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Submitted on 30 Jun 2021
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Titus Lupu, Christophe Sabot, Pierre Tarrès
To cite this version:
E l e c t ro n ic J o f P r o b a bi l i t y Electron. J. Probab. 26 (2021), article no. 96, 1–25. ISSN: 1083-6489 https://doi.org/10.1214/21-EJP657
Inverting the Ray-Knight identity on the line
*Titus Lupu
†Christophe Sabot
‡Pierre Tarrès
§Abstract
Using a divergent Bass-Burdzy flow we construct a self-repelling one-dimensional diffusion. Heuristically, it can be interpreted as a solution to an SDE with a singular drift involving a derivative of the local time. We show that this self-repelling diffu-sion inverts the second Ray-Knight identity on the line. The proof goes through an approximation by a self-repelling jump processes that has been previously shown by the authors to invert the Ray-Knight identity in discrete.
Keywords: self-interacting diffusion; Gaussian free field; isomorphism theorems; local time. MSC2020 subject classifications: Primary 60G15; 60J60; 60K35; 60K37, Secondary 60J55;
81T25; 81T60.
Submitted to EJP on March 5, 2020, final version accepted on June 7, 2021. Supersedes arXiv:1910.06836.
Supersedes HAL:hal-02315386.
1
Introduction and presentation of results
Ray-Knight identity onR
We will construct a continuous self-repelling one-dimensional diffusion, involved in the inversion of the Ray-Knight identity onR. We start by recalling the latter.
Givena ě 0,pφpaqpxqqxPRwill denote a massless Gaussian free field onRconditioned to beaatx “ 0, that is to saypφpaqpxq{
?
2qxě0andpφpaqp´xq{
?
2qxě0are two independent
standard Brownian motions starting froma{?2.
*This work was supported by the French National Research Agency (ANR) grant within the project MALIN (ANR-16-CE93-0003). This work was partly supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). TL acknowledges the support of Dr. Max Rössler, the Walter Haefner Foundation and the ETH Zurich Foundation. PT acknowledges the support of the National Science Foundation of China (NSFC), grant No. 11771293 and of the Australian Research Council (ARC) grant DP180100613.
†CNRS and Sorbonne Université, LPSM, Paris, France. E-mail: titus.lupu@upmc.fr
Theorem 1.1 (Ray-Knight [25, 19, 15, 26, 23, 31]). Fixa ą 0. Letpβtqtě0be a standard
Brownian motion starting from0and let`βtpxqbe its local time process. Letτaβ2{2be the
stopping time τaβ2{2“ inftt ě 0|` β tp0q ą a 2 {2u.
Letpφp0qpxqqxPR be a massless Gaussian free field onRconditioned to be 0atx “ 0,
independent from the Brownian motionβ. Then the field
pφp0qpxq2{2 ` `β
τβ
a2 {2
pxqqxPR
has the same law as the fieldpφpaqpxq2{2qxPR.
The original formulation of Ray [25] and Knight [19] is different. It states that
p`β
τβ
a2 {2
pxqqxě0 is a squared Bessel process of dimension 0, starting froma2{2atx “ 0
(see also [26], Section XI.2). pφp0qpxq2{2qxě0is by definition a squared Bessel process
of dimension1, and by additivity property of squared Bessel processes,pφp0qpxq2{2 ` `β
τβ
a2 {2
pxqqxě0 is a squared Bessel process of dimension1 “ 1 ` 0, starting froma2{2 “
0 ` a2
{2, the same aspφpaqpxq2{2qxě0. In Theorem 1.1 we use a reformulation of the
Ray-Knights identity that generalizes to a much wider setting, such as any discrete electrical network, and continuum setting in dimension 2 and 3 after a Wick renormalization of the square of the GFF [15, 23, 31]. It also makes the connection to Brydges-Fröhlich-Spencer-Dynkin’s isomorphism [6, 11, 12] and Symanzik’s identities in Euclidean Quantum Field Theory [28, 29, 30].
Theorem 1.1 provides a way to couple on the same probability space the triplet
pφp0q, β, φpaqq. We formalize this in the following definition.
Definition 1.2.Fixa ą 0. We say that the tripletpφp0q, β, φpaqqsatisfies a Ray-Knight
coupling if the following conditions are satisfied.
• The process pφp0qpxqqxPR is distributed like a massless Gaussian free field onR conditioned to be0atx “ 0.
• The processpβtqtě0is a standard Brownian motion onRstarting from0.
• The processesφp0qandβ are independent.
• For everyx P R,
φpaqpxq2“ φp0qpxq2` 2`β
τβ
a2 {2
pxq.
• For everyx P Rsuch thatpφpaqq2is strictly positive onr0, xs, respectivelyrx, 0s, one hasφpaqpxq ą 0. For all other
x P R,φpaqpxq “ φp0qpxq.
It follows from Theorem 1.1 thatφpaqin a Ray-Knight coupling is distributed like a
massless Gaussian free field onRconditioned to beaatx “ 0.
Inversion of the Ray-Knight identity
Given a Ray-Knight coupling ofpφp0q, β, φpaqq, we are interested in the conditional law
of the stochastic process`βτβ a2 {2´t
˘
tknowingφ paq.
The Ray-Knight identity of Theorem 1.1 generalizes to discrete electrical networks and symmetric Markov jump processes on them [15, 23, 31]. This is known as second generalized Ray-Knight identity. The inversion in the discrete setting was done in [27, 20]. This inversion involves a nearest neighbor self-repelling jump process on the network. More precisely, the jump rate at timetfrom a vertexx1 to a neighborx2 is
whereCpx1, x2qis a fixed conductance,Ltpxjqis the time spent atxjby the jump process
before timet, andΦis a field on the vertices, considered as an initial condition. In the inversion of Ray-KnightΦis random, distributed as a discrete Gaussian free field. In Section 3 we detail this in the setting of a discrete subset ofR. Also note that this self-repelling jump process with jump rates (1.1) is up to a time change the vertex-diminished jump process (VDJP) studied in [27, 7].
If one takes a one-dimensional fine mesh lattice and renormalizes the jump rates (1.1), then on a purely formal level, without dealing with the convergence or the meaning of the terms involved, one gets the following equation for a continuous self-repelling diffusion: d qXt“ dWt` “ 1 2Bxlogpˇλtpxqq ˇ ˇ ˇ x“ |Xt dt”, ˇλtpxq “ ˇλ0pxq ´ 2ˇ`tpxq. (1.2)
ThereXqtis a continuous stochastic process on an intervalI,ˇλ0a continuous function
fromItop0, `8q,Wtis a standard Brownian motion and`ˇtpxqis the local time process
ofXqt. We will callˇλtthe occupation profile at timet. Our processXqtis defined up to a
finite time
q
T “ suptt ě 0|ˇλtp qXtq ą 0u.
We will also assume that
ż inf I ˇ λ0pxq´1dx “ `8, żsup I ˇ λ0pxq´1dx “ `8 (1.3)
and say thatλˇ0is admissible. This is a condition for not reaching the boundary ofIin
finite time.Xqtis a self-repelling process that tends to avoid places it has visited a lot, yet
we will see that a.s. it will eventually exhaust the occupation profile at some location in finite timeTq. As we will further see, this self-repelling process appears in the inversion
of the Ray-Knight identity in the continuous one-dimensional setting.
The equation (1.2) is not a classical SDE. It is not immediately clear how to make sense of the drift term 1
2Bxlogpˇλtpxqq ˇ ˇ ˇ
x“ |Xt
dt, as x ÞÑ ˇ`tpxq will not be differentiable
for t ą 0, and moreover there will not be a change of scale under which it will be differentiable for allt ą 0. So the problem is not only to solve (1.2) by an approximation scheme, the problem is already to give an appropriate meaning to being a solution to (1.2). The equation (1.2) is also somewhat misleading, as we believe that a solutionXqt
would not be a semi-martingale, admitting an adapted decomposition into a Brownian motion plus a drift term with zero quadratic variation, but with an infinite total variation. See [18, 21] for a discussion on this point.
However, it turns out that the equation (1.2) is in some sense exactly solvable, and in this paper we will give the explicit solution which involves a divergent bifurcating stochastic flow of diffeomorphisms of Rintroduced by Bass and Burdzy in [3]. Our construction here is similar to that of [21], where we introduced a reinforced diffusion constructed out of a different, convergent, Bass-Burdzy flow.
Heuristic reduction to a Bass-Burdzy flow
Next we explain a non-rigorous heuristic derivation of an explicit solution to (1.2). A similar heuristic appears in the introduction to [21].
Assume that fort0 ą 0, X¯tpt0q is a continuous process coinciding withXqtonr0, t0s,
and after timet0continues as a Markovian diffusion with infinitesimal generator
In other words, there is no additional self-repulsion after timet0. Then after time t0,
¯ Xpt0q
t is a scale and time changed Brownian motion. GivenS¯t0 an anti-derivative ofλˇ
´1 t0 ,
p ¯St0p ¯X
pt0q
t qqtět0is a local martingale. By further performing the time change
du “ ˇλt0p ¯X
pt0q t q´2dt
we get a standard Brownian motion.
