Model
Results
Idea of the proof
Occupation time fluctuations of particle
systems
Anna Talarczyk
University of Warsaw
joint work with T. Bojdecki and L. Gorostiza
Anna Talarczyk
Occupation time fluctuations
Model
Results
Idea of the proof
Particle system
Problem
Particle system in Rd
initial distribution of particles – given by a Poisson random
measure with intensity measure ν (or νT )
particle motion – standard, spherically symmetric α-stable
Lévy process
intensity of branching V > 0,
Anna Talarczyk
Occupation time fluctuations
Model
Results
Idea of the proof
Particle system
Problem
Particle system in Rd
initial distribution of particles – given by a Poisson random
measure with intensity measure ν (or νT )
particle motion – standard, spherically symmetric α-stable
Lévy process
intensity of branching V > 0,
Anna Talarczyk
Occupation time fluctuations
Model
Results
Idea of the proof
Particle system
Problem
Particle system in Rd
initial distribution of particles – given by a Poisson random
measure with intensity measure ν (or νT )
particle motion – standard, spherically symmetric α-stable
Lévy process
intensity of branching V > 0,
Anna Talarczyk
Occupation time fluctuations
Model
Results
Idea of the proof
Particle system
Problem
Particle system in Rd
(1 + β) branching mechanism, 0 < β ≤ 1, with probability
generating function
g(s) = s +
1
(1 − s)1+β , 0 < s < 1.
1+β
branching is critical – mean value = 1
if β = 1 – binary branching
if β < 1 – infinite variance branching
particles move and branch independently
Empirical process
Nt : Nt (A) = number of particles in the set A at time t.
Anna Talarczyk
Occupation time fluctuations
Model
Results
Idea of the proof
Particle system
Problem
Particle system in Rd
(1 + β) branching mechanism, 0 < β ≤ 1, with probability
generating function
g(s) = s +
1
(1 − s)1+β , 0 < s < 1.
1+β
branching is critical – mean value = 1
if β = 1 – binary branching
if β < 1 – infinite variance branching
particles move and branch independently
Empirical process
Nt : Nt (A) = number of particles in the set A at time t.
Anna Talarczyk
Occupation time fluctuations
Model
Results
Idea of the proof
Particle system
Problem
Particle system in Rd
(1 + β) branching mechanism, 0 < β ≤ 1, with probability
generating function
g(s) = s +
1
(1 − s)1+β , 0 < s < 1.
1+β
branching is critical – mean value = 1
if β = 1 – binary branching
if β < 1 – infinite variance branching
particles move and branch independently
Empirical process
Nt : Nt (A) = number of particles in the set A at time t.
Anna Talarczyk
Occupation time fluctuations
Model
Results
Idea of the proof
Particle system
Problem
Problem
Z
Tt
Ns
ds,
t ≥0
0
Anna Talarczyk
Occupation time fluctuations
Model
Results
Idea of the proof
Particle system
Problem
Problem
Rescaled occupation time fluctuations
XT (t) =
1
FT
Z
Tt
(Ns − ENs )ds,
t ≥0
0
where FT is a proper norming.
Anna Talarczyk
Occupation time fluctuations
Model
Results
Idea of the proof
Particle system
Problem
Problem
Rescaled occupation time fluctuations
XT (t) =
1
FT
Z
Tt
(Ns − ENs )ds,
t ≥0
0
where FT is a proper norming.
Problem
find a suitable norming FT , such that XT converges in law
as T → ∞ to a nontrivial limit
identification of the limit
properties of the limit
Anna Talarczyk
Occupation time fluctuations
Model
Results
Idea of the proof
Particle system
Problem
Two cases
1
initial intensity measure = ν does not depend on T (only
speed up the time)
2
initial intensity measure = νT = HT ν, HT → ∞ (speed up
the time + high density)
We consider the following initial intensity measures
ν = λ (Lebesgue measure)
HT µ, where µ finite measure, HT → ∞
HT µγ with
µγ (dx) =
dx
,
1 + |x|γ
γ≥0
either HT ≡ 1 or HT → ∞ sufficiently quickly
Anna Talarczyk
Occupation time fluctuations
Model
Results
Idea of the proof
Particle system
Problem
Two cases
1
initial intensity measure = ν does not depend on T (only
speed up the time)
2
initial intensity measure = νT = HT ν, HT → ∞ (speed up
the time + high density)
We consider the following initial intensity measures
ν = λ (Lebesgue measure)
HT µ, where µ finite measure, HT → ∞
HT µγ with
µγ (dx) =
dx
,
1 + |x|γ
γ≥0
either HT ≡ 1 or HT → ∞ sufficiently quickly
Anna Talarczyk
Occupation time fluctuations
Model
Results
Idea of the proof
Particle system
Problem
History and related problems
Deuschel, Wang (1994), α = 2 (Brownian particles),
Lebesgue measure, without branching
Dawson, Gorostiza, Wakolbinger (2001), convergence of
random variable XT (1), β = 1, general β only for “large”
dimensions
Iscoe (1986), for superprocesses, Lebesgue, convergence
of XT (1).
Birkner, Zähle (2007), for branching random walks on a
lattice.
Anna Talarczyk
Occupation time fluctuations
Model
Results
Idea of the proof
Particle system
Problem
Notation
S(Rd ) – the Schwartz space of smooth quickly decreasing
functions
S 0 (Rd ) – the space of tempered distributions on Rd ,
It is convenient to consider XT as a process in S 0 (Rd )
pt – the transition density of the α-stable process,
α
(p̂t (x) = e−t|x| )
Tt – transition semigroup Tt f = pt ∗ f ,
⇒ – convergence in law in C([0, τ ], S 0 (Rd )), for any τ > 0
c
⇒ – convergence of finite dimensional distributions
f
Anna Talarczyk
Occupation time fluctuations
Model
Results
Idea of the proof
Particle system
Problem
Expected value of the occupation time
By the Poisson
R property of the initial distribution:
E hNt , ϕi = Rd Tt ϕ(x)ν(dx)
Proposition 1
Initial intensity measure ν = µγ =
Z
E
0
T
dx
,
1+|x|γ
 1− γ
if
T α





