Decomposition of the loops and excursion theory

Fix some set F ⊂S.

Definition 3.3.1 (excursion outside F). By non-empty excursion outside F, we mean a multiplet of the form ((ξ₁, τ¹, . . . , ξ_k, τ^k), A, B) for some k ∈ N+, ξ₁, . . . , ξ_k ∈ F^c, A, B ∈ F and τ¹, . . . , τ^k ∈ R+. Let T₀ = 0 and T_m = τ¹ +· · · + τ^m for m = 1, . . . , k. Define e : [0, Tk[→ F^c such that e(u) = ξm for u ∈ [Tm−1, Tm[. Therefore, the excursion can be viewed as a path e attached to starting point A and ending point B and it will also be denoted by (e, A, B). By empty excursion, we mean (φ, A, B).

Definition 3.3.2 (excursion measure outside F). Define a family of probability measure ν_F,ex^x,y indexed by x, y ∈F as follows:

ν_F,ex^x,y (ξ₁ =x₁, τ¹ ∈dt¹, . . . , ξ_k =x_k, τ^k ∈dt^k, A=u, B =v)

=δ_(u,v)^(x,y) 1

(R^F)^x_y1{x₁,...,xk∈F^c}Q^x_x1L^x_x¹₂· · ·L^x_x^k−1

k L^x_y^ke^Lxx11t1· · ·e^Lxkxktkdt¹· · · dt^k. and ν_F,ex^x,y [(φ, A, B) = (φ, u, v)] =δ^(x,y)_(u,v)(R^QF^xy)^x_y. Recall that

(R^F)^x_y =







Q^x_y + P

k≥1

P

x1,...,xk∈F^c

Q^x_x1Q^x_x¹₂· · ·Q^xx^k−1k Q^x_y^k for y∈F

0 otherwise.

Define a functionφ^br→exfrom the space of bridges to the space of excursions as follows: Given a bridge γ fromx toy, which is represented by

(x, T₁, γ(T₁), T₂−T₁, . . . , γ(Tp(γ)−1), T_p(γ)−Tp(γ)−1, y =γ(T_p(γ)), T −T_p(γ)), we represent φ^br→ex(γ) by

((γ(T₁), T₂−T₁, . . . , γ(Tp(γ)−1), T_p(γ)−Tp(γ)−1), x, y).

The image measure ofµ^x,y underφ^br→ex, namely φ^br→ex◦µ^x,y, has the following relation with the excursion measure ν_F,ex^x,y :

Proposition 3.3.1.

ν_F,ex^x,y (dγ) = 1

−L^yyR^x_yφ^br→ex◦µ^x,y(dγ, γ(T₁), . . . , γ(T_p(γ)−1)∈F^c)

Define a function ϕ^ex→po_F from the space of non-empty excursions out of F to the space of pointed loops as follows:

ϕ^ex→po_F : ((ξ₁, τ¹, . . . , ξ_k, τ^k), A, B)→(ξ₁, τ¹, . . . , ξ_k, τ^k)

Accordingly, ν_F,ex^x,y induces a pointed loop measure on the space of pointed loops outside of F, which is denoted by the same notationν_F,ex^x,y . The relation with the pointed loop measure is as follows:

Proposition 3.3.2. Let C = {(ξ1, τ¹, . . . , ξn, τⁿ) ∈ {pointed loops} : Q^ξ_ξⁿ₁ > 0}. Then, ϕ^ex→po_F ◦ν_F,ex^x,y is absolutely continuous with respect to µ^p∗. Moreover,

Q^ξ_ξⁿ

1

dϕ^ex→po_F ◦ν_F,ex^x,y

dµ^p∗ ((ξ₁, τ¹, . . . , ξ_n, τⁿ)) = 1{R^x_y>0,ξ1,...,ξn∈F^c}

nQ^x_ξ1Q^ξ_yⁿ R_y^x

Definition 3.3.3(Decomposition of a loop). Letl= (ξ₁, τ¹, . . . , ξ_k, τ^k)^obe a loop visiting F. The pre-trace of the loopl onF is obtained by removing all theξ_m, τ^m such thatξ_m ∈F^c for m = 1, . . . , k . We denote it by P tr_F(l). Suppose the pre-trace on F can be written as (x₁,s¹, . . . ,x_q,s^q)^o. Then we can write the loop l^o in the following form:

