The study of small deviation concerns the asymptotic behavior of the probability of the rare event that a random process stays in a small region D over a certain period of time. From the Probability Theory point of view, the importance of the asymptotic behavior of the small deviation probability lies on the fact that it determines the rate of convergence of many limit laws. From other mathematical point of view, the importance lies on its close connections with many other areas. Besides the well-known connection with metric entropy, here are some examples.

Connection with Geometry. It is not difficult to imagine that the aforementioned probability depends on the "size" of D, or more precisely, the geometry of D. Indeed, a closer look reveals some precise connections with convex geometry [2]. It is also closely related to differential geometry. Without introducing any new terminology, let X(t) be an n-dimensional Brownian motion, and D be the cone generated by an open connected set G on the n-dimensional sphere. Then the asymptotic behavior as T goes to infinity of the probability that the process stays in D before time T is completely determined by the principal eigenvalue of G.

Connection with Differential Equations. Arguably, Gaussian process is one of the most important stochastic processes in Probability and its applications. If X(t) is a centered Gaussian process, and D is a ball in L^2[a,b] with radius r. (Here we view each sample path X(t) as a function of t on [a,b]). Then the asymptotic behavior of the probability of X(t) belongs to D as r goes to 0 is completely determined by the decay rate of the eigenvalues of the Fredholm Operator with the covariance function of X(t) as the kernel. In many applications, such integral equations can be changed into differential equations [4].

Connection with Functional Analysis. In a more general setting, if X(t) is a Gaussian process, and D is a symmetric convex set in C(T), the set of continuous functions on T. (Again we view the sample path as a function of t on T.) Then the asymptotic behavior of the logarithm of the probability of X(t) belongs to a ball of C(T) with radius r is clasely related to the metric entropy absolute convex hull of X(t), viewed as a subset of a Hilbert space [2],[8].

Some recent papers related to small deviation

Integrated Brownian motions and exact L(2) small balls. (with J. Hannig, F. Torcaso). Obtained the exact L^2 small ball rate of integrated Brownian motions. (Ann. Probab. 2003)

Entropy of convex hulls in Hilbert spaces. Small deviation probability method is used to obtain a very sharp upper bound for the entropy of the absolute convex hull, given the entropy rate of the set of extreme points. (Bull. London. Math. Soc. 2004)

L(2) small ball probability of Gaussian processes. (with J. Hannig, Y. Lee, and F. Torcaso). Strengthed Li's comparison theorem. (EJP 2004)

The Laplace transforms via Hadamard Factorization (with J. Hannig, Y. Lee, and F. Torcaso). A method to obtain the exact form of laplace transform, as well as L(2) small balls. (EJP 2004)

Comparison Theorems for Small Deviations of Random Series (with J, Hannig and F. Torcaso). A general comparison theorem about small deviation of probability of positive random series. (J. Theor. Probab. 2004)

On combinatorial dimensions (I) (with R. Blei). Proved the complete independence of two "closely related" measurements of inderdependence. The main idea is to use random selection. (Random Struct. 2005)

Small ball probabilities for the Slepian Gaussian fields. (with W. Li) A Fourier analytic method handeling small deviation of Gaussian fields. (To appear in Trans. Amer. Math. Soc.)

Metric entropy of high dimensional distributions and small ball probability of Brownian sheets. (abstract) (with R. Blei and W. Li) Bounds for the metric entropy of high dimensional distribution are obtained.

Log-level comparison for small deviation probabilities (with W. Li). (To appear in J. Theor. Probab.)

Metric entropy of monotonic functions (with J. Wellner) New bounds for the metric entropy and bracketing entropy of monotonic functions are obtained. As an application, we obtain the convergence rate for the Maximum Likelihood Estimator of a block decreasing density. (Submitted to J. of Multivariate Anal.)