log T




1− d


T α log T
hNt , ϕi dt ∼ (log T )2


T 1− αd





log T



1
Anna Talarczyk
γ < d, γ < α
γ=α<d
γ=d <α
γ=d =α
γ > d, d < α
γ > d, d = α
γ > α, d > α
Occupation time fluctuations
Model
Results
Idea of the proof
Lebesgue measure (γ = 0)
General case (gamma ≥ 0)
Long range dependence
stable random measure
Let M be independently scattered (1 + β)-stable random
measure on Rd+1 with control measure λd+1 (Lebesgue) totally
skewed to the right, i.e. for any A ∈ B(Rd+1 ) such that
0 < λd+1 (A) < ∞, M(A) is (1 + β)-stable, with characteristic
function
π
1+β
1 − i(sgn z) tan (1 + β) ,
exp −λd+1 (A)|z|
2
z ∈ R,
M is σ-additive and M(Aj ), j = 1, 2, ... are independent if Aj are
disjoint.
Anna Talarczyk
Occupation time fluctuations
Model
Results
Idea of the proof
Lebesgue measure (γ = 0)
General case (gamma ≥ 0)
Long range dependence
Theorem 2
Assume that the initial intensity measure is ν = λ (γ = 0), then
(i)
α
β
<d <
α(1+β)
,
β
d
FT1+β = T (2+β− α β) , then XT ⇒ K λξ, where
c
Z
ξt =
Rd+1
Z
t
11[0,t] (r )
pu−r (x)du M(drdx), t ≥ 0,
r
K is a constant.
ξ is well defined for d <
Z
Z
α(1+β)
,
β
1+β
τ
pu (x)du
Rd
since
dx < ∞,
for any τ > 0.
0
Anna Talarczyk
Occupation time fluctuations
Model
Results
Idea of the proof
Z
ξt =
Rd+1
Lebesgue measure (γ = 0)
General case (gamma ≥ 0)
Long range dependence
Z t
11[0,t] (r )
pu−r (x)du M(drdx), t ≥ 0.
r
Remark: If β = 1 (binary branching), then ξ is a centered
Gaussian process with covariance function
sh + t h −
i
1h
(s + t)h + |s − t|h ,
2
h = 3 − d/α (0 < h < 2) (sub-fractional Brownian motion)
Anna Talarczyk
Occupation time fluctuations
Model
Results
Idea of the proof
Lebesgue measure (γ = 0)
General case (gamma ≥ 0)
Long range dependence
Theorem
(ii) d =
α(1+β)
,
β
˜ where ξ˜ is
FT1+β = T log T , then XT ⇒ K λξ,
f
one-dimensional (1 + β)-stable process, totally skewed to
the right with stationary independent increments.
n
o
π
eiz ξ̃t = exp −t |z|(1+β) 1 − i(sgnz) tan (1 + β) .
2
Anna Talarczyk
Occupation time fluctuations
Model
Results
Idea of the proof
(iii) d >
α(1+β)
,
β
Lebesgue measure (γ = 0)
General case (gamma ≥ 0)
Long range dependence
FT1+β = T , then XT ⇒ X , where X is a
f
(1 + β)-stable process with values in S 0 (Rd ), it has
stationary independent increments
Z Eexp{ihX (t), ϕi} = exp −Kt
cβ ϕ(x)Gϕ(x)
Rd
π
V
+ |Gϕ(x)|1+β 1 − i(sgnGϕ(x)) tan (1 + β) dx ,
2
2
where
ϕ ∈ S(Rd ), t ≥ 0,
Z ∞
Z
1
Gϕ(x) =
Ts ϕ(x)ds = Cα,d
ϕ(y )dy ,
d−α
0
Rd |x − y |
(
0 if 0 < β < 1
cβ =
1 if β = 1.
Anna Talarczyk
Occupation time fluctuations
Model
Results
Idea of the proof
Lebesgue measure (γ = 0)
General case (gamma ≥ 0)
Long range dependence
Remark: If d < αβ , α = 2, then there is a.s. local extinction, i.e.
lim Nt (A) = 0
t→∞
a.s. for any bounded A ∈ B(Rd )
Hypothesis: true also for α < 2.
Anna Talarczyk
Occupation time fluctuations
Model
Results
Idea of the proof
Lebesgue measure (γ = 0)
General case (gamma ≥ 0)
Long range dependence
Remark: If d < αβ , α = 2, then there is a.s. local extinction, i.e.
lim Nt (A) = 0
t→∞
a.s. for any bounded A ∈ B(Rd )
Hypothesis: true also for α < 2.
High density
Theorem 2(i’)
Assume that d ≤ αβ , νT = HT λ, HT satisfies
dβ
lim HT−β T 1− α = 0,
T →∞
d
FT1+β = HT T (2+β− α β) , then XT ⇒ K λξ, where ξ as in Theorem
c
2 (i).
Anna Talarczyk
Occupation time fluctuations
Model
Results
Idea of the proof
Lebesgue measure (γ = 0)
General case (gamma ≥ 0)
Long range dependence
Remark. If β = 1, then ξ is a centered Gaussian process with
covariance function