(x1,s¹, y₁¹, t¹₁, . . . , y_m¹₁, t¹_m1,x2,s², y₁², t²₁, . . . , y_m²₂, t²_m2, . . . ,xq,s^q, y₁^q, t^q₁, . . . , y^q_mq, t^q_mq)^o with x_i ∈ F for all i and y_jⁱ ∈ F^c for all i, j (with the following convention: if m_i = 0 for some i = 1, . . . , q, y₁ⁱ, tⁱ₁, . . . , y_mⁱ _i, tⁱ_mi does not appear in the above expression). We will use e_i to stand for (y₁ⁱ, tⁱ₁, . . . , yⁱ_m

i, tⁱ_m

i) with the convention that e_i =φ ifm_i = 0. Define a point measure EF(l) = P

i

δ_(ei,xi,xi+1). Define N_y^x(P trF(l)) =

q

P

i=1

1{xi=x,xi+1=y} with the convention that x_q+1 =x₁. Set q(P tr_F(l)) = P

x,y

N_y^x(P tr_F(l)). In particular, in the case above, we have q(P tr_F(l)) =q if q ≥2and q(P tr_F(l)) = 0 if q= 1.

Remark 5. The pre-trace (x₁,s¹, . . . ,xq,s^q)^o of a loop l on F is not necessarily a loop. We allow x_i =x_i+1 for some i= 1, . . . , q which is prohibited in the definition we gave of a loop.

Definition 3.3.4. The pre-trace of a loop l onF can always be written as follows:

(x₁,s¹₁, . . . ,x₁,s¹_m1,x₂,s²₁, . . . ,x₂,s²_m2, . . . ,x_k,s^k₁, . . . ,x_k,s^k_m

k)^o

with x_i 6=x_i+1 for i= 1, . . . , k with the usual convention that x_k+1 =x₁. Then, l_F, the trace of l onF is defined by

(x₁,t¹ =s¹₁+· · ·+s¹_m1,x₂,s²₁+· · ·+s²_m2, . . . ,x_k,t^k =s^k₁+· · ·+s^k_m

k)^o.

Formally, the trace of l on F is obtained by throwing away the parts out of F and then by gluing the rest in circular order.

By replacingµbyµ^p∗,F and considering the pointed loops, we have the following propositions.

Proposition 3.3.3. Let f be some measurable positive function on the space of excursions and g a positive measurable function on the space of pre-traces on F. Then,

µ(1_{lvisitsF}g(P tr_F(l))e^−hEF(l),fi) = µ(1_{lvisitsF}g(P tr_F(l)) Y

x,y∈F

(ν_F,ex^x,y (e^−f))^{Nyx(P trF(l))}).

Proposition 3.3.4. The image measure µ^p∗_{P tr,F} of the pointed loop measure µ^p∗,F under the map of the pre-trace on F can be described as follows:

• if x₁, . . . , x_q are not identical, then

µ^p∗_{P tr,F}(q(P tr_F(l)) =q,x₁ =x₁,s¹ ∈ds¹, . . . ,x_q=x_q,s^q ∈ds^q)

= 1

p(l_F) Y

x,y

((R^F)^x_y)^{Nyx(P trF(l))}

q

Y

i=1

(−L^x_xⁱ

i)e^Lxixisidsⁱ;

• if x₁ =. . .=x_q =x and q >1, then

µ^p∗_{P tr,F}(q(P tr_F(l)) =q,x₁ =x₁ =· · ·=x_q =x_q=x,s¹ ∈ds¹, . . . ,s^q∈ds^q)

= 1

q((R^F)^x_x)^q

q

Y

i=1

(−L^x_x)e^Lxxsidsⁱ

• if q = 1 and x₁ =x and , then

µ^p∗_{P tr,F}(q(P tr_F(l)) = 1,x=x,s∈ds) = (R^F)^x_x(−L^x_x)e^Lxxsds

| {z }

contribution of the non-trivial loops

+ 1

se^Lxxsds

| {z }

contribution of the trivial loops

where µ^p∗_{P tr,F} is the image measure of the pointed loop measureµ^p∗,F|{loops visiting F}. Proposition 3.3.5. Under the same notation as Definition 3.3.4,

• for k > 1,

µ^p∗_{P tr,F}(x₁ =x₁, . . . ,x_k =x_k, m₁ =q₁, . . . , m_k=q_k,t¹ ∈dt¹, . . . ,t^k ∈dt^k)

= 1

k(LF)^x_x¹₂· · ·(LF)^x_x^k₁

k

Y

i=1

e^(LF)xixitidtⁱ

k

Y

j=1

e^{(Lxjxj−(LF)xjxj)tj}((−L^xx^jj+ (L_F)^xx^jj)t^j)^qj−1 (q_j−1)!

