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Gibbs measures with double stochastic integrals on a path space 8 0 0 2 Volker Betz∗†and Fumio Hiroshima‡§ n a February 1, 2008 J 1 3 ] h p Abstract - h t We investigate Gibbs measures relative to Brownian motion in the case when a the interaction energy is given by a double stochastic integral. In the case when m thedoublestochastic integral isoriginatingfromthePauli-Fierz modelinnonrel- [ ativistic quantum electrodynamics, we prove the existence of its infinite volume 2 limit. v 2 6 3 1 Preliminaries 3 . 7 0 1.1 Gibbs measures relative to Brownian motions 7 0 Gibbs measures relative to Brownian motion appeared in [22], where they have been : v introduced to study a particle system linearly coupled to a scalar quantum field. A i X systematic study of such measures has been started from [6], where by making use of r this measure the spectrum of the so-called Nelson model is investigated. Since then a there has been growing activity and interest in the study of various types of these measures [4, 5, 12, 13, 21]. One way to understand Gibbs measures relative to Brownian motion is to view them as the limit of a one-dimensional chain of unbounded interacting spins, with the distancebetween thespinsgoingtozero. Asasimpleexample, whichwillbeinstructive in what follows, let us take Rd for the spin space, and fix a (finite or infinite) a priori measure ν on Rd as well as smooth, bounded functions V : Rd R and W : R Rd R. 0 → × → ∗Supported by an EPSRC fellowship EP/D07181X/1 †Mathematics Institute, University of Warwick,Coventry CV4 7AL, United Kingdom, e-mail: [email protected] ‡ThisworkisfinanciallysupportedbyGrant-in-AidforScienceResearch(C)17540181fromJSPS. §Faculty of Mathematics, Kyushu University, Hakozaki, Higashi-ku, 6-10-1, Fukuoka 812-8581, Japan, e-mail:[email protected]. 1 2 Double stochastic integrals On the lattice εZ [ T,T] with spacing ε and n = 2T/ε sites, we define the measure νW through ∩ − 1 νW(dx−n...dxn) = ν0(dxi)e−εPiV(xi)−1εPi(xi+1−xi)2+ε2Pi,jW(j−i,xj−xi). (1.1) Z νW |i|≤n Y Here Z normalizes νW to a probability measure, and for finite ε, νW is just a chain νW of interacting spins. However, the scaling becomes very important when ε 0. Then, → formally each spin configuration (x ) εZ [ T,T] becomes a function x( ) on i |i|≤n ∈ ∩ − · [ T,T], and the particular scaling of the quadratic term above gives rise to the term − lim ε (x x )2/ε2 = T (dx(s)/ds)2ds. It is this term that prevents the mea- ε→0 i i+1 − i −T sure νW from being concentrated on more and more rough functions when ε 0, P R → ensuring continuity of x(t) in the limit. Indeed, when ν is chosen as the Lebesgue 0 measure on Rd, it is not difficult to show that part of the normalization along with the quadratic term give converge to Wiener measure , so that in the limit, ε 0, we W → obtain 1 νW(dB) = e−R−TTV(Bs)ds−R−TTdsR−TTdtW(Bt−Bs,t−s)d . (1.2) T Z W νW T Here (B ) is now a Brownian motion, hence we call (1.2) Gibbs measures relative t t≥0 to Brownian motion. Indeed, the measure appearing in [22] is of the above type, and most of the subsequent works cited above have been concerned with measures of the form (1.2). In this paper we study another type of Gibbs measures, arising from a very similar discrete spin system. Namely, let us now define νWM(dx ...dx ) −n n 1 1 = ν (dx )exp ε V(x ) (x x )2 0 i i i+1 i Z − − ε − νWM |i|≤n i i Y X X + (x x ) W (j i,x x )(x x ) . (1.3) i+1 i M j i j+1 j − · − − − ! i,j X Now W is a d d matrix, but otherwise the expression looks very similar to (1.1). The M × crucial point is, however, that now the scaling of