for d <
α
β
1
−sh − t h + [(s + t)h + |s − t|h ],
2
where h = 3 − d/α (2 < h < 2 + 12 ).
(This function is positive definite for (2 < h < 4)).
for d =
α
β
i
1h
(s + t)2 log(s + t) + (s − t)2 log |s − t| −s2 log s−t 2 log t
2
Introducing high density in Theorem 2 does not change the
results, only FT is different.
Anna Talarczyk
Occupation time fluctuations
Model
Results
Idea of the proof
Lebesgue measure (γ = 0)
General case (gamma ≥ 0)
Long range dependence
General case (γ ≥ 0)
dx
initial intensity measure: HT 1+|x|
γ , HT → ∞ sufficiently quickly,
in some cases one can take HT = 1
limits: (1 + β)-stable processes X
dimension d spatial structure
temporal structure
“low”
simple
(X = K λξ)
complicated
(long range dependence)
“critical”
simple
(X = K λξ)
simple
(independent increments)
“large”
complicated
(space inhomog.)
(ξ varies in different cases)
Anna Talarczyk
simple
(independent increments)
Occupation time fluctuations
Model
Results
Idea of the proof
Lebesgue measure (γ = 0)
General case (gamma ≥ 0)
Long range dependence
for low
ξ has the form:
R t dimensions
R
ξt = 0 Rd f (t, r , x)M(drdx)
Critical dimension:
Lebesgue measure case (γ = 0): d =
1+β
β α
finite measure case (with high density) (γ > d): d =
0 ≤ γ ≤ d: d =
2+β
β α
−
γ∨α
β
Anna Talarczyk
Occupation time fluctuations
2+β
1+β
Model
Results
Idea of the proof
Lebesgue measure (γ = 0)
General case (gamma ≥ 0)
Long range dependence
S 0 (Rd ) process
of the form K λξ
X (t) = X (1), t > 0
γ<α
1
d<
1+β
β α
lrd
2
d=
1+β
β α
4
d=
1+β
β α
lrd
3
d>
1+β
β α
5
d>
1+β
β α
γ<d
γ=α
γ<d
one can take HT = 1
if
α
β
+γ <d
α<γ
6
d=
2+β
β α
−
γ
β
γ<d
γ=d
γ>d
7
9
8
10
d=
2+β
1+β α
d=
2+β
1+β α
Anna Talarczyk
Occupation time fluctuations
11
Model
Results
Idea of the proof
Lebesgue measure (γ = 0)
General case (gamma ≥ 0)
Long range dependence
S 0 (Rd ) process
of the form K λξ
X (t) = X (1), t > 0
γ<α
1
d<
1+β
β α
2
d
lrd
d=
lrd
1+β
β α
γ
3
d>
1+β
β α
γ
γ
γ<d
FT1+β = HT T 2+β− α β− α FT1+β=HT T 1− α log T FT1+β = HT T 1− α
γ=α
one can take HT = 1
γ<d
if
α
β
4
d=
1+β
β α
d>
1+β
β α
FT1+β = HT (log T )2 FT1+β = HT log T
+γ <d
α<γ
6
d=
2+β
β α
−
FT1+β = HT log T
γ<d
5
γ
β
11
FT1+β = HT
1+β
7 F
8 d = 2+β α
γ=d
Td =
1+β
2+β− α
(1+β)
log T FT1+β = HT (log T )2
HT T
9
10 d = 2+β α
γ>d
1+β
d
FT1+β=HT T 2+β− α (1+β) FT1+β = HT log T
Anna Talarczyk
Occupation time fluctuations
Model
Results
Idea of the proof
2+β
β
Case 1: Let γ < d, d <
Lebesgue measure (γ = 0)
General case (gamma ≥ 0)
Long range dependence
−
lim T 1−(d−γ)β/α HT−β = 0
γ∨α
β ,
FT1+β = HT T 2+β−(dβ+γ)/α
(only needed if d <
T →∞
α
+ γ)
β
Then XT ⇒ K λξ where ξ
c
Z
Z
ξt =
Rd+1
11[0,t] (r )
−γ
pr (x − y )|y |
1/(1+β)
dy
Rd
Z
t
pu−r (x)du M(drdx),
r
back to the general scheme
Anna Talarczyk
Occupation time fluctuations
t ≥ 0,
Model
Results
Idea of the proof
Lebesgue measure (γ = 0)
General case (gamma ≥ 0)
Long range dependence
Local extinction
Theorem
If α = 2 and d < αβ + γ and the initial intensity measure is µγ ,
then the particle system suffers a.s. local extinction, i.e. for
each bounded Borel set A
P( lim Nt (A) = 0) = 1.
t→∞
Hypothesis: the same is true for α < 2
Theorem
Let 0 < α ≤ 2, initial intensity measure = µγ , γ < α and
d ≥ αβ + γ. Then there is no almost sure local extinction.
Anna Talarczyk
Occupation time fluctuations
Model
Results
Idea of the proof
Lebesgue measure (γ = 0)
General case (gamma ≥ 0)
Long range dependence
Note: there is always extinction in probability for γ > 0 since
 −γ
if γ < d