• for k = 1, q₁ =q > 1,

µ^p∗_{P tr,F}(x₁ =x₁, m₁ =q₁,t¹ ∈dt¹)

= 1

t₁e^(LF)xx11t1dt¹e^{(Lxx11−(LF)xx11)t1}((−L^x_x¹₁ + (L_F)^x_x¹₁)t¹)^q1 q₁!

• for k = 1 and q₁ = 1,

µ^p∗_{P tr,F}(x₁ =x₁, m₁ = 1,t¹ ∈dt¹)

= 1

t₁e^(LF)xx11t1dt¹e^{(Lxx11−(LF)xx11)t1}((−L^x_x¹₁ + (LF)^x_x¹₁)t¹) + 1

t¹e^Lxx11t1dt¹

Proof. The result comes from Proposition 3.3.4 and Proposition 2.3.1.

Combining Proposition 3.3.3 and Proposition 3.3.5, we have the following proposition:

Proposition 3.3.6.

µ(1{l visits F}g(l_F)e^−hEF(l),fi)

=µ(1{l visits F}g(lF) Y

x6=y∈F

ν_F,ex^x,y (e^−f)^Nyx(lF)e

P

x∈F

(L^x_x−(L_F)^x_x)l^x_Fν_F,ex^x,x (1−e^−f)

) Corollary 3.3.7. We see that ν_F,ex^x,y is a probability measure on the space of the excursions from x to y out of F. By mapping an excursion (e, x, y) into the Dirac measure δ_(e,x,y), ν_F,ex^x,y induces a probability measure on M^p({excursions}), the space of point measures over the space of excursions. We will adopt the same notation ν_F,ex^x,y . Choose k samples of the excursions according to ν_F,ex^x,y , namelyex₁, . . . , ex_k, thenP

i

δ_exi has the law(ν_F,ex^x,y )^⊗k. For any β = (β^x, x ∈ F) ∈ R^F+, let NF(β) be a Poisson random measure on the space of excursions with intensity P

x

(−L^x_x + (L_F)^x_x)β^xν_F,ex^x,x . Let l_F → K(l_F,·) be a transition kernel from {the trace of the loop on F} to {point measure over the space of excursions} as follows:

K(l_F,·) = O

x6=y∈F

(ν_F,ex^x,y )^⊗Nyx(lF)O

N_F((l^x_F, x∈F))

Then the joint measure of (l_F,E_F(l)) is µ_F(dl_F)K(l_F,·) where µ_F is the image measure of µ under l→lF. By Proposition 3.2.1, µF is actually the loop measure associated with the trace of the Markov process on F or with L_F equivalently.

Remark 6. K(l_F,·) can also be viewed as a Poisson random measure on the space of excur- sions with intensity P

x

(−L^x_x+ (L_F)^x_x)l^x_Fν_F,ex^x,x + P

x6=y∈F

ν_F,ex^x,y conditioned to have exactlyN_y^x(l_F) excursions from x to y out of F for all x6=y∈F.

Definition 3.3.5. Supposeχ is a non-negative function onS vanishing onF. For an excur- sion (e, A, B), define the real-valued function hχ,·i of the excursion as follows:

hχ,(e, A, B)i= Z

χ(e(t))dt.

Lemma 3.3.8. We see that the excursion measureν_F,ex^x,y varies as the generator changes. Let ν_F,ex^x,y,χ be the excursion measure when L is replaced byL−M_χ. Define(R_χ^F)^x_y as (R^F)^x_y when L is replaced by L−M_χ. Then,

e^−hχ,·i ·ν_F,ex^x,y = (R^F_χ)^x_y (R^F)^x_yν_F,ex^x,y,χ.

In particular,

ν_F,ex^x,y [e^−hχ,·i] = (R^F_χ)^x_y (R^F)^x_y. Accordingly, we have the following corollary,

Corollary 3.3.9.

µ(1{l visits F}g(lF)e

− P

EF(l)

hχ,·i

)

=µ



1{l visits F}g(l_F) Y

x6=y∈F

(R^F_χ)^x_y (R^F)^x_y

!N_y^x(lF)

e

P

x∈F

(L^x_x−(LF)^x_x)l^x_F

1−^(RFχ)

xx (RF)x x



.

No documento Contribution à l’étude des lacets markoviens (páginas 40-45)