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

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Abstract: Suppose that

X = X_{t}, t g e 0

is a supercritical superprocess 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+int_{(0,+infty)}(e^{-lambda y}-1+lambda y)n(x,dy), quad xin E, quadlambda> 0,

where

a i n m a t h c a l B_{b} (E)

,

b i n m a t h c a l B_{b}^{+} (E)

and

n

is a kernel from

E

to

(0, i n f t y)

satisfying

sup_{xin E}int_0^infty y^2 n(x,dy)<infty.

Put

T_{t} f (x) = m a t h b b P_{d e l t a_{x}} < f, X_{t} >

. Let

l a m b d a_{0} > 0

be the largest eigenvalue of the generator

L

of

T_{t}

, and

p h i_{0}

and

h a t {p h i}_{0}

be the eigenfunctions of

L

and

h a t L

(the dural of

L

) respectively associated with

l a m b d a_{0}

. Under some conditions on the spatial motion and the

p h i_{0}

-transformed semigroup of

T_{t}

, we prove that for a large class of suitable functions

f

, we have

lim_{t ightarrowinfty}e^{-lambda_0 t}< f, X_t> = W_inftyint_Ehat{phi}_0(y)f(y)m(dy),quad mathbb{P}_{mu}{-a.s.},

for any finite initial measure

m u

on

E

with compact support, where

W_{i} n f t y

is the martingale limit defined by

W_{i} n f t y : = l i m_{t o i n f t y} e^{- l a m b d a_{0} t} < p h i_{0}, X_{t} >

. Moreover, the exceptional set in the above limit does not depend on the initial measure

m u

and the function

f

.

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