T α
d
−
E hNT , ϕi ∼ T α log T
γ=d

 −d
T α
γ>d
Anna Talarczyk
Occupation time fluctuations
Model
Results
Idea of the proof
Lebesgue measure (γ = 0)
General case (gamma ≥ 0)
Long range dependence
Case 3: Assume γ < d, γ < α, d > α 1+β
β , and
FT1+β = HT T 1−γ/α ,
with HT ≥ 1. Then XT ⇒f X , where X is an S 0 (Rd )-valued
(1 + β)-stable process with independent increments
if β < 1 :
Ee
n
γ
γ
1− α
1− α
= exp t
−s
V
π
1+β
|Gϕ(x)|
1 − i(sgn Gϕ(x)) tan (1 + β) dx
2
2
ihX (t)−X (s),ϕi
Z
Rd
if β = 1, additional term with ϕ(x)Gϕ(x).
back to the general scheme
Anna Talarczyk
Occupation time fluctuations
Model
Results
Idea of the proof
Lebesgue measure (γ = 0)
General case (gamma ≥ 0)
Long range dependence
Case 2: Assume γ < d, γ < α, d = α 1+β
β and
FT1+β = HT T 1−γ/α log T ,
with HT ≥ 1. Then XT ⇒f K λη, where η is a (1 + β)-stable
process with independent, non-stationary increments (for
γ > 0) whose laws are determined by
Eeiz(ηt −ηs ) = exp
n
γ
γ
t 1− α − s1− α |z|1+β
o
π
1 − i(sgn z) tan (1 + β)
2
z ∈ R, t ≥ s ≥ 0.
back to the general scheme
Anna Talarczyk
Occupation time fluctuations
Model
Results
Idea of the proof
Lebesgue measure (γ = 0)
General case (gamma ≥ 0)
Long range dependence
Long range dependence
Case 1: γ < d and d <
α(2+β)
β
−
γ∨α
β .
β=1
Gaussian case.
Let 0 ≤ u < v < s < t. Then
d
Cov (ξv − ξu , ξt+T − ξt+T ) ∼ CT − α .
β<1
We introduce dependence exponent:
For 0 ≤ u < v < s < t, T > 0, z1 , z2 ∈ R let
DT (z1 , z2 ; u, v , s, t)
= | log Eei(z1 (ξv −ξu )+z2 (ξT +t −ξT +s ))
− log Eeiz1 (ξv −ξu ) − log Eeiz2 (ξT +t −ξT +s ) |.
Anna Talarczyk
Occupation time fluctuations
Model
Results
Idea of the proof
Lebesgue measure (γ = 0)
General case (gamma ≥ 0)
Long range dependence
dependence exponent κ is defined as
κ=
inf
inf
z1 ,z2 ∈R 0≤u<v <s<t
sup{γ > 0 : DT (z1 , z2 ; u, v , s, t) = o(T −γ ) as T → ∞}.
For the process ξ in 1
Z tZ
Z
pr (x − y ) |y |
ξt =
0
Rd
−γ
Rd
1
1+β
Z
dy
t
pu−r (x)duM(drdx),
r
we have

