Boundary Value Problems and Markov Processes

Focusing on the interrelations of the subjects of Markov processes, analytic semigroups and elliptic boundary value problems, this monograph provides an exposition of functional methods in stochastic analysis. The author studies a class of boundary value problems for second-order elliptic differential operators, and proves that this class provides a new example of analytic semigroups both in the Lp topology and in the topology of uniform convergence. As an application, one can construct analytic semigroups corresponding to the diffusion phenomenon of a Markovian particle moving continuously in the state space until it 'dies', at which time it reaches the set where the absorption phenomenon occurs. A class of initial-boundary value problems for semilinear parabolic differential equations is also considered.