Stochastic processes connected with harmonic functions
Joanne Elliott, William Feller · Transactions of the American Mathematical Society · 1956
The family of transformations (0.1) Ttf(x) =v-1C° -J--f(y)dy, t > 0define a semi-group of bounded linear transformations from C[-°°, + co ] to itself.The salient feature of this semi-group is that the functionis harmonic in the half-plane t>0.The infinitesimal generator of this semigroup is of the form (0.2) QF(x) = t-IP.I --ay, J _" y -x the integral being taken in the sense of a Cauchy principal value.The precise definition of this generator is found in Theorem 3.3 as a byproduct of the present investigation (2).In [2] the first of the present authors studied a class of semi-groups from C[-a, a] to itself with 0<a< °o , whose infinitesimal generator fla is obtained from (0.2) by truncation: r+a F'(y)--ay;J_" y -x the exact definition of this operator is found in Definition 1.1 of the present paper.These semi-groups share with (0.1) the property of being positivity preserving and norm-not-increasing.Among them there is a minimal semigroup 7~i(a) characterized by the boundary conditions:(0.4) T\a)f(±a) = 0. fCC[-a, +a}.Although the present paper in no way depends on the consideration of