On the Continuity of Measure-Valued Processes

Abstract

Let Y = (52, F, F t, Yt, Px) be a Hunt process with Borel semigroup Pt taking values in a topological Lusin space (E,E)(E is homeomorphic to a Borel subset of a compact metric space and E is its Borel o-field). MF(E) and MS(E) denote the spaces of finite measures and finite signed measures, respectively, on (E,E) with the weak (i.e. weak) topology. The (Y,−λ2/2) — superprocess is a continuous MF(E)-valued Borel strong Markov process x.If mo ∈ MF(E) the law \({Q_{{m_o}}}\) on C([0,∞),MF(E)) of X starting at X0 = mo is uniquely determined by (write Xt(o) for #x222B;o(x)Xt(dx)) $${Q_{{m_o}}}\left( {\exp \left( { - {x_t}\left( \phi \right)} \right)} \right) = \exp \left( { - {m_o}\left( {{v_t}\phi } \right)} \right)\,\,\phi \in bpE$$ (bpE is the set of bounded non-negative measurable functions on E) where Vto is the unique solution of $${V_t}\phi \left( x \right) = {p_t}\phi \left( x \right) - \int_0^t {{p_s}} \left( {{{\left( {{V_{t - s}}\phi } \right)}^2}/2} \right)\left( x \right)\,ds\,t \geqslant 0,\,x\, \in \,E$$ (see Fitzsimmons (1988, (4.7), (2.3)).