Date of Completion

8-5-2016

Embargo Period

8-3-2016

Keywords

stabilization by noise, stochastic differential equations, ergodicity

Major Advisor

Maria Gordina

Associate Advisor

Iddo Ben-Ari

Associate Advisor

Alexander Teplyaev

Field of Study

Mathematics

Degree

Doctor of Philosophy

Open Access

Open Access

Abstract

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.

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