Construction of an Edwards’ probabilitymeasure on \mathcal{C}(\mathbb{R}_{+},\mathbb{R})

Construction of an Edwards’ probability measure on C(R+,R)

[ [ Universität Zürich Institut für Mathematik
Universität Zürich
Winterthurer strasse 190
CH-8057 Zürich
Switzerland
\smonth1 \syear2008\smonth2 \syear2010
\smonth1 \syear2008\smonth2 \syear2010
\smonth1 \syear2008\smonth2 \syear2010
Abstract

In this article, we prove that the measures associated to the one-dimensional Edwards’ model on the interval converge to a limit measure when goes to infinity, in the following sense: for all and for all events depending on the canonical process only up to time , .

Moreover, we prove that, if is Wiener measure, there exists a martingale such that , and we give an explicit expression for this martingale.

[
\kwd
\doi

10.1214/10-AOP540 \volume38 \issue6 2010 \firstpage2295 \lastpage2321 \newproclaimremarkRemark

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Construction of an Edwards’ probability measure

{aug}

A]\fnmsJoseph \snmNajnudel\correflabel=e1]joseph.najnudel@math.uzh.ch

class=AMS] \kwd60F99 \kwd60G30 \kwd60G44 \kwd60H10 \kwd60J65. Edwards’ model \kwdpolymer measure \kwdBrownian motion \kwdpenalization \kwdlocal time.

1 Introduction and statement of the main theorems

Edwards’ model is a model for polymers chains, which is defined by considering Brownian motion “penalized” by the “quantity” of its self-intersections (see also e65 ()). More precisely, for , and , let be Wiener measure on the space , and the corresponding canonical process. The -dimensional Edwards’ model on is defined by the probability measure on such that, very informally,

 Q(d),βT=exp(−β∫T0∫T0δ(X(d)s−X(d)u)dsdu)P(d)T[exp(−β∫T0∫T0δ(X(d)s−X(d)u)dsdu)]⋅P(d)T, (1)

where is a strictly positive parameter, and is Dirac measure at zero.

(In this article, we always denote by the expectation of a random variable under the probability .)

Of course, (1) is not really the definition of a probability measure, since the integral with respect to Dirac measure is not well defined. However, it has been proven that one can define rigorously the measure for , by giving a meaning to (1) (for , the Brownian path has no self-intersection, so the measure has to be equal to ).

In particular, for , one has formally the equality

 ∫T0∫T0δ(X(1)s−X(1)u)dsdu=∫∞−∞(LyT)2dy, (2)

where is the continuous family of local times of (which is -almost surely well defined).

Therefore, one can take the following (rigorous) definition:

 Q(1),βT=exp(−β∫∞−∞(LyT)2dy)P(1)T[exp(−β∫∞−∞(LyT)2dy)]⋅P(1)T.

Under , the canonical process has a ballistic behavior; more precisely, Westwater (see w85 ()) has proven that for , the law of under tends to , where is Dirac measure at and is a universal constant (approximately equal to 1.1).

This result was improved in hhk97 () (see also h98 ()), where van der Hofstad, den Hollander and König show that tends in law to a centered Gaussian variable, which has a variance equal to a universal constant (approximately equal to 0.4; in particular, smaller than one).

Moreover, in hhk03 (), the authors prove large deviation results for the variable under .

In dimension , the problem of the definition of Edwards’ model was solved by Varadhan (see v69 (), lg85 (), r86 ()). In this case, it is possible to give a rigorous definition of , but this quantity appears to be equal to infinity. However, if one formally subtracts its expectation (i.e., one considers the quantity: ), one can define a finite random variable which has negative exponential moments of any order; therefore, if we replace by this random variable in (1), we obtain a rigorous definition of . Moreover, this probability is absolutely continuous with respect to Wiener measure.