Then it is reasonable to assume that near timet0,Xqtis close toX¯tpt0q. The idea is
to let the change of scale depend on time. Assume there is a flow of changes of scales
q
St : I Ñ R, such that Sqtis an anti-derivative of ˇλ´1t , and such thatSqtp qXtqis a local
martingale. Consideruptqthe time change given by
du “ ˇλtp qXtq´2dt,
andtpuqthe inverse time change. Assume that, by analogy with the Markovian case,
q
Stpuqp qXtpuqquě0 is a standard Brownian motionpBuquě0. Letx1ă x2P I. Then
d dup qStpuqpx2q ´ qStpuqpx1qq “ dt du d dt żx2 x1 ˇ λtpxq´1dx “ ˇλtp qXtq2 d dt żt 0 1x 1ă |Xsăx22ˇλsp qXsq ´2ds “ 2ˇλtp qXtq2λˇtp qXtq´21x 1ă |Xtăx2 (1.4) “ 21x 1ă |Xtăx2 “ 21 q Stpuqpx1qăBuă qStpuqpx2q.
This implies that dud Sqtpuqpxqis of form d qStpuqpxq
du “ 1SqtpuqpxqąBu´ 1SqtpuqpxqăBu` f puq,
for some functionf puqnot depending onx P I. Further, it is reasonable to assume that the left and the right sides ofXqtplay symmetric roles, and thusf puq ” 0. Then, we get
that
@x P I,d qStpuqpxq
du “ 1SqtpuqpxqąBu´ 1SqtpuqpxqăBu.
This is an equation studied by Bass and Burdzy in [3]. In the sequel we will constructXqt
out of the flow of solutions to the above equation. Note that if in the equation (1.2), one replaced the 1
2 in front of 1 2Bxlogpˇλtpxqq ˇ ˇ ˇ x“ |Xt dt
by a different positive constant, one would not get an as simple explicit solution. Indeed, the cancellation of powers ofˇλtp qXtqas in (1.4) would not occur.
Construction of a self-repelling diffusion out of a divergent Bass-Burdzy flow
The divergent Bass-Burdzy flow is given by the differential equation
dYu du “ " 1 ifYuą Bu, ´1 ifYuă Bu, (1.5) whereBuis a standard Brownian motion starting from0. The behavior at times when
Yu “ Buis not specified. It is shown in [3] that given an initial condition, there is a.s.
a unique solution defined for all positive times that is Lipschitz continuous. Moreover, these Lipschitz continuous solutions form a flow of increasingC1diffeomorphisms of
R,
Define
ˇ
ξu“ p qΨuq´1pBuq.
p qΨuquě0satisfies a bifurcation property [3]: there is a finite random valueybifPR, such
that fory ą ybif, Ψqupyq ą Bu forularge enough, andlim`8Ψqupyq “ `8, fory ă ybif, q
Ψupyq ă Bu forularge enough and lim`8Ψqupyq “ ´8, andtu ě 0| qΨupybifq “ Buuis
unbounded. Moreover,
ybif“ lim uÑ`8
ˇ ξu.
The processp ˇξuquě0admits [3, 18] a family of local timesΛqupyqcontinuous inpy, uq, such
that for anyf bounded Borel measurable function onRandu ě 0,
żu 0 f p ˇξvqdv “ ż R f pyqqΛupyqdy.
Moreover, these local times are related to the spatial derivative of the flow as follows:
B
ByΨqupyq “ 1 ` 2qΛupyq.
For allu ě 0,Λqupyq,
B
ByΨqupyqand B Byp qΨuq
´1pyqare locally1{2 ´ εHölder continuous iny.
Next we give the construction ofXqtout of the flowp qΨuquě0.
Definition 1.3. Letx0P I. Let be the change of scale
q S0pxq “ żx x0 ˇ λ0prq´1dr, x P I,
andSq0´1the inverse change of scale. Consider the change of timetpuqfromutot(and uptqthe inverse time change) given by
dt “ ˇλ0p qS0´1p ˇξuqq2p1 ` 2qΛup ˇξuqq´2du. (1.6) Let q T “ ż`8 0 ˇ λ0p qS0´1p ˇξuqq2p1 ` 2qΛup ˇξuqq´2du.
SetXqt“ qS0´1p ˇξuptqq, fort P r0, qT q.
We will callpBuquě0the driving Brownian motion ofXqt.
Note that q T “ ż`8 0 ˇ λ0p qS0´1p ˇξuqq2p1 ` 2qΛup ˇξuqq´2du “ ż R ż`8 0 ˇ λ0p qS0´1pyqq 2 p1 ` 2qΛupyqq´2duΛqupyqdy “ 1 2 ż R ˇ λ0p qS0´1pyqq 2 p1 ´ p1 ` 2qΛ`8pyqq´1qdy ď 1 2psupuě0 ˇ ξu´ inf uě0 ˇ ξuq sup
r qS´10 pinfuě0ξˇuq, qS´10 psupuě0ξˇuqs
ˇ λ20.
Sinceξˇuconverges at`8and thus has a bounded range,T ă `8q a.s. The processXqt
has local times
ˇ
Indeed, forf a measurable bounded function onI, żt1 0 f p qXtqdt “ żt1 0 f p qS0´1p ˇξuptqqqdt “ żupt1q 0 f p qS0´1p ˇξuqqˇλ0p qS0´1p ˇξuqq2p1 ` 2qΛup ˇξuqq´2du “ ż R żupt1q 0 f p qS´10 pyqqˇλ0p qS0´1pyqq 2 p1 ` 2qΛupyqq´2duΛqupyqdy “ 1 2 ż R f p qS0´1pyqqˇλ0p qS0´1pyqq 2
p1 ´ p1 ` 2qΛupt1qpyqq´1qdy
“ 1 2 ż I f pxqˇλ0pxqp1 ´ p1 ` 2qΛupt1qp qS0pxqqq ´1 qdx. Set ˇ λtpxq “ ˇλ0´ 2ˇ`tpxq “ ˇλ0pxqp1 ` 2qΛuptqp qS0pxqqq´1.
A posteriori, the change of time (1.6) is
dt “ ˇλtp qXtq2du.
We see that for allt P r0, qT qandx P I,λˇtpxq ą 0. Note thatXq
q T “ p qS0q ´1py bifq. Moreover, lim tÑ qT ˇ λtp qXtq “ lim uÑ`8 ˇ λ0pybifqp1 ` 2qΛupybifqq´1“ 0,
aslimuÑ`8Λqupybifq “ `8(see Section 4 in [18]).
Also note that if one setsSqt“ qΨuptq˝ qS0, thenSqtp qXtq “ Buptq, andp qS
t^ qTp qXt^ qTqqtě0is
a local martingale. Moreover,
B
BxSqtpxq “ ˇλ0pxq
´1
p1 ` 2qΛuptqp qS0pxqqq “ ˇλtpxq´1.
So,Sqtis a time-dependent change of scale indeed satisfying the properties postulated
previously in our heuristic.
Statement of the results
Now, let us see whyp qXt, ˇλtqcan be interpreted as solution to the equation (1.2), with
initial conditionpx0, ˇλ0q. We will give an explanation in terms of discrete approximations.
Let Jpnq “ 2´n
Z X I. LetXqtpnq be a continuous time discrete space self-interacting
nearest neighbor jump process onJpnq, defined by the jumps rates fromxtox ` σ2´n,
σ P t´1, 1u, at timet, equal to 22n´1 ˇ λpnqt px ` σ2´nq12 ˇ λpnqt pxq 1 2 , (1.7) where ˇ λpnqt pxq “ ˇλ0pxq ´ 2ˇ`pnqt pxq, `ˇ pnq t pxq “ 2 n żt 0 1 | Xpnqs “xds. Let q Tpnq ε “ suptt ě 0|ˇλ pnq t p qX pnq t q ą εu, ε ą 0, t pnq BJpnq“ inftt ě 0| qX pnq
t P tminJpnq, max Jpnquu.
We introduce the stopping timetpnq
BJpnqto avoid considering what happens afterXq
pnq t hits
the boundary of the domainJpnq.
If there were no self-interaction, that is to say in (1.7)λˇpnqt were replaced byλˇ0, the
process would converge in law asn Ñ `8to a solution of the SDE
dXt“ dWt` 1 2Bxlogpˇλ0pxqq ˇ ˇ ˇ x“Xt dt.
Theorem 1.4. With the notations above, for allε ą 0the family of process p qTpnq ε ^ t pnq BJpnq, qX pnq t^ qTεpnq^tpnq BJpnq , ˇλpnq t^ qTεpnq^tpnq BJpnq pxqqxPJpnq,tě0 converges in law asn Ñ `8to p qTε, qXt^ qT ε, ˇλt^ qTε pxqqxPI,tě0,
whereXqtis given by Definition 1.3 and q
Tε“ suptt ě 0|ˇλtp qXtq ą εu,
provided thatXq0pnqconverges toXq0. In particular,
tpnq
BJpnq ą qT
pnq ε
with probability converging to1. The convergence in law is for the topology of uniform convergence on compact subsets of I ˆ r0, `8q. The spatial processes on Jpnq are
considered to be linearly interpolated outsideJpnq.
Next we state how our self-repelling diffusion is related to the inversion of the Ray-Knight identity of Theorem 1.1.
Theorem 1.5. Leta ą 0andpφpaqpxqqxPRbe a massless Gaussian free field onR condi-tioned to beaatx “ 0. LetIpφpaqqbe the connected component of0intx PR|φpaqpxq ą 0u.
Forx P Ipφpaq
q, set λˇ˚
0pxq “ φpaqpxq2. Then a.s. λˇ˚0 satisfies the condition (1.3). Let
p qX˚
t, ˇλ˚tpxqqxPIpφpaqq,0ďtď qT˚ be the process, distributed conditionally onpφpaqpxqqxPR, as
the self repelling diffusion onIpφpaqq, starting from0, with initial occupation profileλˇ˚ 0,
following Definition 1.3. Let be the triple
pφp0qpxq2, βt, φpaqpxq2qxPR,0ďtďτβ a2 {2
,
jointly distributed as in the Ray-Knight coupling (Definition 1.2). Let be
q Tβ,a“ τaβ2{2´ suptt P r0, τ β a2{2s|φ p0q pβtq “ 0and@s P r0, tq, βs‰ βtu.
