Theoretical and Numerical Studies of Nonlocal Equations Derived from Stochastic Differential Equations with Levy Noises

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Stochastic effects are ubiquitous in complex systems in science and engineering. Although random mechanisms may appear to be very small or very fast, their long time impact on the system evolution may be delicate or even profound, which has been observed in, for example, stochastic bifurcation, stochastic resonance and noise-induced pattern formation. The research team will study the complex systems under uncertainty by developing numerical methods to be simulated on computers and answer fundamental questions about the average quantities of the systems. The investigators will deliver the following broader impact outcomes: (1) Two graduate students (including one female underrepresented minority) will receive education and training and (2) they will continue to recruit and nurture underrepresented students in STEM. Additionally, the resulting computational tools have broad applications in areas ranging from biology to geophysics. The software resulting from the proposed project will be made publicly available.Mathematical modeling of complex systems under uncertainty often leads to stochastic differential equations (SDEs). Fluctuations appeared in the SDEs are often non-Gaussian (e.g., Levy motions) rather than Gaussian (e.g., Brownian motion). Compared with systems with Gaussian noises, quantifying the impacts of non-Gaussian Levy fluctuations are much less understood. The researchers will develop convergent and efficient numerical techniques for investigating the deterministic macroscopic quantities that can help understand the dynamics of SDEs with Levy noises, in particular the mean exit time, escape probability, and probability density function. They will also answer theoretical questions with regard to the well-posedness and the regularity of the solutions to these nonlocal equations. Building upon previously developed methods in the one-dimensional case, the team will focus its effort on two-dimensional systems. The proposed project will provide not only broadly applicable computational techniques to solve integro-differential equations with singular integrands, but also theoretically address the central questions pertaining to the solutions. The theory will help to provide insight into fundamental issues in quantifying the impacts of non-Gaussian Levy fluctuations in dynamical systems.