In dimension 3 (the most difficult case), subtracting the expectation (this technique is also called “Varadhan renormalization”) is not sufficient to define Edwards’ model. However, by a long and difficult construction, Weswater (see w80 (), w82 ()) has proven that it is possible to define the probability ; this construction has been simplified by Bolthausen in b93 () (at least if is small enough). Moreover, the measures are mutually singular, and singular with respect to Wiener measure.

The behavior of the canonical process under , as , is essentially unknown for and . One conjectures that the following convergence holds:

 Q(3),βT[∥XT∥]→DTν,

where depends only on and , and where is equal to for and approximately equal to for (see h98 (), Chapter 1).

At this point, we note that all the measures considered above are defined on finite interval trajectories [exactly, on ].

An interesting question is the following: is it possible to define Edwards’ model on trajectories indexed by ?

More precisely, if is Wiener measure on and the corresponding canonical process, is it possible to define a measure (for all ) such that, informally,

 Q(d),β=exp(−β∫∞0∫∞0δ(X(d)s−X(d)u)dsdu)P(d)[exp(−β∫∞0∫∞0δ(X(d)s−X(d)u)dsdu)]⋅P(d)?

In this article, we give a positive answer to this question in dimension one. The construction of the corresponding measure is analogous to the construction given by Roynette, Vallois and Yor in their articles about penalisation (see rvy06j (), rvy05 (), rvy061 (), rvy062 ()).

More precisely, let us replace the notation by for the standard Wiener measure and the notation by for the canonical process. If is the natural filtration of , and if for all , the measure is defined by

 QβT=exp(−β∫∞−∞(LyT)2dy)P[exp(−β∫∞−∞(LyT)2dy)]⋅P,

where is the jointly continuous version of the local times of (-almost surely well defined), the following theorem holds.

Theorem 1.1

For all , there exists a unique probability measure such that for all , and for all events ,

 QβT(Λs)⟶T→∞Qβ(Λs). (3)

Theorem 1.1 is the main result of our article.

Let us remark that if () and , then , since is, by definition, absolutely continuous with respect to . Hence, if Theorem 1.1 is assumed, is equal to zero.

Therefore, the restriction of to is absolutely continuous with respect to the restriction of to , and there exists a -martingale such that, for all ,

 Qβ|Fs=Dβs⋅P|Fs.

In our proof of Theorem 1.1, we obtain an explicit formula for the martingale . However, we need to define other notation before giving this formula.

Let be the measure on , defined by , and let be the set of functions from to such that

 ∫∞0[g(x)]2ν(dx)<∞,

equipped with the scalar product

 ⟨g|h⟩=∫∞0g(x)h(x)ν(dx).

The operator defined from to by

 [K(g)](x)=2g′′(x)+2g′(x)x−xg(x) (4)

is the infinitesimal generator of the process killed at rate at level , where is a Bessel process of dimension two; it is a Sturm–Liouville operator, and there exists an orthonormal basis of , consisting of eigenfunctions of , with the corresponding negative eigenvalues: where is in the interval .

Moreover, the functions are analytic and bounded (they tend to zero at infinity, faster than exponentially), and is strictly positive (these properties are quite classical, and they are essentially proven in h98 (), Chapters 2 and 3; see also hlac ()).

Now, for , let us denote by a process from to such that:

• is a squared Bessel process of dimension zero, starting at .

• is an independent squared Bessel process of dimension two.

Moreover, let be a continuous function with compact support from to , and let be a strictly positive real such that for all . We define the following quantities:

 Aβ,M+(f) = ∫∞0dlE[e∫M−∞[−β(Yyl+f(y))2+ρβ2/3Yyl]dye0(β1/3YMl)], Aβ,M−(f) = Aβ,M+(˜f),

where is defined by , and

 Aβ,M(f)=Aβ,M+(f)+Aβ,M−(f).

With this notation, we can state the following theorem, which gives an explicit formula for the martingale .