Then the couple
p qX˚
t, φpaqpxq 2
qxPR,0ďtď qT˚
has the same distribution as
pβτβ a2 {2´t
, φpaq
pxq2qxPR,0ďtď qTβ,a.
The notationp qX˚
t, ˇλ˚tpxq, qT˚qis reserved to the case of the initial occupation profile
ˇ λ˚
0pxq “ φpaqpxq2, so as to to avoid confusion with the case of genericˇλ0.
Note that Theorem 1.5 also trivially implies that the triple
p qX˚
t, φpaqpxq 2, ˇλ˚
tpxqqxPR,0ďtď qT˚ (1.8)
has the same distribution as
pβτβ a2 {2´t , φpaq pxq2, φp0q pxq2` 2`β τβ a2 {2´t pxqqxPR,0ďtď qTβ,a. (1.9)
Moreover, since the law ofpβτβ a2 {2´t
q0ďtďτβ a2 {2
is the same as that ofpβtq0ďtďτβ a2 {2
, we have that (1.8) is also distributed as
pβt, φpaqpxq2, φp0qpxq2` 2` β τβ
a2 {2
where Tβ,a“ inftt P r0, τaβ2{2s|φ p0q pβtq “ 0and@s P pt, τ β a2{2s, βs‰ βtu.
We prefer the time-reversed presentation (1.9) as we imagine the Brownian path
pβtq0ďtďτβ a2 {2
being reconstructed from its end by starting from the final condition
pφpaqpxq2qxPRforpφp0qpxq2` 2`βtpxqqxPR.
One could also extend the definition of the self-repelling diffusion to metric graphs, where it is again related to the inversion of the Ray-Knight identity. The proof would be essentially the same as for an interval. We won’t detail it here. A metric graph is obtained by replacing in an undirected graph each edge by a continuous line segment of certain length, corresponding to the resistance of the edge. For background on Markovian (non-selfinteracting) diffusions on metric graphs, see [4, 14, 36]. The Gaussian free field on metric graphs was introduced in [22], in relation with isomorphism theorems.
Other works on self-interacting diffusions in dimension one
Now let us review some other works on self-interacting diffusions in dimension one and their relations to ours. Two other Bass-Burdzy flows appeared in construction of self-interacting diffusions. First, in [34] it was shown that the flow of solutions to
dYu
du “ 1YuąBu
was related to the Brownian first passage bridge conditioned by its family of local times and to the Brownian burglar [35]. There the problem is similar to ours, i.e. constructing a Brownian motion with some conditioning on its family of local times, yet it is different and the processes obtained are different. Then, in [21] we constructed a linearly reinforced diffusion onRout of the flow of solutions to
dYu
du “ ´1YuąBu` 1YuăBu,
that is to say the signs are opposite to those in (1.5). The reinforced diffusion in [21] can be considered as a dual of the self-repelling diffusion in the present paper.
Our self-repelling diffusion is different from the Brownian polymer models studied in [10, 8, 9, 24, 32, 5, 17], as here the interaction of the moving particle with the occupation profile occurs locally, at zero range, and is not an average over positive ranges. In other words, we do not mollify the occupation profile prior to taking its derivative. Also, for that reason, we do not expect our process to be a semi-martingale as the above Brownian polymer models, but only a Dirichlet process in the sense of Föllmer [16], admitting an adapted decomposition into a continuous local martingale and zero quadratic variation drift, with the drift term not necessarily of bounded variation (see also [18] and [21]). Our process is also different from the true self-repelling motion (TSRM) introduced by Tóth and Werner in [33], as our process, unlike the TSRM, has the Hölder regularity of a Brownian motion and does not exhibit a2{3scaling exponent. We do not know if our process is related to the continuum directed random polymer introduced by Alberts, Khanin and Quastel in [1].
Organization of the article
the limit exists a priori and is a Brownian path. On the other hand one can construct out of the discrete space processes discrete analogues of the divergent Bass-Burdzy flow converging to the latter. Further, the proof of Theorem 1.4 for general occupation profile will follow out of a path transformation as in Proposition 2.1 (1).
The reason why we do not proceed directly to the proof of Theorem 1.4 for general occupation profile is that we need tightness and need the ratio
ˇ λpnq t^ qTεpnq px ` 2´nq ˇ λpnq t^ qTεpnq pxq
to converge to1asn Ñ `8, uniformly inpx, tqon compact subsets. Forˇλ˚
0 “ pφpaqq2this
is achieved by embedding the discrete space self-repelling processes into a Brownian motion.
Our article is organized as follows. In Section 2 we will give some properties of our self-repelling diffusion. In Section 3 we will recall how the self-repelling jump processes of Theorem 1.4 appear in the inversion of the Ray-Knight identity on discrete subsets of
R. This is a result obtained in [20]. Using this we will prove in Section 4 the Theorem 1.5 and the particular case of Theorem 1.4 whenλˇ0pxq “ ˇλ˚0pxq “ φpaqpxq
2. In Section 5
we will prove Theorem 1.4 in general.
2
Elementary properties of the self-repelling diffusion
First, we give some elementary properties ofp qXt, ˇλtpxqqxPI,0ďtď qT. They are
straigh-forward and come without proofs. For proofs of analogous statements, see Proposition 2.4 in [21].
Proposition 2.1.(1) Let I andI‚ be two open subintervals of
R. Let be λˇ0 and ˇλ‚0
two admissible initial occupation profiles onI, respectivelyI‚, andp qX
t, ˇλtpxqqxPI,0ďtď qT,
p qX‚
t, ˇλ‚tpxqqxPI‚,0ďtď qT‚ the corresponding self-repelling diffusions, starting from x0 P
I, respectively x‚
0 P I‚. One can go from one to the other by a deterministic path
transformation. More precisely, let
q S0pxq “ żx x0 ˇ λ0prq´1dr, x P I, Sq‚0pxq “ żx x‚ 0 ˇ λ‚ 0prq´1dr, x P I‚.
Lett ÞÑ θ‚ptqbe the change of time
dθ‚
ptq “ ˇλ‚
0pp qS‚0q´1˝ qS0p qXtqq2ˇλ0p qXtq´2dt.
Then then processpp qS‚
0q´1˝ qS0p qXpθ‚q´1ptqq, ˇλpθ‚q´1ptqp qS0´1˝ qS0‚pxqqq
xPI‚,0ďtďθ‚p qT q has the
same law asp qX‚
t, ˇλ‚tpxqqxPI‚,0ďtď qT‚.
(2)(Strong Markov property) For anyT stopping time for the natural filtration of
p qXt^ qTqtě0, such thatT ă qT a.s., the processp qXT `t, ˇλT `tpxqqxPI,0ďtď qT ´T, conditional on
the past beforeT, is a self-repelling diffusion with inital occupation profileλˇT.
(3) Let a ă b P I such that a ă x0 ă b. Let ta,b be the first time t that Xqt hits
a or b. We consider the stopping time ta,b ^ qT. Given a Brownian motion pBuquě0.
Let UÒ qS0pbq P p0, `8s the first time B
u ´ u hits Sq0pbq, whenever this happens. Let
UÓ qS0paqP p0, `8sthe first timeB
u` uhitsSq0paq, whenever this happens. Then
Ppta,b ă qT , qXta,b“ bq “PpU
Ò qS0pbqă UÓ qS0paqq,
Ppta,bă qT , qXta,b “ aq “PpU
Ppta,bą qT q “ PpUÒ qS0pbq“ UÓ qS0paq“ `8q.
In particular,
Ppta,bă qT , qXta,b “ bq ěPpta,bă qT , qXta,b “ aq
if and only if żb x0 ˇ λ0prq´1dr “ qS0pbq ď | qS0paq| “ żx0 a ˇ λ0prq´1dr.
Next we state that our self-repelling diffusion depends continuously on the initial occupation profile.
Lemma 2.2. LetpIkqně0be a sequence of open subintervals ofRsuch that
lim
nÑ`8inf Ik “ inf I P r´8, `8q, nÑ`8lim sup Ik“ sup I P p´8, `8s.
On eachIk we considerλˇI0k an admissible occupation profile, and we assume that for all
Kcompact subset ofI, lim kÑ`8KXIsupk|ˇλ0´ ˇλ Ik 0 | “ 0. Letp qXIk t , ˇλ Ik
t pxqqxPIk,0ďtď qTIk be the self-repelling diffusion onIk with initial occupation
profileλˇIk
0 . We assume thatlimkÑ`8Xq0Ik “ qX0P I. Then, ask Ñ `8,
p qXIk t^ qTIk, ˇλ Ik t^ qTIkpxqqxPIk,tě0 converges in law to p qXt^ qT, ˇλt^ qTpxqqxPI,tě0,
where the convergence is for the uniform topology int P r0, `8q, and uniform on compact subsets ofIforx.
Proof. This is an immediate consequence of Definition 1.3 and Proposition 2.1 (1). If moreover all of the processesp qXIk
t , ˇλ Ik
t pxqqxPIk,0ďtď qTIk and p qXt^ qT, ˇλt^ qTpxqqxPI,tě0 are
constructed of the same driving Brownian motionpBuquě0 (Definition 1.3), then the
convergence is a.s. Indeed, one uses the same processp ˇξuquě0, and the change of scale
and change of time functions involved in the construction ofp qXIk
t , ˇλ Ik
t pxqqxPIk,0ďtď qTIk
converge ask Ñ `8.