the term involving W is different. M The ε2 which ensured convergence to a double Riemann integralis goneby sandwiching ε2W between (x x )/ε and (x x )/ε, and replaced by the increments of the M i+1 i j+1 j − − spins themselves. Since these increments will eventually converge to Brownian motion increments, as discussed above, they are of order √ε, so the scaling is indeed different. So after taking the limit ε 0, we informally obtain → 1 νWM(dB) = e−R−TTV(Bs)ds−R−TTR−TTdBs·WM(Bt−Bs,t−s)dBtd . (1.4) T Z W νWM T As a consequence, taking the limit ε 0 yields a double stochastic integral in place of → the double Riemann integral (1.2). Double stochastic integrals 3 1.2 Definition of double stochastic integrals From now on we assume that d = 3 and specify the pair potential W = W = M W(X,t) = (W (X,t)) given by µν 1≤µ,ν≤3 ϕˆ(k) 2 W (X,t) := | | e−ω(k)|t|eik·Xδ⊥ (k)d3k, (1.5) µν 2ω(k) µν Z where δ⊥(k) = (δ⊥ (k)) is given by µν 1≤µ,ν≤3 k k δ⊥ (k) := δ µ ν. (1.6) µν µν − k 2 | | Measures of the type (1.4) with pair potential (1.5) appear in the study of the so-called non-relativistic quantum electrodynamics, and have been introduced on a formal level in[10,14,24]. However wenoticethattherearesomedifficulties intheexpression (1.4): For t > s, the integrand is not adapted to the natural filtration F = σ(B ;r T), T r ≤ so as a stochastic integral or any of its obvious transformations the double stochastic integral such as (1.4) does not make sense. So the right-hand side of (1.4) is just an informal symbol. In [16, Definition 4.1] and [18, (4.2)], however, the firm mathematical definition of (1.4) has been given through a Gaussian random process associated with an Euclidean quantum field. We outline it below. A Gaussian random process AE(f) labeled by f 3L2(R3+1) on some probability space (Q ,Σ ,µ ) is introduced, which has mean E E E zer∈o⊕and covariance E [A(f)A(g)] = q(f,g) given by µE 1 q(f,g) := fˆ(k,k ) δ⊥(k)gˆ(k,k )d3kdk (1.7) 0 0 0 2 · Z for f,g 3L2(R3+1), where ˆdenotes the Fourier transformation. Let ∈ ⊕ t K = 3 j ϕ( B )dBµ (1.8) t ⊕µ=1 s ·− s s Z0 be the 3L2(R3+1)-valued stochastic integral defined in the similar way as standard ⊕ stochastic integrals, where j : L2(R3) L2(R3+1) denotes the isometry satisfying s → (j f,j g) = (fˆ,e−|t−s|ωgˆ) . (1.9) s t L2(R3+1) L2(R3) See Subsection 3.1 for the details. Definition 1.1 Let W be the pair potential defined in (1.5). The dounle stochastic integral is defined by t t dB W(B B ,t s)dB := q(K ,K ). (1.10) s t s t t t · − − Z0 Z0 We would like to express (1.10) as an iterated stochastic integral in this paper. 4 Double stochastic integrals 1.3 Main results Let us define the Wiener measure on X := C(R;R3), cf. also [23, p. 39, Remark 1.]. W Let H = (1/2)∆. Suppose that f ,...,f L∞(R3) with compact support. Then 0 1 n−1 − ∈ there exists a measure on X such that W (f ,e−(t1−t0)H0f f e−(tn−tn−1)H0f ) 0 1 n−1 n L2(R3) ··· = f (B )f (B ) f (B )d . (1.11) 0 t0 1 t1 ··· n tn W ZX A path with respect to this measure is denoted by B (w) = w(t) for w X. Note t that Wiener measure is not a probability measure, indeed it has infinite m∈ass. If Px,t0 W denotes the measure of standard Brownian motion starting from x R3 at time t , 0 ∈ then f (B ) f (B )d = dx f (B ) f (B )dPx,t0. 