# Mathematical Psychology and Psychophysiology

By Stephen Grossberg | Go to book overview

SIAM-AMS Proceedings Volume 13 1981

A "Psychological" Proof that Certain Markov Semigroups Preserve Differentiability

M. FRANK NORMAN

1. Introduction. Unlike the preceding paper [1], the present one is directed toward a purely mathematical result. This result is linked to genetics because it can play a role in analyzing genetic models. It is linked to psychology because my proof makes essential use of a psychological model. However, the result itself is a fairly general theorem about a well-studied class of mathematical objects. The objects I have in mind are the semigroups of operators associated with certain diffusions.

For our purposes, a diffusion, X(t), is a real-valued strong Markov process with continuous sample paths. (See [2, Chapter 2] for background information and terminology regarding diffusions.) We assume that the process is defined for all t ≥ 0 and that its state space, I = [r0, r1], is closed and bounded. A central role in the description and analysis of a finite Markov chain, Xn, is played by its transition matrix, P, with components
Pxy = P(Xn+1 = yXn = x).

Associated with this matrix is a transformation of functions (regarded as column vectors) defined by matrix multiplication,

or, equivalently, Pƒ(x) = E(ƒ(Xn+1)∣Xn = x). Both the transition matrix and the transition operator can be generalized to diffusions, but the transition operator, defined by
Ttƒ(x) = E(ƒ(Xt+s)∣Xs = x),
is far more tractable and thus occupies a more prominent place in the theory of diffusions.

____________________
1980 Mathematics Subject Classification. Primary 47D07, 35K15, 35K65; Secondary 60F05, 60J60, 60J70.

-197-

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