3
Inversion of the Ray-Knight identity on a discrete subset
Consider the triple
pφp0qpxq2, βt, φpaqpxq2qxPR,0ďtďτβ a2 {2
,
jointly distributed as in the Ray-Knight coupling (Definition 1.2). LetIpφpaqq be the
connected component of0intx PR|φpaqpxq ą 0u.
Let`βtpxqbe the family of local times of the Brownian motionβt. LetJ‚be a countable
discrete subset ofR, containing0, unbounded in both directions. Consider the change of time QJ‚,βptq “ ÿ xPJ‚ `βtpxq. DefineXJ‚ q “ βpQJ‚,βq´1pqq, wherepQJ ‚,β
q´1 is the right-continuous inverse ofQJ‚,β
. It is a nearest neighbor Markov jump process onJ‚, with jump rate from a vertexx
1to a
neighborx2equal to the conductance
Cpx1, x2q “
1 2|x2´ x1|
Givenx P J‚,λJ‚ q pxqwill denote λJq‚pxq “ φp0qpxq2` 2`β pQJ‚,βq´1pqqpxq “ λJ ‚ 0 pxq ` 2 żq 0 1XJ‚ r “xdr. Next, forq ě 0,OJ‚
q will denote a function from pairs of neighbor vertices inJ‚to
t0, 1u. Givenx1ă x2two neighbors inJ‚, we will say that the edgetx1, x2uis open (at
timeq) ifOJ‚ q ptx1, x2uq “ 1and closed ifOJ ‚ q ptx1, x2uq “ 0. OJ ‚ q ptx1, x2uq is defined as follows: OJ‚ 0 ptx1, x2uq “ 1φp0qpxq2has no zeroes onrx 1,x2s, OJ‚ q ptx1, x2uq “ 1φp0qpxq2`2`β pQJ‚ ,βq´1 pqqpxqhas no zeroes onrx1,x2s ,
By construction,pOJq‚qtě0is a family non-decreasing inq.
Next we state that the joint process pXqJ‚, λJ
‚
q , OJ
‚
q qqě0 is Markovian and give the
transitions rates. For details we refer to Theorem 8 in [20].
Proposition 3.1.pXqJ‚, λJ
‚
q , OJ
‚
q qqě0is a Markov process. Letx1andx2be two neighbors
inJ‚. IfXJ‚
q “ x1, then:
• XJ‚
q jumps tox2with ratep2|x2´ x1|q´1.OJ
‚
q ptx1, x2uqis then set to1(if it was not
already). • In caseOJ‚ q ptx1, x2uq “ 0,OJ ‚ q ptx1, x2uqis set to1withoutXJ ‚
q jumping with rate
1 2|x2´ x1| λJq‚px2q 1 2 λJ‚ q px1q 1 2 expp´|x2´ x1|´1λJ ‚ q px1q 1 2λJ ‚ q px2q 1 2q. (3.2) Forx P J‚, λJq‚pxq “ λJ0‚pxq ` 2 żq 0 1XJ‚ r “xdr. Proof. IfXJ‚
q jumps through the edgetx1, x2u, thenβtcrosses the interval delimited byx1
andx2, and then the local time ofβton this interval is positive, and thusOJ
‚
q ptx1, x2uq “ 1
after the jump.
As described in Section 2 in [20] and in particular in Theorem 8, the conditional probability thatOJ‚ q ptx1, x2uq “ 0, givenpXJ ‚ r , λJ ‚ r pxqqxPJ‚,0ďrďq, and thatOJ ‚ 0 ptx1, x2uq “ 0, and thatXJ‚
r has not crossed the edgetx1, x2ubefore timeq, equals
exppCpx1, x2qλJ ‚ 0 px1q 1 2λJ ‚ 0 px2q 1 2 ´ Cpx1, x2qλJ ‚ q px1q 1 2λJ ‚ q px2q 1 2q,
whereCpx1, x2qis given by (3.1). Thus, the rate (3.2) is obtained as
lim ∆qÑ0` 1 ∆q ˆ 1 ´exppCpx1, x2qλ J‚ 0 px1q 1 2λJ ‚ 0 px2q 1 2 ´ Cpx1, x2qpλJ ‚ q px1q ` 2∆qq 1 2λJ ‚ q px2q 1 2q exppCpx1, x2qλJ ‚ 0 px1q 1 2λJ0‚px2q 1 2 ´ Cpx1, x2qλJq‚px1q 1 2λJq‚px2q 1 2q ˙ .
The fieldsφp0qandφpaqrestricted toJ‚are discrete Gaussian free fields onJ‚. The
triple
pφp0qpxq2, XqJ‚, φpaq
pxq2qxPJ‚,0ďqďQJ‚,βpτβ a2 {2q
satisfies the Ray-Knight identity on the discrete networkJ‚. So in [20] one can find
a procedure inverting this Ray-Knight identity in the discrete setting. It corresponds to a time reversal of the process of Proposition 3.1 from stopping time QJ‚,βpτβ
a2{2q.
the introduction of the variables OJ‚
q , as in [20], allows for a simpler expression of
the inversion procedure. This is related to the fact thatφp0qpx
1qφp0qpx2q ą 0whenever
OJ‚
0 ptx1, x2uq “ 1. The inversion procedure that does not keep track of the variables
OJ‚
q is presented in [27], and it involves more complicated expressions with conditional
expectations of relative signs.
LetJbe a finite subset ofRcontaining0. Let us consider the continuous time discrete space self-repelling nearest neighbor jump process onJ, which has been introduced in [20]. LetˇλJ
0 be a positive function onJ. We consider the processp qXqJ, ˇλJqpxqqxPJ,qě0,
whereXqqJis a nearest neighbor jump process onJ, starting from0, with time-dependent
jump rates fromx1to a neighborx2inJgiven by
1 2|x2´ x1| ˇ λJ qpx2q 1 2 ˇ λJ qpx1q 1 2 , (3.3) and ˇ λJqpxq “ ˇλJ0pxq ´ 2 żq 0 1 | XJ r“xdr. (3.4)
Let beQqJbe a the random time coupled top qXqJ, ˇλJqpxqqxPJ,qě0 in the following way. IfJ
is reduced tot0u, then we setQqJ“ 0. Otherwise,QqJis the first timeqwhen the integral żq 0 ÿ x„ |JXJ r ˆ ˇ λJrpxq 1 2 | qXJ r´ x|ˇλJrp qXrJq 1 2 pexpp| qXrJ´ x|´1λˇJrpxq 1 2λˇJ rp qXrJq 1 2q ´ 1q´1 ˙ dr (3.5)
hits an independent exponential random variable of mean1. The notationx„ qJ XJ r means
thatxis a neighbor ofXqrJinJ. We will further explain where the definition ofQqJcomes
from. Note that a.s. the timeQqJfires before one of theˇλJqpxqreaches0. This is due to
the fact that
@K ą 0, ż 0 1 r1{2pexppKr 1{2q ´ 1q´1 dr “ `8.
Next we describe the processpX( Jq‚, ˘λJq‚,O( J‚
q qqě0 introduced in Section 3.3 in [20].
(
OJ‚
q is a function from pairs of neighbor vertices inJ‚ to t0, 1u. Given x1 ă x2 two
neighbors inJ‚, we set
(
OJ0‚ptx1, x2uq “ 1φpaqpxq2has no zeroes onrx1,x2s.
(
XJ‚
q is a nearest neighbor jump process onJ‚.
( XJ‚ 0 “ 0. Forx P J‚, ˘ λJq‚pxq “ φpaqpxq2´ 2 żq 0 1 ( XJ‚ r “x dr “ ˘λJ0‚pxq ´ 2 żq 0 1 ( XJ‚ r “x dr.
Letx1andx2be two neighbors inJ‚. If
( XJ‚ q “ x1and ( OJ‚ q ptx1, x2uq “ 1, then:
• X( Jq‚ jumps tox2with rate
1 2|x2´ x1| ˘ λJ‚ q px2q 1 2 ˘ λJ‚ q px1q 1 2 . • O( J‚
q ptx1, x2uqis set to0with rate
1 |x2´ x1| ˘ λJ‚ q px2q 1 2 ˘ λJ‚ q px1q 1 2 ` expp|x2´ x1|´1λ˘J ‚ q px1q 1 2˘λJ ‚ q px2q 1 2q ´ 1˘´1. ( XJ‚
qě0jumps instantaneously jumps tox2or stays inx1depending on which of the
The processpX( J‚ q , ˘λJ ‚ q , ( OJ‚ q qqě0is defined up to time ( QJ‚ “ suptq ě 0|˘λJ‚ q p ( XJq‚q ą 0u. By construction,X( J‚
q is always in the same connected component induced by open edges
inO( J‚
q as the vertex0.p
(
OJ‚
q q0ďqďQ(J‚ is a non-increasing family. It is easy to see that a.s.
(
XJ‚
q p
(
QJ‚
q “ 0and the edges adjacent to0are closed inO( J‚
(
QJ‚.
