**Course No.****：**21522Z

**Period****：**20

**Credits****：**1

**Course Category****：**Advanced Course

**Aims & Requirements:**

Probability is both a fundamental way of viewing the world, and a core mathematical discipline, alongside geometry, algebra and analysis. Without any doubt, probability theory has become one of the most fascinating, fast growing and main stream areas of mathematics. Probability estimates of rare events play a fundamental role in my applications and we will develop techniques for estimating the rare events of the type that positive random variables take smaller values. In the past twenty years signi cant developments in the area have transformed our understanding of rare events

and have the potential to signi cantly expand the applicability of new techniques in the general setting. The three main goals for our lectures are to systematically study the existing techniques and applications which are spread over various topics, to further develop new methods of estimating small value problems for Gaussian and closely related stochastic partial di erential equations, and to reformulate and investigate problems in other areas of mathematics from the point of view of small value problems. The guiding philosophy of these lectures is that analysis of concrete processes is the most e ective way to explain even the most general methods or abstract principles. Many open problems and conjectures at the undergraduate/graduate level will be discussed.

**Primary Coverage****：**

The purpose of these lectures is to present the state of the art of various powerful techniques on estimating small value probabilities. Major applications include exit time and boundary crossing asymptotics in probability and statistics, smoothness of density via Malliavin calculus, regularity properties of stochastic partial di erential equations.

One special feature is that, at the end of each lecture, several open and important problems related to techniques discussed will be o ered. This will ensure that the participants can thinking and working on interesting problems.

Part 1. Introduction, overview and applications. We rst de ne the small value

(deviation) probability in several setting, which basically studies the asymptotic rate of approaching zero for rare events that positive random variables take smaller values.Many applications are given. Bene ts and di erences of various formulations of small value probabilities are examined in details, together with connections to related elds.

Part 2. Basic estimates and equivalent transformations. We rst formulate several equivalent results for small value probability, including negative moments, exponential moments, Laplace transform and Taubirean theorems. The basic techniques involved are various useful inequalities, motivated from large deviation estimates. Some re nement of known results are given, including to some classical inequalities. Applications to regularity and smoothness of probability laws via small value estimates of the determinant

of Malliavin matrix are discussed in the setting of stochastic (partial) di erential equations.

Part 3. Techniques associated with independent variables. We start with probabilistic arguments for algebraic properties of small value probabilism, such as independent sums and products. These estimates are non-asymptotic and hence they can be applied are in the setting of conditional probability. Separate treatments are analyzed for exponential and power decay rates. A newly discover symmetrization inequality is proved by Fourier analytic method.

Part 4. Blocking techniques for the sup-norm. We rst present the vary useful blocking techniques for the maximum of the absolute value of partial sums in both upper and bound setting. The lower bound is more involved since the end position of each block has to be controlled also. The resulting estimates play a critical role in many strong limit theorems for sample paths. Similar techniques are applied to weighted and/or controlled sup-norms for Brownian motion and stable processes. Applications to the two-sided exit time are indicated.

Part 6. Small deviation (ball) estimates for sums of correlated Gaussian elements.We treat the sum of two not necessarily independent Gaussian random vectors in a separable Banach space. The main ingredients are Anderson’s inequality and the weaker correlation inequality developed by the lecturer. Various applicants are provided to show the power of the method. As a direct consequence, under the sup-norm or Lp-norm,Brownian motion and Brownian bridge have exact the same small ball behavior at the

log level, and so do Brownian sheets and various tied down Brownian sheets including Kiefer process and solutions of SPDEs.

**References:**

Lecture notes and slides will be available at the instructor’s home page:

http://www.math.udel.edu/~wli/

**Author：**Wenbo Li