doi.org/10.1016/j.crma.2009.04.027

R. Bafico, On the convergence of the weak solutions of stochastic differential equations when the noise intensity goes to zero, Boll. Unione Mat. Ital., 5, no. 1, 308 – 324 (1980).

R. Bafico, P. Baldi, Small random perturbations of Peano phenomena, Stochastics, 6, No. 3-4, 279 – 292 (1982), https://doi.org/10.1080/17442508208833208 DOI: https://doi.org/10.1080/17442508208833208

R. Buckdahn, Y. Ouknine, M. Quincampoix, On limiting values of stochastic differential equations with small noise intensity tending to zero, Bull. Sci. Math., 133, 229 – 237 (2009), https://doi.org/10.1016/j.bulsci.2008.12.005 DOI: https://doi.org/10.1016/j.bulsci.2008.12.005

V. S. Borkar, K. Suresh Kumar, A new Markov selection procedure for degenerate diffusions, J. Theor. Probab., 23, No. 3, 729 – 747 (2010), https://doi.org/10.1007/s10959-009-0242-6 DOI: https://doi.org/10.1007/s10959-009-0242-6

F. Delarue, F. Flandoli, The transition point in the zero noise limit for a 1D Peano example, Discrete Contin. Dyn. Syst., 34, No. 10, 4071 – 4083 (2014), https://doi.org/10.3934/dcds.2014.34.4071 DOI: https://doi.org/10.3934/dcds.2014.34.4071

F. Delarue, F. Flandoli, D. Vincenzi, Noise prevents collapse of Vlasov – Poisson point charges, Commun. Pure and Appl. Math., 67, No. 10, 1700 – 1736 (2014), https://doi.org/10.1002/cpa.21476 DOI: https://doi.org/10.1002/cpa.21476

F. Delarue, M. Maurelli, Zero noise limit for multidimensional SDEs driven by a pointy gradient (2019), arXiv preprint arXiv:1909.08702.

N. Dirr, S. Luckhaus, M. Novaga, A stochastic selection principle in case of fattening for curvature flow, Calc. Var. Part. Different. Equat., 13, No. 4, 405 – 425 (2001), https://doi.org/10.1007/s005260100080 DOI: https://doi.org/10.1007/s005260100080

H. J. Engelbert, W. Schmidt, Strong Markov continuous local martingales and solutions of one-dimensional stochastic differential equations (Part III), Math. Nachr., 151, No. 1, 149 – 197 (1991), https://doi.org/10.1002/mana.19911510111 DOI: https://doi.org/10.1002/mana.19911510111

A. Friedman, Partial differential equations of parabolic type, Prentice-Hall, New York (1964).

S. Herrmann, Phénomène de Peano et grandes déviations. (French) , C. R. Acad. Sci. Paris, Ser. I, ´ 332, No. 11, 1019 – 1024 (2001), https://doi.org/10.1016/S0764-4442(01)01983-8 DOI: https://doi.org/10.1016/S0764-4442(01)01983-8

N. Ikeda, S. Watanabe, Stochastic differential equations and diffusion processes, North-Holland Math. Library, 24, North-Holland Co. Publ., Amsterdam etc. (1981).

I. Karatzas, S. E. Shreve, Brownian motion and stochastic calculus, Springer, New York (1988), https://doi.org/10.1007/978-1-4684-0302-2 DOI: https://doi.org/10.1007/978-1-4684-0302-2

I. G. Krykun, S. Ya. Makhno, The Peano phenomenon for Ito equations, J. Math. Sci., 192, No. 4, 441 – 458 (2013), https://doi.org/10.1007/s10958-013-1407-5 DOI: https://doi.org/10.1007/s10958-013-1407-5

A. Kulik, Ergodic behavior of Markov processes, de Gruyter, Berlin, Boston (2017). DOI: https://doi.org/10.1515/9783110458930

A. Kulik, I. Pavlyukevich, Moment bounds for dissipative semimartingales with heavy jumps, http://arxiv.org/abs/2004.12449.

I. Pavlyukevich, A. Pilipenko, Generalized selection problem with Levy noise ´ (2020), arXiv preprint arXiv: 2004.05421.

A. Pilipenko, F. N. Proske, On a selection problem for small noise perturbation in the multidimensional case, Stoch. and Dyn., 18, No. 6 (2018), 23 p., https://doi.org/10.1142/S0219493718500454 DOI: https://doi.org/10.1142/S0219493718500454

A. Pilipenko, F. N. Proske, On perturbations of an ODE with non-Lipschitz coefficients by a small self-similar noise, Statistics & Probab. Lett., 132, 62 – 73 (2018), https://doi.org/10.1016/j.spl.2017.09.005. DOI: https://doi.org/10.1016/j.spl.2017.09.005

D. Trevisan, Zero noise limits using local times, Electron. Commun. Probab., 18, No. 31 (2013), 7 pp., https://doi.org/10.1214/ECP.v18-2587 DOI: https://doi.org/10.1214/ECP.v18-2587

A. Veretennikov, On strong solutions and explicit formulas for solutions of stochastic integral equations, Sb. Math., 111(153), No. 3, 387 – 403 (1981).