Date of Completion
stabilization by noise, stochastic differential equations, ergodicity
Field of Study
Doctor of Philosophy
D. Herzog and J. Mattingly have shown that a complex-valued polynomial ODE with finite-time blow-up solutions may be stabilized by the addition of complex-valued Brownian noise. In this paper, we extend their results to two-dimensional complex-valued systems of coupled ODEs with finite-time blow-up solutions. We show analytically and numerically that stabilization can be achieved in our setting by adding a suitable Brownian noise, and that the resulting systems of SDEs are ergodic. For one of the systems, the proof uses the Girsanov theorem to induce a time change from that two-dimensional complex system to a quasi-one-dimensional complex system similar to the one studied by Herzog and Mattingly.
Shum, Fan Ny, "Stabilization by Noise of Systems of Complex-Valued ODEs" (2016). Doctoral Dissertations. 1216.