Well-posedness of Kimura Equation in Genetic Drift


主讲人:陈新富 西南财经大学特聘教授




主讲人介绍:陈新富,现为西南财经大学特聘教授。1980年至1986年就读北京大学数学本科、硕士研究生,1991年获美国明尼苏达大学博士,师从美国艺术与科学院院士A. Friedman。曾获得Sloan研究奖及多次美国自然科学基金。陈教授研究领域广泛,包括非线性抛物型和椭圆型偏微分方程、自由边值问题、界面动力学等,取得了一系列国际同行认可的重要成果。这些成果发表在 Arch. Ration. Mech. Anal., J. Differential Geom., Math. Ann., Trans. Amer. Math. Soc., SIAM J. Math. Anal., SIAM J. Appl. Math., Calc. Var. Partial Differential Equations等一流数学期刊上。

内容介绍:We present a rigorous derivation of the continuum Kimura equation from a discrete Wright–Fisher genetic drift model. We show that boundary conditions are not needed for and cannot be imposed on the resulting degenerate diffusion problem. To this end, we reformulate the concept of weak solutions. In doing so, we find that the extension of the Kimura equation to the whole space should be the continuum limit that carries over the biologically relevant statistic information from the discrete model; namely, the conservation laws embedded in the discrete model are now self-contained in the continuum problem, without imposing any extra boundary conditions or integral constraints. We establish a well-posedness (existence, uniqueness, regularity, stability) theory and especially prove the analyticity of the solution. Our arguments build an intrinsic connection between the genetic fixation probability and a stochastic process with two absorbing barriers.