0 t0 ··· n tn W 0 t0 ··· n tn W ZX ZR3 ZC([t0,∞);R3) Let ψ L2(R3) be a nonnegative function and we fix it throughout this paper. In the ∈ case of (1.2), the existence of the weak limit of the measure on X, 1 dνTW := Z ψ(B−T)ψ(BT)e−R−TTdsR−TTdtW(Bt−Bs,t−s)e−R−TTV(Bs)dsdW, νW T asT hasbeeninvestigated forvariouskindsofVandW, andthelimitingmeasure, νW, p→rov∞ed to be useful to study the ground state ϕ of some particle system linearly ∞ g coupledtoascalarquantum field. Namely forasuitableoperator , wecanexpress the O expectation (ϕ , ϕ ) as f dνW with some integrand f . So, beyond the existence g O g X O ∞ O of a measure of the form (1.4), one is interested in the limit as T , at least along R → ∞ a subsequence. In other words, one would like to prove the tightness of the family of measures (1.4). This is by no means an easy task, given that there are very few good general estimates on single stochastic integrals, let alone double integrals. The purpose of our present paper is to point out that there is at least one special case where there is a comparatively easy way to construct both the finite volume Gibbs measure and the infinite volume limit, namely the case when W = W = (W ) M µν 1≤µ,ν≤3 is given in (1.5). Fortunately, this special case is the one that motivated the whole theory of Gibbs measures with double stochastic integrals. The main results in this paper are (1) we give an iterated stochastic integral expression of (1.10); (2) we show the tightness of the family of measures 1 ψ(BT)ψ(B−T)e−R−TTV(Bs)ds−α2R−TTR−TTdBs·W(Bt−Bs,t−s)dBtd Z W T for a general class of V including the Coulomb potential V(x) = 1/ x , and arbitrary − | | values of coupling constant α R. ∈ Double stochastic integrals 5 There has been recent progress both of the above topics: M. Gubinelli and J. L˝orinczi [13] employ the concepts of stochastic currents rough paths in order to define (1.4) rigorously for finite volume, and use a cluster expansion in order to construct the infinite volume limit. While these are impressive results, the techniques used are rather advanced, and the use of cluster expansion comes with strong assumptions on single site potentials V and coupling constants. The advantage of our methods is that we can avoid some restrictions needed in [13]; in particular we need not restrict to single site potentials that grow faster than quadratically at infinity, and we need no small coupling constant in front of the double stochastic integral. In particular our results include the Coulomb potential which is the most reasonable single site potential. On the other hand, of course the range of potentials W that is treated in [13] is much greater than ours. The paper is organized as follows: In Section 2 we will construct the finite volume Gibbs measure as the marginal of a measure with single stochastic integral on a larger state space. This construction is well known [25], but has not been carried out rigorously so far. In Section 3, we rely on the detailed results available about the Pauli-Fierz model [20, 11] in order to show that our family of Gibbs measures is tight, giving the existence of an infinite volume measure. While we expect that the general method of enlarging the state space should allow us to define and prove infinite volume limits for many more models than just Pauli-Fierz, this is not all straightforward. We will comment on this issue at the end of Section 3. 2 Iterated expression of finite volume measures In this section we will specify the measure µ that we are working with, and identify T it as the marginal of another measure ν on a larger state space. Let us start by T introducing an infinite dimensional Ornstein-Uhlenbeck process which will serve as the reference measure for the auxiliary degrees of freedom. Put ω(k) = k 2 +m2 (2.1) | | for m 0, and let X (f) be the Gausspian random process on a probability space s (Q,Σ, ≥)labeledbymeasurable functionf = (f ,f ,f )withmeanzeroandcovariance 1 2 3 G given by 1 E [X (f)X (g)] = d3k