Some Research on Stochastic Transport Type Equations and Distribution Dependent Backward Stochastic Differential Equations


Student thesis: Doctoral Thesis

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Award date12 Jul 2018


This thesis focuses on the fields of stochastic transport differential equations and distribution dependent backward stochastic differential equations. In this work, some rigorous mathematical tools for the study of these two interrelated projects are developed. Since the influences of internal, external or environmental noise in the governing models are ubiquitous in the real world, the development of new mathematical techniques in the stochastic setting is of paramount practical interest, not only in the pursuit of more accurate models, but also in the attempts at achieving a given goal.

In the first part, we consider an abstract stochastic transport equation,
where some differential or integral operators are involved. The additional multiplicative noise in the governing equations is commonly used to account for empirical and physical uncertainties in applications ranging from climatology to turbulence theory. We obtain the existence, uniqueness and stability of the solutions in a somewhat abstract framework so that the results can be applied to some special instances, such as the stochastic Euler-Poincaré equations and Camassa-Holm type equations. For some particular examples, like the Camassa-Holm with linear noise, we will further consider the global existence and blow-up of the solutions. Moreover, the effect of the noise on the solutions will also be investigated.

In the second part, we consider backward stochastic differential equation (BSDE), where the driven term depends on the distribution law of the solutions. The dependence on the distribution often arises from the mean-field theory in mathematical physics. We first establish the existence and uniqueness of the solution when the driven term is not Lipschitz continuous. Then we consider the reconstruction of the many-body system from a mean field BSDE and prove that the reconstruction is unique in law. After that, we combine a forward SDE with a BSDE to investigate the kinetic Landau equation.