d


if

 α
κ=

d
d −γ


1+β−
if

α
α+d
Anna Talarczyk
α = 2 or β >
d −γ
,
d +α
α < 2 and β ≤
Occupation time fluctuations
d −γ
.
d +α
Model
Results
Idea of the proof
Lebesgue measure (γ = 0)
General case (gamma ≥ 0)
Long range dependence
general scheme
Cases 7 and 9: γ ≥ d, d < α 2+β
1+β
The limit process is of the form K λζ, where
Z t
Z
1/(1+β)
ζt =
11[0,t] (r )pr
(x)
pu−r (x)du M(drdx),
Rd+1
r
Dependence exponent of ζ is
κ=
Anna Talarczyk
d
α
Occupation time fluctuations
t ≥ 0.
Model
Results
Idea of the proof
Lebesgue measure (γ = 0)
General case (gamma ≥ 0)
Long range dependence
Other comments
The same results hold for superprocesses (some
differences if β = 1 in “large” dimensions)
Questions
probabilistic interpretation of the two long range
dependence regimes for γ < d in “low” dimensions
interpretation of the “critical” dimension
Anna Talarczyk
Occupation time fluctuations
Model
Results
Idea of the proof
Idea of the proof of convergence
1
Let X̃ (t)
t∈R+
be an S 0 (Rd )-valued process. Define
S 0 (Rd+1 )-valued random variable X̃ by:
D
E
Z
1
hX (t), Φ(·, t)i dt,
X̃ , Φ =
Φ ∈ S(Rd+1 )
0
d
In C([0, 1], S 0 (R
D )): E
D
E
convergence X̃T , Φ ⇒ X̃ , Φ , in law ∀Φ ∈ S(R d+1 )
+ tightness of hXT , ϕiT ≥2 in C([0, 1], R), ∀ϕ ∈ S(Rd )
=⇒ convergence of XT ⇒ X in law in w C([0, 1], S 0 (Rd )).
(Remark: in general convergence of finite dimensional
disributions of XT < convergence in law of X̃T )
Anna Talarczyk
Occupation time fluctuations
Model
Results
Idea of the proof
2
3
D
E
Convergence of X̃ , Φ – is reduced to te case Φ ≥ 0.
The limit process is totally skewed to the right ⇒ one can
use Laplace transform.
We show
( Z
)
Z T
1
Ee−hX̃T ,Φi = exp −
vT (x, T )νT (dx) +
E
hNs , Φi ds ,
FT
Rd
0
where νT – initial intensity measure, vT satisfies a certain
integral equation.
(Remark: a similar formula holds for finite dimensional
distributions: (hXT (t1P
), ϕ1 i , . . . , hXT (tn ), ϕn i)
– by approximating nj=1 δtj ϕj by Φm ∈ S(Rd+1 )).
Anna Talarczyk
Occupation time fluctuations
Model
Results
Idea of the proof
4
Ee−hX̃T ,Φi → Ee−hX̃ ,Φi ,
5
Φ ≥ 0.
Tightness (if the limit process is continuous)
β = 1 one can study
k
E |hXT (t2 ) − XT (t1 ), ϕi|
