Probabilistic method in PDE is equally used in Pure and Applied Mathematics research. This is regarded as a very powerful tool by the researchers working on the theory of differential equations. However, as the topic demands expertise on both PDE and probability theory, an initiative to teach the topic as a structured course is vastly absent globally, including in India. There is hardly any lecture note or a course accessible for the mathematics students. There is no book for mathematics students which focuses on this topic and assembles all important aspects suitable for an introduction to this topic. Young researchers like PhD students or junior postdoctoral fellows, who aspire to learn the topic, resorts on several different books and study by themselves, which often consumes considerable amount of their productive time. To change the present discouraging scenario and to boost up research on this very powerful and vibrant topic, I have designed this introductory course. I have offered this once informally at IISER Pune and then officially at Justus-Liebig University, Giessen, Germany for research students. This course, although an advanced one, attracts students with a background of PDE, Probability Theory, Mathematical Finance, or Mathematical Physics. This course allows a researcher to confidently take up an original research problem in the related field. This course content is mainly based on two different books, one on stochastic calculus and another on semigroup theory. Many theorems would be proved in the lectures with greater details than the reference books.
INTENDED AUDIENCE : Doctoral students or researchers in the area of partial differential equations or stochastic processes who wish to learn the probabilistic techniques in PDE for solving research problems either in pure or applied mathematics such as Mathematical Finance or Mathematical Physics.PREREQUISITES : Appropriate for students with MSc in Pure Mathematics with specialization in Analysis and/or Probability Theory. Prerequisite courses: Measure Theory, Functional Analysis, Probability Theory, Stochastic Processes, INDUSTRY SUPPORT : This course develops tools to solve deterministic Evolution Problem arising in physical scenarios with random noise. The evolution problems include, for example, heat equation and option price equations. Therefore, the cutting edge R&D sectors of Finance industry should value this course.
Week 1 : Mathematical formulation of stochastic processesWeek 2 : Brief review of L2 theory of stochastic integrationWeek 3 : Ito’s formulaWeek 4 : Probabilistic method in Dirichlet problemWeek 5 : Further topics of Dirichlet problem and Probabilistic method in heat equationWeek 6 : Further topics of Probabilistic method in heat equationWeek 7 : Feynman Kac formulaWeek 8 : Stochastic differential equationsWeek 9 : PDE with general elliptic operatorsWeek 10 : Feynman Kac formula and its abstraction with semigroup theoryWeek 11 : Mild solution to linear evolution problemsWeek 12 : Mild solution to semilinear evolution problem