Let beJ˚“J‚X Ipφpaqq,Ipφpaqqbeing as in Theorem 1.1. We consider the process
p qXJ˚ q , ˇλJ
˚
q pxqqxPJ˚,0ďqď qQJ˚ following the definition (3.3), (3.4) and (3.5), with J “ J˚,
q
X0J˚ “ 0, andXqJ
˚
q “ φpaqpxq2, x P J˚. By construction,
(
XJq‚ takes values inJ˚. One
can couplep ( XJ‚ q , ˘λJ ‚ q , ( OJ‚ q q0ďqďQ(J‚andp qXJ ˚ q , ˇλJ ˚
q pxqqxPJ˚,0ďqď qQJ˚ such that on the event
q QJ˚
‰ 0(i.e.J˚ not reduced tot0u),
q QJ˚ “ suptq ě 0| ( OJ‚ q “ ( OJ‚ 0 u, and @q P r0, qQJ˚s, qXJ˚ q “ ( XJq‚. q QJ˚
is the first timeqwhen one more edge ofJ‚is closed inO(J‚
q . Note that after time
q QJ˚ , the processesXqJ ˚ q and ( XJ‚
q do not coincide anymore.
Proposition 3.2 (Lupu-Sabot-Tarrès [20], Proposition 3.4). With the notations above, the
process pX( Jq‚, ˘λJ ‚ q , ( OJq‚q0ďqďQ(J‚
has the same law as the time-reversed process
pXQJ‚J‚,β pτa2 {2β q´q, λ J‚ QJ‚,βpτβ a2 {2q´q , OJ‚ QJ‚,βpτβ a2 {2q´q q0ďqďQJ‚,βpτβ a2 {2q . In particular, by considering pX( Jq‚, ˘λJ ‚ q , ( OJ‚ q q0ďqď ( QJ‚ up to time QqJ ˚ we get the following: Corollary 3.3. LetTqJ ˚,β,a
be0ifJ˚is reduced tot0u, and otherwise,
q
TJ˚,β,a“ τaβ2{2´suptt P r0, τ
β
a2{2s|βtP pminJ˚, max J˚q, φp0qpβtq “ 0and@s P r0, tq, βs‰ βtu.
Then, the joint law of
pJ˚, φpaq pxq, XQJ‚J‚,β pτa2 {2β q´qqxPJ˚,0ďqďQJ‚,βpτβ a2 {2q´Q J‚,βpτβ a2 {2´ qT J˚,β,aq
is the same as the joint law of
pJ˚, φpaq
pxq, qXqJ˚qxPJ˚,0ďqď qQJ˚.
Proof. The identity comes from Proposition 3.2 and the fact that, in case J˚ is not
4
Convergence for squared GFF initial occupation profile
We use the notations of the previous section. First we will check that the condition (1.3) is satisfied byˇλ˚
0pxq “ φpaqpxq 2.
Lemma 4.1. A.s. we have that
ż
inf Ipφpaqq
φpaq
pxq´2dx “ `8,
żsup Ipφpaqq
φpaq
pxq´2dx “ `8.
Proof. Conditional oninf Ipφpaqq,pφpaqpinf Ipφpaqq ` xq{?2q
0ďxď| inf Ipφpaqq|{2is absolutely
continuous with respect to a Bessel3process starting from0. So we only need to check that givenpρpxqqxě0a Bessel 3 process starting from0,
ż
0
ρpxq´2dx “ `8.
Forh ą 0, letχρhdenote the first “time”xwhenρpxqreaches the levelh. Then,
żχρ1 0 ρpxq´2dx “ ÿ kě0 żχρ 2´k χρ 2´k´1 ρpxq´2dx.
By the strong Markov property ofρ, the sum on the right-hand side is a sum of positive independent terms. Moreover, by Brownian scaling satisfied by ρ, these terms are identically distributed. So the sum is a.s. infinite.
Now we considerJ‚“
Zn“ 2´nZ, andJ˚“Z˚n “ZnX Ipφpaqq. Let
Xtpnq“ βpQZn,βq´1p2ntq. Lemma 4.2. The process
pXpnq p2´nQZn,βpτβ a2 {2q´tq , φpaqpxq2´ 2`β τβ a2 {2 pxq ` 2`β pQZn,βq´1pQZn,βpτβ a2 {2q´2 ntqpxqq x P Z˚ n, 0 ď t ď p2´nQZn ,β pτaβ2{2q ´ 2´nQZn ,β pτaβ2{2´ qTZ ˚ n,β,aqq, (4.1)
interpolated linearly outsidex P Z˚
n, converges a.s. in the uniform topology to
pβτβ a2 {2´t , φpaqpxq2 ´ 2`β τβ a2 {2 pxq ` 2`β τβ a2 {2´t
pxqqxPIpφpaqq,0ďtď qTβ,a (4.2)
asn Ñ `8.
Proof. One needs to show that, on one hand, asn Ñ `8, a.s.pQZn,βq´1p2ntqconverges
totuniformly onr0, τaβ2{2s, and on the other handTqZ
˚
n,β,aconverges a.s. toTqβ,a.
The first convergence comes from the fact that
2´n ÿ xP2´nZ `βtpxq converges to t “ ż R `βtpxqdx
uniformly on compact intervals of time.
The second convergence comes from the fact thatpminZ˚
n, max Z˚nqis a non-decreasing
sequence of intervals converging toIpφpaqq, and thus, a.s., fornlarge enough,
q Tβ,a“
LetXqt˚pnq“ qXZ
˚ n
2ntbe the self-repelling jump process onZ˚n, accelerated by the factor
2n. It is the same process as in Theorem 1.4, but with a random initial occupation profile
ˇ λ˚pnq0 pxq “ ˇλZ˚n 0 pxq “ φpaqpxq 2 , x P Z˚ n.
We will show thatXqt˚pnqconverges in law asn Ñ `8to our self-repelling diffusion. For
this we will use a method that appears in [21], and construct a discrete analogue of the divergent Bass-Burdzy flow.
Proposition 4.3. Givenφpaq, let
q
Xt˚pnqbe the process onZ˚
ndefined above, and
ˇ λ˚pnqt pxq “ φpaqpxq2´ 2n`1 żt 0 1 | X˚pnqs “xds, x P Z ˚ n.
Then, asn Ñ `8, the process
p qX˚pnq t^2´nQqZ˚n, ˇλ ˚pnq t^2´nQqZ˚npxqqxPZ ˚ n,tě0, (4.3)
interpolated linearly outsidex P Z˚
n, converges in law to the self repelling diffusion
p qX˚ t^ qT˚, ˇλ
˚
t^ qT˚pxqqxPIpφpaqq,tě0
withXq0˚“ 0and the initial occupation profileλˇ˚0pxq “ φpaqpxq2.
Before proceeding to the proof of Proposition 4.3, let us explain how it implies Theorem 1.5.
Proof of Theorem 1.5. On one hand, according to Proposition 4.3,
p qXt˚pnq, ˇλ ˚pnq t pxqqxPZ˚ n,0ďtď2´nQqZ ˚ n (4.4) converges in law to p qX˚ t, ˇλ˚tpxqqxPIpφpaqq,0ďtď qT˚. (4.5)
On the other hand, according to Corollary 3.3, (4.4) has the same distribution as (4.1). According to Lemma 4.2, (4.1) in turn converges a.s. to (4.2). This means that (4.5) has the same distribution as (4.2), which is exactly what we want.
Proof of Proposition 4.3. From Corollary 3.3 and Lemma 4.2 we already know that the process (4.3) has a limit in law, but we want another description of the limit, which we will obtain by convergence. We will need the triple
pφp0qpxq2, βt, φpaqpxq2qxPR,0ďtďτβ a2 {2
,
jointly distributed as in the Ray-Knight coupling (Definition 1.2). We will also assume that all of theXqt˚pnqare defined on the same probability spaces, embedded inβtas in
Corollary 3.3.
We introducep qSt˚pnqq
0ďtď2´nQqZ
˚
n a family of mapsR Ñ R, parametrized byt. For a
givenn, the family is characterized by the following: • For allxsuch thatxandx ` 2´nare in
Z˚
n, and for allt P r0, 2´nQqZ
˚ ns, q S˚pnqt px ` 2´nq ´ qS ˚pnq t pxq “ 2´nλˇ ˚pnq t pxq´ 1 2λˇ˚pnq t px ` 2´nq´ 1 2. • Sq0˚pnqp0q “ 0. • For everyx P Z˚ n,t ÞÑ qS ˚pnq
• For eachx P Z˚ n,t ÞÑ qS
˚pnq
t pxqis continuous.
• For eacht,Sqt˚pnqis interpolated linearly between points ofZ˚n.
• Belowmin Z˚
nand abovemax Z˚n,x ÞÑ qS ˚pnq
t pxqhas constant slope1.
By construction,x ÞÑ qSt˚pnq is continuous strictly increasing. We see Sqt˚pnq as a
time-dependent change of scale. It has been constructed in such a way that the process
p qSt˚pnqp qXt˚pnqqqtis a local martingale; see Lemma 4.5 further below.
Forx P Ipφpaqqandt P r0, τβ a2{2q, set ¯ S˚ tpxq “ żx 0 pφpaqprq2´ 2`β τβ a2 {2 ` 2`β τβ a2 {2´t q´1dr. x ÞÑ ¯S˚
tpxq is an increasing diffeomorphism fromIpφpaqq to R. Clearly, we have the
following
Lemma 4.4. A.s. Sqt˚pnqpxq ´ qSt˚pnqp0qconverges toS¯t˚pxq ´ ¯S˚tp0quniformly forpx, tqin
compact subsets of Ipφpaq
q ˆ r0, τaβ2{2q. Similarly, a.s. py, tq ÞÑ p qS
˚pnq
t q´1py ` qS ˚pnq t p0qq
converges topy, tq ÞÑ p ¯S˚
tq´1py ` ¯St˚p0qquniformly on compact subsets ofR ˆ r0, τ β a2{2q. Let beMtpnq“ qS ˚pnq t p qX ˚pnq t q. Lett ˚pnq BZ˚
n be the first time
q
Xt˚pnqhitsmin Z˚n ormax Z˚n.
