Diffusion Processes in q-Representation

Abstract

In this chapter we will assume a (non-negative) measurable function ϕ(t, x) is given (but not$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varphi } \left( {t,x} \right)!$), and concentrate on the problem of constructing diffusion processes starting from an arbitrary point (s, x) with an additional drift coefficient σ^Tσ∇(logϕ(t, x)); namely we will not fix an initial distribution. It is clear, then, that we cannot handle our problem in the framework of the variational method (in the p-representation) formulated in the preceding chapter based on Csiszar’s projection theorem for lack of a fixed initial distribution, or in other words, lack of the function $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varphi } \left( {t,x} \right)!$. If a transition density p(s, x; t, x) with creation and killing is given and if ϕ(t, x) is p-harmonic and positive, then the harmonic transformation induces an additional drift term σ^Tσ∇(log ϕ(t, x)) as we have discussed in Chapter 2. A mathematical problem encountered with ϕ(t, x) which vanishes on a subset will be reduced to defining a multiplicative functional properly (cf. (6.5)) and to verifying the normality condition under a mild integrability condition (cf. (6.4)).