β ≤ 1. We show: there exist h, ε, r > 0 such that
P (|hXT (t2 ) − XT (t1 ), ϕi| ≥ δ)
Z δ1 1 − Re Ee−iθhXT (t2 )−XT (t1 ),ϕi dθ
≤ Cδ
0
1+ε
C(ϕ) h
≤
t2 − t1h
δr
for all 0 ≤ t1 ≤ t2 , δ > 0.
Anna Talarczyk
Occupation time fluctuations
Model
Results
Idea of the proof
Ad 3
Let
1
ΨT (x, s) =
FT
1
Z
Φ(x, t)dt
s
T
(ηs )s≥0 – standard α-stable process
g – probability generating function of the branching mechanism
N x – empirical process for a tree starting from one particle at
site x
Since N0 is a Poisson random measure with intensity νT , we get
Z R
−hX̃T ,Φi
− 0T hNsx ,ΨT (·,s)ids
Ee
= exp
Ee
− 1 νT (dx)
Rd
(Z Z
)
T
× exp
Rd
Ts ΨT (·, s)(x)dsνT (dx)
0
Anna Talarczyk
Occupation time fluctuations
(L)
Model
Results
Idea of the proof
Let
wT (x, r , t) := Ee−
RT
0
hNsx ,ΨT (·,r +s)ids
.
Conditioning with respect to the first branching
−
wT (x, r , t) = e−Vt Ee
|
Rt
0
ΨT (ηsx ,r +s)ds
{z
}
=: hT (x, r , t)
Z
+V
t
e
0
−V (t−s)
−
Ee
|
t−s
R
ΨT (ηux ,r +u)du
0
x
g(wT (ηt−s
, r + t − s, s)) ds
{z
}
=: kT (x, r , s, t − s)
Anna Talarczyk
Occupation time fluctuations
(1)
Model
Results
Idea of the proof
By the Feynman Kac Formula:
(
∂
∂
∂t hT (x, r , t) = ∆α + ∂r − ΨT (x, r ) hT (x, r , t)
hT (x, r , 0) = 1
(
∂
∂
∂σ kT (x, r , s, σ) = ∆α + ∂r − ΨT (x, r ) kT (x, r , t)
kT (x, r , s, 0) = g(wT (x, r , s))
This and (1) gives
∂

 ∂t wT (x, r , t) = ∆α +
∂
∂r
− ΨT (x, r ) wT (x, r , t)
V
+ 1+β
(1 − wT (x, r , t))1+β


wT (x, r , 0) = 1
Anna Talarczyk
Occupation time fluctuations
Model
Results
Idea of the proof
Writing the equation in the mild form and substituting
vT (x, t) = 1 − wT (x, T − t, t), (0 ≤ t ≤ T ) we obtain
Z
t
Tt−s (ΨT (·, T − t)(1 − vT (·, s))
vT (x, t) =
0
V
1+β
(vT (·, t))
(x)ds,
−
1+β
From (L) we obtain
Ee−hX̃T ,Φi
(Z "
Z
−vT (x, T ) +
= exp
Rd
(Z
)
TT −s ΨT (·, T − s)(x)ds νT (dx)
0
Z
= exp
Rd
#
T
T
h
i
)
TT −s ΨT (·, T − s)vT (·, s) + (vT (·, s))1+β (x)dsνT (dx)
0
Anna Talarczyk
Occupation time fluctuations
Model
Results
Idea of the proof
Usually (always for β < 1, and if β = 1 – in case of low
dimensions)
lim Ee−hX̃T ,Φi =
(
)
Z s
1+β
Z T
V
lim exp
TT −s
Ts−u ΨT (·, T − u)
(x)νT (dx)
1+β 0
T →∞
0
T →∞
Anna Talarczyk
Occupation time fluctuations