Diffusion semigroup on manifolds with time-dependent metrics

Let $L_t:=\Delta_t +Z_t$, $t\in [0,T_c)$ on a differential manifold equipped with time-depending complete Riemannian metric $(g_t)_{t\in [0,T_c)}$, where $\Delta_t$ is the Laplacian induced by $g_t$ and $(Z_t)_{t\in [0,T_c)}$ is a family of $C^{1,1}$-vector fields. We first present some explicit criteria for the non-explosion of the diffusion processes generated by $L_t$; then establish the derivative formula for the associated semigroup; and finally, present a number of equivalent semigroup inequalities for the curvature lower bound condition, which include the gradient inequalities, transportation-cost inequalities, Harnack inequalities and functional inequalities for the diffusion semigroup

A probabilistic method for gradient estimates of some geometric flows

In general, gradient estimates are very important and necessary for deriving convergence results in different geometric flows, and most of them are obtained by analytic methods. In this paper, we will apply a stochastic approach to systematically give gradient estimates for some important geometric quantities under the Ricci flow, the mean curvature flow, the forced mean curvature flow and the Yamabe flow respectively. Our conclusion gives another example that probabilistic tools can be used to simplify proofs for some problems in geometric analysis.Comment: 22 pages. Minor revision to v1. Accepted for publication in Stochastic Processes and their Application

Financing Development: The Role of Information Costs

How does technological progress in financial intermediation affect the economy? To address this question a costly-state verification framework is embedded into a standard growth model. In particular, financial intermediaries can invest resources to monitor the returns earned by firms. The inability to monitor perfectly leads to firms earning rents. Undeserving firms are financed, while deserving ones are under funded. A more efficient monitoring technology squeezes the rents earned by firms. With technological advance in the financial sector, the economy moves continuously from a credit-rationing equilibrium to a perfectly efficient competitive equilibrium. A numerical example suggests that finance is important for growth.

Financing Development: The Role of Information Costs

To address how technological progress in financial intermediation affects the economy, a costly-state verification framework is embedded into the standard growth model. The framework has two novel ingredients. First, firms differ in the risk/return combinations that they offer. Second, the efficacy of monitoring depends upon the amount of resources invested in the activity. A financial theory of firm size results. Undeserving firms are over financed, deserving ones under funded. Technological advance in intermediation leads to more capital accumulation and a redirection of funds away from unproductive firms toward productive ones. With continued progress, the economy approaches its first-best equilibrium. An extended version of the paper containing some quantitative analysis is available at: http://ssrn.com/abstract=996263financial intermediation, economic development, costly state verification, firm size

Weak Poincar\'e Inequality for Convolution Probability Measures

By using Lyapunov conditions, weak Poincar\'e inequalities are established for some probability measures on a manifold $(M,g)$. These results are further applied to the convolution of two probability measures on $\R^d$. Along with explicit results we study concrete examples