e−ω(k)|t−s|fˆ(k) δ⊥(k)gˆ(k). (2.2) G s t 2ω(k) · Z HerefˆdenotestheFouriertransformoff andweassume thatfˆ /√ω,gˆ /√ω L2(R3), µ ν ∈ µ,ν = 1,2,3. Remark 2.1 Let Y (f) be the Gaussian random process on (Q ,Σ ,µ ) defined by s E E E Y (f) := AE(j (fˆ/√ω)∨). (2.3) s s Then Y (f) is mean zero and its covariance is s E [Y (f)Y (f)] = E [X (f)X (g)]. (2.4) µE s t G s t 6 Double stochastic integrals Hence Y (f) and X (f) are isomorphic as Gaussian random processes. s s Wewill nowcouple totheWiener measure . Forthiswe usea coupling function G W ϕ with the assumption below: Assumption (A): (1) ϕˆ(k) = ϕˆ( k) = ϕˆ(k) and √ωϕˆ,ϕˆ/ω L2(R3). − ∈ (2) ϕˆ is rotation invariant, i.e., ϕˆ(Rk) = ϕˆ(k) for all R O(3). ∈ Let us now define the quantity T J (X) := X (ϕ( B )) dB . (2.5) [0,T] s s s ·− · Z0 The proper definition of J reads [0,T] n J (X) := lim X (ϕ( B )(B B )), (2.6) [0,T] (j−1)T/n (j−1)T/n jT/n (j−1)T/n n→∞ ·− − j=1 X where the right hand side strongly converges in L2(X Q; Px,0). This is proved × G ⊗ W by showing that the right-hand side of (2.6) is Cauchy by making use of (2.2). In the same way, we can define T J (X) := X (ϕ( B )) dB , (2.7) [−T,T] s s s ·− · Z−T where (B ) denotes the 3-dimensional Brownian motion on the whole time line R. t t∈R The coupling between the Gaussian process and Brownian motion is given by the measure ν on X Q with × 1 T dν = exp iα X (ϕ( B )) dB ψ(B )ψ(B )d d , (2.8) T s s s −T T Z ·− · W ⊗ G T (cid:18) Z−T (cid:19) where ψ L2(R3) is an arbitrary nonnegative function, Z the normalizing constant, T ∈ andαisacouplingconstant. Inordertoguaranteethatthedensityin(2.8)isintegrable with respect to , we chose the boundaryfunction ψ to be of rapiddecrease at infinity. W We are now in the position to define our finite volume Gibbs measure. We will introduce an on-site potential V which we take Kato-decomposable [7], i.e. we require that the negative part V is in the Kato class while the positive part V is the locally − + Kato class [23]. This ensures e.g. that t supE exp V(B )ds < . (2.9) x PWx,0(cid:20) (cid:18)−Z0 s (cid:19)(cid:21) ∞ Double stochastic integrals 7 Definition 2.2 Let V : R3 R be Kato-decomposable and α R a coupling constant. → ∈ Then the measure µV on X is defined through T 1 dµVT := Z e−R−TTV(Bs)dsEG[dνT] T 1 T = ψ(B−T)ψ(BT)e−R−TTV(Bs)dsEG exp iα Xs(ϕ( Bs)) dBs d . Z ·− · W T (cid:20) (cid:18) Z−T (cid:19)(cid:21) (2.10) We want to show that the measure µV we just defined is a Gibbs measure with T double stochastic integral as given in Section 1. The key to doing this is the fact that we will be actually able to calculate the Gaussian integral exp(iJ (X))d (X), Q [−T,T] G and thus are left with an expression involving Brownian motion paths only. In doing R so, we will set α = 1 for a simpler notation. Let us give the heuristic presentation first. By the standard formula we have 1 E [eiJ[−T,T]] = exp E [J2 ] (2.11) G −2 G [−T,T] (cid:18) (cid:19) and formally, by Remark 2.1, we have 1 T T E [J2 ] for=mal dB W(B B ,t s)dB , (2.12) G [−T,T] 2 s · t − s − t Z−T Z−T where W is given in (1.5). As it stands, there are problems with the right-hand side of formal expression (2.12), mainly because the integrand is not adapted. The resolution is to use symmetry of W and break up the integral into two parts, one where s < t and one where s > t, which are then proper iterated Itô integrals. This leaves the diagonal part, which gives a non-vanishing contribution by the unbounded variation of B . t We define the iterated stochastic integral S by T ϕˆ(k) 2 T s S := d3k| | eik·BsdB e−ω(k)(s−r)e−ik·Brδ⊥(k)dB + T s r 2ω(k) · Z Z−T Z−T T ϕˆ(k) 2 + d3k| | (2.13) 3 