Theorem 1.2

For all and for all continuous and positive functions with compact support, the quantity is finite, different from zero, and does not depend on the choice of such that outside the interval ; therefore, we can write: . Moreover, for all , the density of the restriction of to , with respect to the restriction of to , is given by the equality

 Dβs=eρβ2/3s⋅Aβ(L∙+Xss)Aβ(0), (5)

where denotes the function [which depends on the trajectory ] such that for all .

{remark*}

The independence of with respect to (provided the support of is included in ) can be checked directly by using the fact that

 (exp(∫x0[−β(Yyl)2+ρβ2/3Yyl]dy)e0(β1/3Yxl))x≥0 (6)

is a martingale, property which can be easily proven by using the differential equation satisfied by .

For , and , let us now define the following quantity:

 K(μ)l(v)=αl(v)eμvE[e−2∫v0V(l/2,v)udu], (7)

where denotes the density of the first hitting time of zero of a Brownian motion starting at (or equivalently, the density of the last hitting time of of a standard Bessel process of dimension 3), and is the bridge of a Bessel process of dimension 3 on , starting at and ending at .

To simplify the notation, we set

 Kl(v)=K(0)l(v).

Moreover, let us consider, for , the function defined by

 (8)

for .

With this notation, Theorem 1.2 is a essentially a consequence of the two propositions stated below.

Proposition 1.3

When goes to infinity,

 eρTP[e−∫∞−∞[LyT+f(y)]2dy\mathbh1XT∈[0,M]]⟶0. (9)
Proposition 1.4

When goes to infinity,

 eρTP[e−∫∞−∞[LyT+f(y)]2dy\mathbh1XT≥M]⟶KA1,M+(f)<∞, (10)

where is a universal constant (in particular, does not depend on and ).

Moreover, for all , and the constant is given by the formula

 K=∫∞0eρv⟨χv|e0⟩dv<∞.

In the proof of these two propositions, we use essentially the same tools as in the papers by van der Hofstad, den Hollander and König. In particular, for , Propositions 1.3 and 1.4 are consequences of Proposition 1 of hhk97 ().

However, for a general function , it is not obvious that one can deduce directly our results from the material of hhk97 () and hhk03 (), since for , one has to deal with the family of local times of the canonical process on the intervals , and as for , but also on the support of . Moreover, some typos in hhk97 () make the argument as written incorrect. For this reason, we present a proof of this result in a different way than was done in hhk97 ().

The next sections of this article are organized as follows. In Section 2, we prove that Propositions 1.3 and 1.4 imply Theorems 1.1 and 1.2; in Section 3, we prove Proposition 1.3. The proof of Proposition 1.4 is split into two parts: the first one is given in Section 4; the second one, for which one needs some estimates of different quantities, is given in Section 6, after the proof of these estimates in Section 5. In Section 7, we make a conjecture on the behavior of the canonical process under the limit measure .

2 Proof of Theorems 1.1 and 1.2 by assuming Propositions 1.3 and 1.4

Let us begin to prove the following result, which is essentially a consequence of Brownian scaling.

Proposition 2.1

Let us assume Propositions 1.3 and 1.4. For any positive continuous function with compact support included in , and for all ,

 eρβ2/3TP[e−β∫∞−∞[LyT+f(y)]2dy]⟶Kβ1/3Aβ,M(f)<∞, (11)

when goes to infinity.

{pf}

Propositions 1.3 and 1.4 imply

 eρTP[e−∫∞−∞[LyT+f(y)]2dy\mathbh1XT≥0]⟶T→∞KA1,M+(f)<∞.

Now, and have the same law; hence,

 eρTP[e−∫∞−∞[LyT+f(y)]2dy\mathbh1XT≤0] =eρTP[e−∫∞−∞[L−yT+f(y)]2dy\mathbh1−XT≤0] =eρTP[e−∫∞−∞[LyT+f(−y)]2dy\mathbh1XT≥0]⟶T→∞KA1,M+(˜f) =KA1,M−(f),

which is finite.

