For printing - MSP

Thus, by the Itô formula, we have
F t (z) =
d
1
t
−t
EIGENVALUES ASYMPTOTICS 7
s
dw
i,j=1
0
i s
0
d
1
dw
i,j=1
0
i s
s
0
d
1
dw
i,j=1
0
i s
s
0
1 s
+ 1
2 t3/2
−t
+t
d
i,j=1
0
1
d
i,j=1
− 1
2 t3/2
+ 1
2 t
d
i=1
0
d
i,j=1
1
0
Z i s
1 − s ds
Z i s
1 − s ds
1
0
∂jai(z + √ tZu)dw j u
∂jai(z + √ tZu) Zj u
1 − u du
0
s
0
Z i s
1 − s ds
∂ 2 j ai(z + √ tZu)du
∂jai(z + √ tZu)dw j u
∂jai(z + √ tZu) Zj u
1 − u du
s
∂iai(z + √ tZu)ds.
0
∂ 2 j ai(z + √ tZu)du
Since Z i s is the Gaussian random variable of mean 0 and variance s(1 − s),
we have
E[|Z i s| 2m ] = (2m − 1)!!(s(1 − s)) m for m = 1, 2, . . . .
Using this equality, the Hölder inequality and (A.2), we can prove the lemma.
Lemma 3.3. For every x ∈ R n , put A |x| = − 1
2 ∆y + c|x| 2p |y| 2q on L 2 (R m )
where c is a positive constant and let e −tA |x|(y, y ′ ) be the kernel of e −tA |x|
and J(t; |x|, y) = e −tA |x|(y, y). Then, we have following:
(i) There exist constants Cj (j = 1, 2, 3) such that
(3.5)
(ii) For every λ > 0,
(3.6)
|J(t; 1, y)| ≤ C1t −m/2 −C2t|y| 2q
e + e −C3|y| 2 /t .
J(t; |x|, y) = λ −m J(λ −2 t; |λ (1+q)/p x|, λ −1 y) for all t > 0, (x, y) ∈ R n × R m .
Proof. For (i), see Matsumoto [6, Lemma 3.1]. For (ii), it follows from the
Feynman-Kac formula that
J(t; |x|, y) = (2πt) −m/2 E e −t R 1
0 c|x|2p |y+ √ tYs| 2q ds .