Strong law of large numbers for supercritical superprocesses under second moment condition

Consider a supercritical superprocess X = {X (t) , t a (c) 3/4 0} on a locally compact separable metric space (E,m). Suppose that the spatial motion of X is a Hunt process satisfying certain conditions and that the branching mechanism is of the form psi(x,lambda) = -a(x)lambda + b(x)lambda(2) + integral((0,+infinity)) (e(-lambda y) - 1 + lambda y)n(x,dy), x is an element of E, lambda > 0, where , , and n is a kernel from E to (0,+a) satisfying sup (xaE) a << (0) (+a) y (2) n(x, dy) < +a. Put . Suppose that the semigroup {T (t) ; t a (c) 3/4 0} is compact. Let lambda (0) be the eigenvalue of the (possibly non-symmetric) generator L of {T (t) } that has the largest real part among all the eigenvalues of L, which is known to be real-valued. Let I center dot (0) and be the eigenfunctions of L and (the dual of L) associated with lambda (0), respectively. Assume lambda (0) > 0. Under some conditions on the spatial motion and the I center dot (0)-transform of the semigroup {T (t) }, we prove that for a large class of suitable functions f, lim(t ->+infinity) e(-lambda 0t) < f,X-t > = W-infinity integral(E) (phi) over cap (0) (y)f(y)m(dy), P-mu-a.s., for any finite initial measure A mu on E with compact support, where W (a) is the martingale limit defined by . Moreover, the exceptional set in the above limit does not depend on the initial measure A mu and the function f.