Electronic Journal of Probability

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

Alejandro Gomez, Jong Jun Lee, Carl Mueller, Eyal Neuman, and Michael Salins

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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

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

Received: 7 March 2017
Accepted: 16 August 2017
First available in Project Euclid: 13 September 2017

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Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 60H15: Stochastic partial differential equations [See also 35R60]

uniqueness blowup stochastic differential equations wave equation white noise stochastic partial differential equations

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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

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