We introduce the stopping timet˚pnq
BZ˚n
to avoid considering what happens afterXqt˚pnqhits
the boundary of the domainZ˚ n.
Lemma 4.5. The processpMpnq
t^t˚pnq
BZ˚n
^2´nQqZ
˚
nqtě0is a local martingale in the filtration of
pφpaq, qX˚pnq t^2´nQqZ ˚ n, t ˚pnq BZ˚n1t ˚pnq BZ˚n ďtq.
Proof. Indeed, consider the following stopping times for the above filtration:t˚pnqk the first timeXqt˚pnqperformskjumps, and
q Tε˚pnq“ suptt ě |ˇλ ˚pnq t p qX ˚pnq t q ą εu. (4.6) Then|Mpnq t^t˚pnqk ^ qTε˚pnq^t˚pnq BZ˚n ^2´nQqZ ˚ n|is bounded by k2´n pmin Z˚ n ˇ λ˚pnq0 ^ εq´1. Moreover, sup kPN,εą0 t˚pnqk ^ qTε˚pnq^ t ˚pnq BZ˚ n ^ 2 ´n q QZ˚n “ t˚pnq BZ˚ n ^ 2 ´n q QZ˚n a.s. To see thatpMpnq t^t˚pnqk ^ qTε˚pnq^t˚pnq BZ˚n
^2´nQqZ˚nqtě0is a martingale, observe that at time t, if
q
Xt˚pnq“ x PZ˚
nztminZ˚n, max Z˚nu,Xqt˚pnqjumps left with rate
22n´1 ˇ λ˚pnqt px ´ 2´nq12 ˇ λ˚pnqt pxq12 ,
and thenMtpnqdecreases by
andXqt˚pnq“ x PZ˚njumps right with rate 22n´1 ˇ λ˚pnqt px ` 2´nq12 ˇ λ˚pnqt pxq12 ,
and thenMtpnqincreases by
2´nλˇ˚pnq t px ` 2´nq´ 1 2λˇ˚pnq t pxq´ 1 2,
so the average variation ofMtpnqis0.
Next we will apply a time-change which will makepMpnq
t^t˚pnq
BZ˚n
^2´nQqZ
˚
nqtě0into a
mar-tingale with normalized variance. Let be
Upnq ptq “ żt 0 1 2 ˇ λ˚pnq s p qXs˚pnqq´ 3 2 ´ ˇ λ˚pnq s p qXs˚pnq´ 2´nq´ 1 2 ` ˇλ˚pnq s p qXs˚pnq` 2´nq´ 1 2 ¯ ds. LetUqpnq“ Upnqpt˚pnq BZ˚n ^ 2 ´n q QZ˚
nq. By considering the rate of jumps and the size of jumps
ofMtpnq, we immediately get the following:
Lemma 4.6.The processppMpnq
t^t˚pnq BZ˚n ^2´nQqZ ˚ nq 2 ´ Upnqptq ^ qUpnqq tě0is a local martingale
in the filtration ofpφpaq, qX˚pnq
t^2´nQqZ ˚ n, t ˚pnq BZ˚ n1t ˚pnq BZ˚n ďtq. Let be Zpnq u “ M pnq pUpnqq´1puq. Lemma 4.7.pZpnq
u^ qUpnqquě0 is a martingale in the filtration of pφ
paq, Zpnq
u , qUpnq1Uqpnqďuq.
Moreover, for any0 ď u1ă u2,
ErpZupnq 2^ qUpnq ´ Zpnq u1^ qUpnq q2|φpaq, pZpnq u q0ďuďu1, qU pnq1 q Upnqďu1s “
Eru2^ qUpnq´ u1^ qUpnq|φpaq, pZupnqq0ďuďu1, qU
pnq1 q Upnqďu
1s, (4.7)
or equivalently, the processppZpnq
u^ qUpnqq
2
´ u ^ qUpnqq
uě0is a martingale in the filtration of
pφpaq, Zupnq, qUpnq1Uqpnqďuq.
Proof. First not that, sincepMpnq
t^t˚pnqk ^ qTε˚pnq^t˚pnq BZ˚n ^2´nQqZ˚nqtě0is a bounded martingale, so ispZpnq u^Upnqpt˚pnq k ^ qT ˚pnq ε q^ qUpnq
quě0. Moreover, with the sizes of jumps and the jump rates,
one sees thatdUtpnqis the average squared variation ofMtpnqduringdt. So after the time change, forZupnq, ErpZupnq 2^Upnqpt˚pnq k ^ qT ˚pnq ε q^ qUpnq ´ Zpnq u1^Upnqpt˚pnq k ^ qT ˚pnq ε q^ qUpnq q2|φpaq, pZupnqq0ďuďu1, qU pnq1 q Upnqďu 1s “ Eru2^ Upnqpt ˚pnq k ^ qT ˚pnq ε q ^ qUpnq´ u1^ Upnqpt ˚pnq k ^ qT ˚pnq ε q ^ qUpnq|φpaq, pZupnqq0ďuďu1, qU pnq1 q Upnqďu1s. For a fixedu ě 0,pZpnq u^Upnqpt˚pnq k ^ qT ˚pnq k´1q^ qU pnqqkě1is a martingale parametrized byk P N ˚. It converges a.s. toZpnq u^ qUpnqand is bounded inL
2, so the convergence is also in
L2. It
follows thatpZpnq
Forε ą 0andn P N˚, we considerTq˚pnq
ε the time defined by (4.6). Let bep rZupn,εqquě0
the process, which up to timeUpnqp qT˚pnq
ε q ^ qUpnqcoincides withZupnq, and after that time
continues as a standard Brownian motion starting fromZpnq
Upnqp qTε˚pnqq^ qUpnq
, conditional of that value independent of everything else.
Lemma 4.8. Asn Ñ `8, the pair pφpaq, rZupn,εqquě0 converges in law, for the uniform
convergence on compact subsets, topφpaq, Buquě0, wherepBuquě0is a standard Brownian
motion starting from0, independent ofφpaq.
Proof. The convergence ofp rZupn,εqquě0topBuquě0 follows from Theorem 1.4, Section 7.1
in [13]. To apply it, we use the following: • p rZupn,εqquě0is a martingale.
• pp rZupn,εqq2´ uquě0is a martingale by Lemma 4.7.
• The jumps ofp rZupn,εqquare bounded by2´npminZ˚ n ˇ λ˚pnq0 ^ εq´1, and in particular lim nÑ`8E ” max uě0p rZ pn,εq u ´ rZ pn,εq u´ q 2ı “ 0.
The independence ofpBuquě0fromφpaqfollows from the fact that the above listed three
conditions hold after conditioning byφpaq.
We stress that in Lemma 4.8 we neither requirepBuquě0to be defined on the same
probability space as the Xqt˚pnq and pφp0qpxq2, βt, φpaqpxq2qxPR,0ďtďτβ a2 {2
, nor the conver-gence to be in probability.
Let be, fort P r0, qTβ,a
s, U ptq “ żt 0 pφpaqpβτβ a2 {2´s q2´ 2`β τβ a2 {2 pβτβ a2 {2´s q ` 2`β τβ a2 {2´s pβτβ a2 {2´s qq´2ds, and q
Tεβ,a“ suptt ě 0|φpaqpβτβ
a2 {2´s q2´ 2`β τβ a2 {2 pβτβ a2 {2´s q ` 2`β τβ a2 {2´s pβτβ a2 {2´s q ą εu.
Clearly, we have the following:
Lemma 4.9. For allε ą 0, a.s., Tqε˚pnq converges to Tqεβ,a, Upnqptq ^ Upnqp qTε˚pnqq ^ qUpnq
converges toU ptq^U p qTβ,a
ε quniformly onr0, `8q, andpUpnqq´1puq^ qT ˚pnq ε ^t˚pnq BZ˚ n^2 ´n q QZ˚ n
converges toU´1puq ^ qTβ,a
ε uniformly onr0, `8q.
Next, foru P r0, qUpnqq, we define
q Ψpnq u pyq “ qS ˚pnq pUpnqq´1puq˝ p qS ˚pnq 0 q´1pyq, y PR.
By simple computation, we have the following:
Lemma 4.10. Foru P r0, qUpnqqsuch thatZpnq
u “ Zupnq´, we have the following expressions
and bounds for B
• ifΨqpnqu pyq ě qS0˚pnqp qX ˚pnq pUpnqq´1puq` 2´nq, B BuΨq pnq u pyq “ 2ˇλ˚pnq pUpnqq´1puqp qX ˚pnq pUpnqq´1puq` 2 ´nq´12 ˇ λ˚pnqpUpnqq´1puqp qX ˚pnq pUpnqq´1puq´ 2´nq´ 1 2 ` ˇλ˚pnq pUpnqq´1puqp qX ˚pnq pUpnqq´1puq` 2´nq´ 1 2 ; • ifΨqpnqu pyq P p qS0˚pnqp qX ˚pnq pUpnqq´1puq´ 2 ´nq, Zpnq u q, 0 ą B BuΨq pnq u pyq ą ´2ˇλ˚pnq pUpnqq´1puqp qX ˚pnq pUpnqq´1puq´ 2´nq´ 1 2 ˇ λ˚pnq pUpnqq´1puqp qX ˚pnq pUpnqq´1puq´ 2´nq´ 1 2 ` ˇλ˚pnq pUpnqq´1puqp qX ˚pnq pUpnqq´1puq` 2´nq´ 1 2 ; • ifΨqpnqu pyq ď qS0˚pnqp qXpU˚pnqpnqq´1puq´ 2 ´nq, B BuΨq pnq u pyq “ ´2ˇλ˚pnq pUpnqq´1puqp qX ˚pnq pUpnqq´1puq´ 2 ´nq´12 ˇ λ˚pnq pUpnqq´1puqp qX ˚pnq pUpnqq´1puq´ 2´nq´ 1 2 ` ˇλ˚pnq pUpnqq´1puqp qX ˚pnq pUpnqq´1puq` 2´nq´ 1 2 .
