An Essay On Probabilistic Approach For Some Classes Of Quasi-linear Partial Differential Equations

It is well known that the solutions of linear second order parabolic partial differ-ential equations(abbreviated as PDEs) can be formulated as a functional of solutions of some stochastic differential equations (abbreviated as SDEs). This kind of interpre-tation has important applications both in the theory of PDEs and that of SDEs, such as large deviation, optimal control theory, martingale problem, variational and quasi variational inequality, etc. This dissertation is mainly concerned with the probability approach to the second order quasi-linear PDEs, whose well-posedness in Sobolev spaces are focused on.This dissertation consists of five parts, which is organized as follows:In Chapter1, we briefly review the origin and development history of the prob-ability method used in solving PDEs, and emphatically describe the promotion of such method to the second order quasi-linear PDEs in recent researches, including the difficulties encountered. Based on these works, the main idea is concentrated on the well-posedness of the second order quasi-linear PDEs in Sobolev space.In Chapter2, some basic concepts and lemmas associated with the research theme are reviewed in the dissertation, such as the Sobolev space, stochastic flows, the related knowledge of stochastic analysis, Malliavin calculus etc.Chapter3is devoted to studing a probabilistic approach to the second order quasi-linear parabolic PDEs. For the initial value problem of second order quasi-linear parabolic PDEs, we prove the existence and uniqueness of its local solution with consideration of its connection with the SDEs and probabilistic representation of solutions, under the conditions that the coefficient is an unbounded function which belongs to k order continuous differentiable space, and that the initial value is in Sobolev space Wk,p(k≥3). In the case of constant and non-degenerate diffusion coefficient, we also prove the global existence, setting certain conditions on the ex-ternal force. Through the vanishing viscosity method, an application of this research results is given. We establish the results of the existence and uniqueness of the local solution in Sobolev space for one-order quasi-linear hyperbolic PDEs and the con-vergence rate of the solution is given. Due to the case that the diffusion coefficient remains constant, adding same conditions on the external force, we also establish the global existence result. Chapter4is further concentrated on studing the local well-posedness of the quasi-linear partial integral-differential equations, probabilistic representation for-mula of which is also established by means of Friedman theory and SDEs driven by Brown motion and pure jump Levy process, basing on the connection between sec-ond order quasi-linear PDEs and SDEs. Under the conditions that the coefficient is unbounded and continuous differential function and that the initial data is in Sobolev space, we also prove the local existence in Sobolev space. However, the Bismut for-mula for SDEs driven by Brown motion and pure jump Levy process can only obtain an exponential estimate, rather than linear estimate, which leads to some difficulties of obtaining the global existence, we don’t give a further study on the proof. As a simple application of this result, the solvability results between the SDEs only driv-en by pure jump Levy process and a class of quasi-linear partial integro differential equations is established.Finally, in Chapter5, we summarize the obtained results and give a prospect for future work, including the remaining and other extensive problems.