Therefore,

 eρTP[e−∫∞−∞[LyT+f(y)]2dy]⟶T→∞KA1,M(f)<∞.

Now, let us set: . By Brownian scaling, and have the same law. Consequently,

 eρα2TP[e−β∫∞−∞[LyT+f(y)]2dy] =eρα2TP[e−α∫∞−∞[LyαTα2+αf(y)]2dy] =eρα2TP[e−∫∞−∞[LzTα2+αf(zα−1)]2dz]⟶T→∞KA1,Mα(fα) <∞,

where , defined by , has a support included in .

Therefore, Proposition 2.1 is proven if we show that .

Now, by change of variable and scaling property of squared Bessel processes,

 A1,Mα+(fα) = ∫∞0dlE[e∫Mα−∞[−(Yyl+αf(yα−1))2+ρYyl]dye0(YMαl)] = ∫∞0dlE[eα∫M−∞[−(Yzαl+αf(z))2+ρYzαl]dze0(YMαl)] = ∫∞0dlE[eβ∫M−∞[−(Yzlα−1+f(z))2+ρα−1Yzlα−1]dze0(αYMlα−1)] = α∫∞0dlE[eβ∫M−∞[−(Yzl+f(z))2+ρα−1Yzl]dze0(αYMl)] = αAβ,M+(f).

By replacing by , one obtains

 A1,Mα−(fα)=αAβ,M−(f), (13)

and by adding (2) and (13),

 A1,Mα(fα)=αAβ,M(f),

which proves Proposition 2.1.

At this point, we remark that does not depend on (as written in Theorem 1.2), since does not appear in the left-hand side of (11).

Now, let be in . One has, for all ,

 LyT=Lys+˜Ly−XsT−s,

where is the continuous family of local times of the process .

Therefore, for all ,

 P[e−β∫∞−∞(LyT)2dy|Fs] = P[e−β∫∞−∞(Lys+˜Ly−XsT−s)2dy|Fs] = P[e−β∫∞−∞(Ly+Xss+˜LyT−s)2dy|Fs].

Under and conditionally on , is fixed and by Markov property, is a standard Brownian motion.

Hence, if we assume Propositions 1.3 and 1.4, we obtain, by using Proposition 2.1,

 eρ(T−s)α2P[e−β∫∞−∞(LyT)2dy|Fs]⟶T→∞KαAβ(L∙+Xss).

Moreover,

 eρ(T−s)α2P[e−β∫∞−∞(LyT)2dy|Fs] ≤ eρ(T−s)α2P[e−β∫∞−∞(˜LyT−s)2dy|Fs] = eρ(T−s)α2P[e−β∫∞−∞(LyT−s)2dy] ≤ 2KαAβ(0)<∞,

if is large enough. On the other hand,

 eρTα2P[e−β∫∞−∞(LyT)2dy]⟶T→∞KαAβ(0),

and for large enough,

 eρTα2P[e−β∫∞−∞(LyT)2dy]≥K2αAβ(0).

Now, for all and , is different from zero (as written in Theorem 1.2), since it is the integral of a strictly positive quantity. Therefore,

and for fixed and large enough

Consequently, for all and , by dominated convergence

 P[\mathbh1ΛsP[e−β∫∞−∞(LyT)2dy|Fs]P[e−β∫∞−∞(LyT)2dy]]⟶T→∞P[\mathbh1Λseρα2sAβ(L∙+Xss)Aβ(0)].

Hence,

 QβT(Λs)⟶T→∞P(\mathbh1ΛsDβs),

where is defined by (5):

 Dβs=eρβ2/3s⋅Aβ(L∙+Xss)Aβ(0).

This convergence implies Theorems 1.1 and 1.2.