Forε ą 0, letΨrpn,εqu pyqbe defined as follows. Foru P r0, Upnqp qTε˚pnqq^ qUpnqs,Ψrpn,εqu pyq “
q
Ψpnqu pyq. Foru ą Upnqp qTε˚pnqq ^ qUpnq,Ψrpn,εqu pyqis a divergent Bass-Burdzy flow driven by
r
Zupn,εq(which is then a Brownian motion) satisfying
r Ψpn,εq u pyq ´ rΨ pn,εq Upnqp qTε˚pnqq^ qUpnq pyq “ żu Upnqp qT˚pnq ε q^ qUpnq p1 r Ψvpn,εqpyqą rZpn,εqv ´ 1Ψrpn,εqv pyqă rZvpn,εqqdv. Lemma 4.11. For allε ą 0, asn Ñ `8, the family
pφpaqpxq, rZpn,εq
u , rΨpn,εqu pyq, p rΨpn,εqu q´1pyqqxPR,yPR,uě0 (4.8)
converges in law to, for the topology of uniform convergence on compact subsets, to
pφpaqpxq, Bu, qΨupyq, p qΨuq´1pyqqxPR,yPR,uě0,
where pBuquě0 is a standard Brownian motion starting from 0, independent ofφpaq,
p qΨuquě0is the divergent Bass-Burdzy flow driven bypBuquě0, andpp qΨuq´1quě0the inverse
flow.
Proof. For this, first we will show the tightness of the family. For the tightness of the functionsp rΨpn,εqu pyqqyPR,uě0, we use that, foru ď Upnqp qTε˚pnqq ^ qUpnq,
r Ψpn,εq u pyq “ p qS ˚pnq pUpnqq´1puq˝ p qS ˚pnq 0 q´1pyq ´ qS ˚pnq pUpnqq´1puqp0qq ´ p qS˚pnq pUpnqq´1puqp qX pnq˚ pUpnqq´1puqq ´ qS ˚pnq pUpnqq´1puqp0qq ` rZ pn,εq u ,
each term having a limit in law by Lemmas 4.4, 4.8 and 4.9, and that after time
Upnqp qT˚pnq
Further, because of the identities and bounds of Lemma 4.10, any subsequential limit of (4.8) is of form
pφpaqpxq, Bu, ¯Ψupyq, p ¯Ψuq´1pyqqxPR,yPR,uě0,
wherepBuquě0is a standard Brownian motion starting from0, independent ofφpaq, and
¯ Ψupyq “ żu 0 p1Ψ¯ vpyqąBv ´ 1Ψ¯vpyqăBvqdv, (4.9)
and thus by the uniquennes proved in [3], Theorem 2.3,p ¯Ψuquě0is the divergent Bass
Burdzy flow driven bypBuquě0. To get (4.9), we used that
ˇ λ˚pnq t^ qTε˚pnq px ` 2´nq ˇ λ˚pnq t^ qTε˚pnq px ´ 2´nq “ φpaqpx ` 2´nq2 ´ 2`β τβ a2 {2 px ` 2´nq ` 2`β pQZn,βq´1pQZn,βpτβ a2 {2q´2 nt^ qT˚pnqqpx ` 2 ´nq φpaqpx ´ 2´nq2´ 2`β τβ a2 {2 px ´ 2´nq ` 2`β pQZn,βq´1pQZn,βpτβ a2 {2q´2 nt^ qT˚pnqqpx ´ 2 ´nq
a.s. converges to1asn Ñ `8, uniformly intand uniformly forxin compact subsets of
Ipφpaqq.
We are now ready to finish the proof of the Proposition 4.3. By construction,
q X˚pnq pUpnqq´1puq^ qTε˚pnq^t˚pnq BZ˚n ^2´nQqZ ˚ n “ p qS ˚pnq 0 q ´1 ˝p rΨpn,εq u^Upnqp qT˚pnq ε q^ qUpnq q´1p rZpn,εq u^Upnqp qT˚pnq ε q^ qUpnq q.
We have that the process pp qS0˚pnqq´1 ˝ p rΨupn,εqq´1p rZupn,εqqquě0 converges in law to the
processpp qS˚
0q´1˝ p qΨuq´1pBuqquě0, which appears in Definition 1.3, and out of which one
constructsXqt˚pnqby the change of time
U˚ptq “ żt
0
ˇ
λ˚sp qXs˚q´2ds, t P r0, qT˚q.
We will also denote
q T˚
ε “ suptt ě 0|ˇλ˚tp qXt˚q ą εu.
We use the fact that, asn Ñ `8, the joint processes
p qT˚pnq ε ^ t ˚pnq BZ˚ n ^ 2 ´n q QZ˚n, Upnqp qT˚pnq ε q ^ qUpnq, q X˚pnq t^ qTε˚pnq^t˚pnq BZ˚n ^2´nQqZ ˚ n, ˇλ ˚pnq t^ qTε˚pnq^t˚pnq BZ˚n ^2´nQqZ ˚ npxq, Upnq ptq ^ Upnqp qT˚pnq ε q ^ qUpnq, pUpnqq´1puq ^ qTε˚pnq^ t ˚pnq BZ˚ n ^ 2 ´n q QZ˚ nq xPZ˚n,tě0,uě0 (4.10) converges a.s. to p qTεβ,a, U p qTεβ,aq, βpτβ a2 {2´tq^pτ β a2 {2´ qT β,a ε q, φ paq pxq2´ 2`β τβ a2 {2 pxq ` 2`β pτa2 {2β ´tq^pτa2 {2β ´ qTεβ,aq pxq,
U ptq ^ U p qTεβ,aq, pU q´1puq ^ qTεβ,aqxPIpφpaqq,tě0,uě0.
If we add to the family (4.10) the processes pφp0qpxq2, βt, φpaqpxq2qxPR,0ďtďτβ a2 {2
and
pp qS˚pnq0 q´1˝ p rΨupn,εqq´1p rZupn,εqqquě0, we get a tight family which has subsequential limits
• Tqε˚“ qTεβ,a, • Xqt˚“ β pτβ a2 {2´tq fort ď qT˚ ε, • `β τβ a2 {2 pxq ´ `β τβ a2 {2´t
pxqis the local time process ofXqt˚fort ď qTε˚.
So we get the equality in law between
p qX˚ t, φpaqpxq 2 qxPR,0ďtď qT˚ ε and pβpτβ a2 {2´tq , φpaqpxq2qxPR,0ďtď qTβ,a ε .
Takingε Ñ 0, we get the equality in law between
p qX˚ t, φpaqpxq2qxPR,0ďtď qT˚ and pβpτβ a2 {2´tq , φpaqpxq2qxPR,0ďtď qTβ,a.
This finishes our proof.
Note that a posteriori, once the above identity in law established, one can show that the Brownian motionpBuquě0driving the self repelling diffusionp qXt˚qxPR,0ďtď qT˚ can be
constructed on the same probability space aspφp0qpxq2, βt, φpaqpxq2qxPR,0ďtďτβ a2 {2
, and the convergence ofpZpnq
u^Upnqp qTε˚pnqq^ qUpnq
quě0topBu^U˚p qT˚
εqquě0can be upgraded from in law
as in Lemma 4.8 to almost sure. However, in our proof we avoid using that a priori, and only rely on the convergence in law.
Combining Theorem 1.5 and Proposition 2.1 (1) one immediately gets the following:
Corollary 4.12. Let be the triple
pφp0qpxq2, βt, φpaqpxq2qxPR,0ďtďτβ a2 {2
,
jointly distributed as in the Ray-Knight coupling (Definition 1.2) and letIpφpaqqbe the
connected component of0intx PR|φpaqpxq ą 0u. LetIbe another, deterministic,
subin-terval ofRandˇλ0an admissible initial occupation profile onI. Letp qXt, ˇλtpxqqxPI,0ďtď qT
be the self-repelling diffusion onIwith initial occupation profileˇλ0, starting fromx0P I.
Let q S˚ 0pxq “ żx 0
φpaqprq´2dr, x P Ipφpaqq,
q S0pxq “ żx x0 ˇ λtprq´1dr, x P I.
Lett ÞÑ θptqbe the change of time
dθptq “ ˇλ0p qS0´1˝ qS ˚ 0pβτβ a2 {2´t qq2φpaqpβτβ a2 {2´t q´4dt.
Then then process
p qS´10 ˝ qS˚0pβτβ a2 {2´θ ´1ptqq, ppφ paq q2´ 2`β τβ a2 {2 ` 2`β τβ a2 {2´θ ´1ptqqpp qS ˚ 0q´1˝ qS0pxqqqxPI,0ďtďθp qTβ aq
has the same law asp qXt, ˇλtpxqqxPI,0ďtď qT.
