We consider nonnegative solutions of a parabolic equation in a cylinder D×I, where D is a noncompact
domain of a Riemannian manifold and I = (0,T ) with 0<T ∞or I = (−∞, 0). Under the assumption
[SSP] (i.e., the constant function 1 is a semismall perturbation of the associated elliptic operator on D),
we establish an integral representation theorem of nonnegative solutions: In the case I = (0,T ), any nonnegative
solution is represented uniquely by an integral on (D × {0}) ∪ (∂MD × [0,T )), where ∂MD is
the Martin boundary of D for the elliptic operator; and in the case I = (−∞, 0), any nonnegative solution
is represented uniquely by the sum of an integral on ∂MD × (−∞, 0) and a constant multiple of a
particular solution. We also show that [SSP] implies the condition [SIU] (i.e., the associated heat kernel is
semi-intrinsically ultracontractive).
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