3 Proof of Proposition 1.3

If is a continuous function from to with compact support included in , one has

 P[e−∫∞−∞[LyT+f(y)]2dy\mathbh1XT∈[0,M]] ≤ P[e−∫∞−∞(LyT)2dy\mathbh1XT∈[0,M]] = P[e−T3/2∫∞−∞(Ly1)2dy\mathbh1X1∈[0,M/√T]]

by scaling properties of Brownian motion.

Hence, the right-hand side of (3) is decreasing with , which implies (for )

 eρTP[e−∫∞−∞[LyT+f(y)]2dy\mathbh1XT∈[0,M]] ≤eρT∫TT−1duP[e−∫∞−∞(Lyu)2dy\mathbh1Xu∈[0,M]] ≤eρ∫TT−1duP[e∫∞−∞[−(Lyu)2+ρLyu]dy\mathbh1Xu∈[0,M]]

by using the equality

 ∫RρLyudy=ρu.

By dominated convergence, Proposition 1.3 is proven if we show that

 ∫∞0duP[e∫∞−∞[−(Lyu)2+ρLyu]dy\mathbh1Xu∈[0,M]]<∞. (15)

In order to estimate the left-hand side of (15), we need the following lemma.

Lemma 3.1

For every positive and measurable function on

 ∫∞0P[G(Xu,L∙u)]du=∫Rda∫∞0dlE[G(a,Y∙l,a)],

where the law of the process is defined in the following way:

• for , is a squared Bessel process of dimension zero, starting at ;

• for , is an independent inhomogeneous Markov process, which has the same infinitesimal generator as a two-dimensional squared Bessel process for and the same infinitesimal generator as a zero-dimensional squared Bessel process for ;

• for , has the same law as .

{pf}

For , let be a standard Brownian motion, an independent Brownian motion starting at , and let us denote by the inverse local time of at level 0, and the first time when reaches zero.

By l98 () and by88 (), for every process on the space , which is progressively measurable with respect to the filtration ,

 ∫∞0duE[Fu(B)]=∫∞0dl∫∞−∞daE[Fτl+T(a)0(Z(l,a))], (16)

where is a process such that for and for .

By applying (16) to the process defined by , and by using Ray–Knight theorems, one obtains Lemma 3.1.

An immediate application of this lemma is the following equality:

 ∫∞0duP[e∫∞−∞[−(Lyu)2+ρLyu]dy\mathbh1Xu∈[0,M]] (17) =∫∞0dl∫M0daE[e∫∞−∞[−(Yyl,a)2+ρYyl,a]dy].

In order to majorize this expression, let us prove another result, which is also used in the proof of Proposition 1.4.

Lemma 3.2

For all , and for all measurable functions from to , the following equality holds:

 E[e∫∞0[−(Yyl,0)2+μYyl,0]dyg(∫∞0Yyl,0dy)]=∫∞0K(μ)l(v)g(v)dv, (18)

where is defined by (7).

In particular,

 E[e∫∞0[−(Yyl,0)2+ρYyl,0]dy]=¯K(ρ)l,

where

 ¯K(ρ)l=∫∞0K(ρ)l(v)dv.

Moreover, is bounded by a universal constant and

 ∫∞0¯K(ρ)ldl<∞.
{pf}

The process is a local martingale with bracket given, for , by

 ⟨Yl,0,Yl,0⟩y=4∫y0Yxl,0dx.

Therefore,

 Yyl,0=2B(l/2)∫y0Yxl,0dx

where is a Brownian motion starting at . Moreover, since stays at zero when it hits 0, the hitting time of zero for is . Hence, the change of variable gives

 ∫∞0(μ−Yyl,0)Yyl,0dy=∫S0(μ−2B(l/2)s)ds.

Therefore, one has the equalities

 E[e∫∞0[−(Yyl,0)2+μYyl,0]dyg(∫∞0Yyl,0dy)] =E[e∫S0(μ−2B(l/2)s)dsg(S)] =∫∞0eμvg(v)E[e−∫v02B(l/2)sds|S=v]P[S∈dv].

Now, this formula implies (18), since the density at of the law of is equal to