Remark 4.13. Note that the processpβτβ a2 {2´t
q0ďtďτβ a2 {2
has the same law aspβtq0ďtďτβ a2 {2
5
Convergence for general initial occupation profile
In the sequel I, Xqt, ˇλt will denote the general setting,Xqt˚ andλˇ˚t being reserved
for the case ˇλ˚
0pxq “ φpaqpxq
2. Next we show that a discrete space nearest neighbor
self-repelling jump process as in Corollary 3.3, but with general initial occupation profile, can be embedded into a continuous self-repelling diffusion.
Proposition 5.1. LetJbe a finite subset ofRcontaining0. LetˇλJ
0 be a positive function
onJ. Let bep qXJ
q, ˇλJ0pxqqxPJ,0ďqď qQJ,the nearest neighbor self-repelling jump process onJ
introduced previously ( (3.3), (3.4), (3.5)), starting from0. LetqBJthe first timeqwhen
q XJ
q reachesmin Jormax J.
Letϕ “ ϕλˇJ0 be the Gaussian free fieldφpaq, witha “ ˇλJ 0p0q
1
2, conditioned onφpaqbeing
positive onrminJ, max Js, and onφpaqpxq “ ˇλJ 0pxq
1
2 for allx P J. In other words,ϕ{
? 2is obtained by interpolating between valuesˇλJ
0pxq
1 2{
?
2for consecutive pointsx P Jwith independent Brownian bridges conditioned on staying positive, and by adding below
min Jand abovemax Jtwo independent Brownian motions, the first one time-reversed, starting fromλˇJ 0pminJq 1 2{ ? 2and fromλˇJ 0pmaxJq 1 2{ ? 2respectively.
LetIpϕqbe the connected component of0in the non-zero set ofϕ. Denote by
p qXtϕ, ˇλϕtpxqqxPIpϕq,0ďtď qTϕ
the process, which conditional onϕ, is distributed as the self-repelling diffusion onIpϕq, starting from0, with initial occupation profileλˇϕ0pxq “ ϕpxq2,Tqϕbeing the first time one
of theλˇϕtpxqreaches0. LettϕBJbe the first timetwhenXqtϕreachesmin Jormax J.
Let be
QJ,ϕptq “ÿ
xPJ
ˇ `ϕtpxq,
where`ˇϕtpxq “ pˇλϕ0pxq ´ ˇλtϕpxqq{2is the local time process ofXqtϕ. DenotepQJ,ϕq´1 the
right-continuous inverse ofQJ,ϕ. Then the process
p qXϕ
pQJ,ϕq´1pqq, ˇλ
ϕ
pQJ,ϕq´1pqqpxqqxPJ,0ďqďQJ,ϕp qTϕq^QJ,ϕptϕBJq (5.1)
has the same law as
p qXqJ, ˇλJ0pxqqxPJ,0ďqď qQJ^q
BJ. (5.2)
Proof. ForpˇλJ0pxqqxPJzt0unot fixed, but random, distributed aspφpaqpxq2qxPJzt0u,φpaqbeing
conditioned on being positive onrminJ, max Js, the identity in law is a direct consequence of Corollary 3.3 and Theorem 1.5. To conclude that the identity in law disintegrated according the values ofpˇλJ
0pxqqxPJzt0ualso holds, it is sufficient to show that both sides of
the identity, (5.1) and (5.2), are continuous with respect topˇλJ
0pxqqxPJzt0u. The continuity
of the law of (5.2) with respect topˇλJ
0pxqqxPJzt0uis clear from the construction. As for
(5.1), first the law ofpϕpxqqxPrmin J,max Js, hence the law ofpˇλϕ0pxqqxPrmin J,max Js, depends continuously onpˇλJ
0pxqqxPJzt0u, and second, according to Lemma 2.2, the law of (5.1)
depends continuously onpˇλϕ0pxqqxPrmin J,max Js.
Proof of Theorem 1.4. We will first consider the case of I bounded. Without loss of generality, we assume that 0 P I and Xq0 “ 0. We also slightly simplify by taking
q
X0pnq“ qX0“ 0for alln. Using the notations of Proposition 5.1, let beJpnq“ 2´nZ X I
andϕpnqthe conditioned GFF interpolating betweenpˇλ 0pxq 1 2q xPJpnq. By Proposition 5.1, we can take q Xtpnq“ qXϕpnq pQJpnq,ϕpnqq´1p2ntq, ˇλ pnq t pxq “ qX ϕpnq pQJpnq,ϕpnqq´1p2ntqpxq, (5.3) t ď 2´nQJpnq,ϕpnq p qTϕpnq^ tϕ pnq BJpnqq, wheretϕBpnq
Jpnq is the first timeXq
ϕpnq
Lemma 5.2. Asn Ñ `8, pϕpnqpxqqxPI converges in probability to pˇλ0pxq
1
2qxPI for the
topology of uniform converge on compact subsets ofI.
Proof. Indeed, givenK a compact subinterval of I andn large enough so thatK Ď rminJpnq, max Jpnqs, one will obtainϕpnqby first interpolating linearly between the values
ofpˇλ0pxq
1 2q
xPJpnq, then by adding of order2n independent bridges from0to0of duration
2´n, each conditioned by a positivity event. The minimal probability of an event by which
we condition will converge to 1 with n. Moreover, for an unconditioned bridge, the probability to deviate more thanεfrom0isOpexpp´k2nε2
qq, for a constantk ą 0. This beats the2nfactor.
Lemma 5.3. As n Ñ `8, the process p qXϕpnq
t^ qTϕpnq^tϕpnq BJpnq , ˇλϕpnq t^ qTϕpnq^tϕpnq BJpnq pxqqxPI,tě0
con-verges in law top qXt^ qT, ˇλt^ qTpxqqxPI,tě0.
Proof. Indeed, by Lemma 5.2,pˇλϕ0pnqpxqqxPI converges in probability to pˇλ0pxqqxPI for
the topology of uniform convergence on compact subsets, the law of the self-repelling diffusion depends continuously on the initial occupation profile (Lemma 2.2), and the range ofp qXt^ qTqtě0is a.s. a compact subinterval ofI.
Lemma 5.4. Asn Ñ `8, simultaneously with the convergence in law of Lemma 5.3, we have thatt ÞÑ 2´nQJpnq,ϕpnq
pt ^ qTϕpnq
^ tϕ
pnq
BJpnqqconverges in law tot ÞÑ t ^ qT for the
uniform topology.
Proof. To simplify, we will assume here that all the
p qXϕpnq t^ qTϕpnq^tϕpnq BJpnq , ˇλϕpnq t^ qTϕpnq^tϕpnq BJpnq pxqqxPI,tě0
andp qXt^ qT, ˇλt^ qTpxqqxPI,tě0live on the same probability space, constructed from the same
driving Brownian motionpBuquě0, independent of theϕpnq. This is always possible to do.
Write 2´nQJpnq,ϕpnqpt ^ qTϕpnqq “ 2´n´1 ÿ xPJpnq pˇλϕ0pnqpxq ´ ˇλϕpnq t^ qTϕpnqpxqq “ 2´n´1 ÿ xPJpnq pˇλϕ0pnqpxq ´ ˇλ0pxq ´ ˇλϕ pnq t^ qTϕpnqpxq ` ˇλt^ qTpxqq `2´n´1 ÿ xPJpnq pˇλ0pxq ´ ˇλt^ qTpxqq. We have that 2´n´1 ÿ xPJpnq pˇλ0pxq ´ ˇλt^ qTpxqq
converges a.s. tot ^ qT, uniformly ofr0, `8q. Moreover,
ˇ ˇ2´n´1 ÿ xPJpnq pˇλϕ0pnqpxq ´ ˇλ0pxq ´ ˇλ ϕpnq t^ qTϕpnqpxq ` ˇλt^ qTpxqq ˇ ˇ ď p1 ` |I|q1 2xPJmaxpnq,sě0|ˇλ ϕpnq 0 pxq ´ ˇλ0pxq ´ ˇλϕ pnq s^ qTϕpnqpxq ` ˇλs^ qTpxq|,
|I|being the length ofI, and the right-hand side converges in probability to0. Finally,
tϕBpnq
Jpnqą qT
ϕpnq
Lemma 5.5. Asn Ñ `8, the process p qXpnq t^2´nQJpnq,ϕpnqp qTϕpnq^tϕpnq BJpnqq , ˇλpnq t^2´nQJpnq,ϕpnqp qTϕpnq^tϕpnq BJpnqq pxqqxPJpnq,tě0
converges in law top qXt^ qT, ˇλt^ qTpxqqxPI,tě0.
Proof. This follows from (5.3), Lemma 5.3 and the convergence of
t ÞÑ 2´nQJpnq,ϕpnq
pt ^ qTϕpnq^ tϕ
pnq
BJpnqq
in law tot ÞÑ t ^ qT (Lemma 5.4).
To finish the proof of Theorem 1.4, observe that by Lemma 5.5,
ˇ λpnq 2´nQJpnq,ϕpnqp qTϕpnq^tϕpnq BJpnqq p qXpnq 2´nQJpnq,ϕpnqp qTϕpnq^tϕpnq BJpnqq q converges in probability toˇλ q Tp qXTqq “ 0, thusTq pnq ε ă 2´nQJ pnq,ϕpnq p qTϕpnq^ tϕ pnq BJpnqqwith probability converging to 1.
Finally, ifIis unbounded, it is enough to consider an increasing family of bounded subintervals ofIwhich at the limit givesI, as the range ofXqt^ qT
ε is a.s. bounded.
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