## Electronic Journal of Probability

### On uniqueness and blowup properties for a class of second order SDEs

#### Abstract

As the first step for approaching the uniqueness and blowup properties of the solutions of the stochastic wave equations with multiplicative noise, we analyze the conditions for the uniqueness and blowup properties of the solution $(X_t,Y_t)$ of the equations $dX_t= Y_tdt$, $dY_t = |X_t|^\alpha dB_t$, $(X_0,Y_0)=(x_0,y_0)$. In particular, we prove that solutions are nonunique if $0<\alpha <1$ and $(x_0,y_0)=(0,0)$ and unique if $1/2<\alpha$ and $(x_0,y_0)\neq (0,0)$. We also show that blowup in finite time holds if $\alpha >1$ and $(x_0,y_0)\neq (0,0)$.

#### Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 72, 17 pp.

Dates
Accepted: 16 August 2017
First available in Project Euclid: 13 September 2017

https://projecteuclid.org/euclid.ejp/1505268103

Digital Object Identifier
doi:10.1214/17-EJP95

Mathematical Reviews number (MathSciNet)
MR3698741

Zentralblatt MATH identifier
06797882

#### Citation

Gomez, Alejandro; Lee, Jong Jun; Mueller, Carl; Neuman, Eyal; Salins, Michael. On uniqueness and blowup properties for a class of second order SDEs. Electron. J. Probab. 22 (2017), paper no. 72, 17 pp. doi:10.1214/17-EJP95. https://projecteuclid.org/euclid.ejp/1505268103

#### References

• [1] M. Abramowitz and I. A. Stegun, Handbook of mathematical functions: with formulas, graphs, and mathematical tables, Dover Books on Mathematics, Dover Publications, 1965.
• [2] A. S. Cherny and H.J. Engelbert, Singular stochastic differential equations, Lecture Notes in Mathematics, vol. 1858, Springer Berlin Heidelberg, 2005.
• [3] P. L. Chow, Nonlinear stochastic wave equations: Blow-up of second moments in ${L}^2$-norm, The Annals of Applied Probability 19 (2009), no. 6, 2039–2046.
• [4] D. A. Dawson, Measure-valued Markov processes, École d’été de probabilités de Saint-Flour, XXI-1991 (Berlin, Heidelberg, New York) (P. L. Hennequin, ed.), Lecture Notes in Mathematics, no. 1180, Springer-Verlag, 1993, pp. 1–260.
• [5] K. Ito and H. P. Jr. McKean, Diffusion processes and their sample paths, Springer-Verlag, Berlin, Heidelberg, New York, 1974.
• [6] P. Mörters and Y. Peres, Brownian motion, Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 2010.
• [7] C. Mueller, Long time existence for the heat equation with a noise term, Probab. Theory Related Fields 90 (1991), 505–518.
• [8] C. Mueller, The critical parameter for the heat equation with a noise term to blow up in finite time, Ann. Probab. 28 (2000), no. 4, 1735–1746.
• [9] C. Mueller, L. Mytnik, and E. Perkins, Nonuniqueness for a parabolic spde with $\frac{3} {4}-\epsilon$ Hölder diffusion coefficients, Ann. Probab. 42 (2014), no. 5, 2032–2112.
• [10] C. Mueller and G. Richards, Can solutions of the one-dimensional wave equation with nonlinear multiplicative noise blow up?, Open Prob. Math 2 (2014), 1–4.
• [11] C. Mueller and R. Sowers, Blow-up for the heat equation with a noise term, Probab. Theory Related Fields 97 (1993), 287–320.
• [12] L. Mytnik and E. Perkins, Pathwise uniqueness for stochastic heat equations with Hölder continuous coefficients: the white noise case, 2011, pp. 1–96.
• [13] E. Perkins, Dawson-Watanabe superprocesses and measure-valued diffusions, Lectures on probability theory and statistics (Saint-Flour, 1999), Lecture Notes in Math., vol. 1781, Springer, Berlin, 2002, pp. 125–324.
• [14] L.C.G. Rogers and D. Williams, Diffusions, Markov processes, and martingales, vol. 2: Ito Calculus, John Wiley and Sons, Chichester, New York, Brisbane, Toronto, Singapore, 1987.
• [15] Andreas Winkelbauer, Moments and absolute moments of the normal distribution,https://arxiv.org/pdf/1209.4340.pdf (2012).
• [16] T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ. 11 (1971), 155–167.