Default functions and Liouville type theorems based on symmetric diffusions

Default functions appear when one discusses conditions which ensure that a local martingale is a true martingale. We show vanishing of default functions of Dirichlet processes enables us to obtain Liouville type theorems for subharmonic functions and holomorphic maps. Default functions were introduced in [7] and it is known that vanishing of the default function of a local martingale implies that it is a true martingale. Positivity of the default function then indicates the singularity of local martingale. Such local martingales are called strictly local martingales. Recently strictly local martingales are playing important roles in the theory of financial bubbles (cf. [19]), so the notion of default function becomes important in mathematical finance area. We consider them in a different context. In mathematical analysis of subharmonic functions it is classical and natural to consider the functions along Brownian motions. A stochastic process derived from a subharmonic function composed with Brownian motion is a local submartingale. Then if we know that the process is a true submartingale, which follows from vanishing of the default function of the submartingale, it effects simpleness and clearness in analysis of subharmonic functions. In this paper we intend to show that this probabilistic notion plays effective roles in some analysis such as L-Liouville type theorems of subharmonic functions and Liouville type theorems for functions satisfying some nonlinear differential inequalities. It covers and extends the precedent results about L-Liouville theorem for subharmonic functions 2000 Mathematics Subject Classification. Primary 31C05; Secondary 58J65.