2ω(k) Z S is the well-defined expression that will replace (2.10). The above line of reasoning T and(2.13)arenotnew[25],except that(2.13)isusuallynotwrittenoutbutinsteadjust referred to as the double stochastic integral with the diagonal removed. Nevertheless, (2.13) can be considered as known. However, the derivation above is mathematically not rigorous, since the ill-defined expression (2.12) appears along the way. To avoid this, one has to derive (2.13) directly from E [eiJ[−T,T]]. This is what we do in the next G theorem. Theorem 2.3 For almost every w X, we have ∈ E [eiJ[−T,T]] = e−ST. (2.14) G 8 Double stochastic integrals Proof: Let us replace the time interval [ T,T] with [0,T] for notational convenience. − We employ (2.6) and use dominated convergence to get n E [eiJ[0,T]] = lim E exp i X (ϕ( B )) δB G n→∞ G " ∆j ·− ∆j · j!# j=1 X n 2 1 = lim exp E X (ϕ( B )) δB , n→∞ −2 G " ∆j ·− ∆j · j#  j=1 X   where we set δB = B B and ∆ = (j 1)T/n, j = 1,...,N. Now j jT/n (j−1)T/n j − − n 2 E X (ϕ( B )) δB G ∆j ·− ∆j · j " # j=1 X ϕˆ(k) 2 n n = d3k | | e−|∆j−∆l|ω(k)eik(B∆j−B∆l)δBj δ⊥(k)δBl 2ω(k) · Z j=1 l=1 XX n ϕˆ(k) 2 j−1 = 2 d3k | | e−∆jω(k)eikB∆j e+∆lω(k)e−ikB∆lδBj δ⊥(k)δBl (2.15) 2ω(k) · j=1 Z l=1 X X n ϕˆ(k) 2 + δB d3k | | δ⊥(k) δB . (2.16) j j · 2ω(k) j=1 (cid:18)Z (cid:19) X For the diagonal term in the last line above we note that ϕˆ(k) 2 2 ϕˆ(k) 2 | | δ⊥ (k)d3k = δ | | d3k 2ω(k) µν µν3 2ω(k) Z Z bytherotationinvarianceofϕˆ. Nowasn , n δB 2 T,while n δBµ 2 → ∞ j=1| j| → j=1| j| → T/3 for all µ = 1,2,3, for almost every w X. Thus for almost every w X, we find ∈ P P∈ n ϕˆ(k) 2 2T ϕˆ(k) 2 lim δB d3k | | δ⊥(k) δB = d3k | | . j j n→∞ · 2ω(k) 3 2ω(k) j=1 (cid:18)Z (cid:19) Z X For the off-diagonal term, we start by noting that by the definition of the Itô integral for locally bounded functions f,g : R R3 R, we can see that × → t s 2 E ds f(s,B ) g(r,B )dBµ < . PW0,0 "Z0 (cid:12) s Z0 r r(cid:12) # ∞ (cid:12) (cid:12) Hence thestochastic integralofρµ(cid:12)(s) = f(s,B ) sg(r,B(cid:12))dBµ exists forallµ = 1,2,3, (cid:12) s 0 r(cid:12) r and it holds that for each k R3, ∈ R n ∆j lim f(∆ ,B )δB δ⊥(k) g(r,B )dB n→∞ j ∆j j · r r j=1 (cid:18) Z0 (cid:19) X T s = f(s,B )dB δ⊥(k) g(r,B )dB (2.17) s s r r · Z0 (cid:18)Z0 (cid:19) Double stochastic integrals 9 strongly in L2(X;P0,0). By the independence of Brownian increments and the fact W that E [(δB )2] = 1/n, E [δB ] = 0, we can estimate the L2(X;P0,0)-difference of P0,0 j P0,0 j W W W (2.17) and the off-diagonal term: 2 n ∆j j E f(∆ ,B )δB δ⊥(k) g(r,B )dB g(∆,B )δB PW0,0 "j=1 j ∆j j · Z0 r r − l=1 l ∆l l!# X X 2 2 1 3 n ∆j j = δ (k)E f(∆ ,B )2 g(r,B )dBν g(∆ ,B )δBν 3n ν=1 νν PW0,0 j=1 j ∆j Z0 r r − l=1 l ∆l l !  X X X  2  2 1 3 n ∆j j f 2 δ (k) E g(r,B )dBν g(∆,B )δBν . (2.18) ≤ k k∞ 3n ν=1 νν j=1 PW0,0"Z0 r r − l=1 l ∆l l # X X X Then the right-hand side above converges to zero as n , and together with (2.17) → ∞ we find n j lim f(∆ ,B )δB δ⊥(k) g(∆,B )δB n→∞ j ∆j j · l ∆l l j=1 l=1 X X s = f(s,B )dB δ⊥(k) g(r,B )dB (2.19) s s r r · Z Z0 in L2(X;P0,0). By putting f(t,x) = eik·xe−ω(k)t and g(t,x) = e−ik·xeω(k)t in (2.19), we W can see that (2.15) converges to the off-diagonal part of S . Then the proof is finished. T qed Remark 2.4 It is interesting that we know that e−ST = E [eiJ[−T,T]] 1 almost G | | | | ≤ surely. This is not obvious from the iterated integral representation e−ST. Let us summarize: Proposition 2.5 Let µV be the measure on X from Definition 2.2. Then T 1 dµVT = Z ψ(B−T)ψ(BT)e−α2SˆTe−R−TTV(Bs)dsdW, T where Sˆ is defined by S with the diagonal part removed: T T ϕˆ(k) 2 T s Sˆ := d3k| | eik·BsdB e−ω(k)(s−r)e−ik·Brδ⊥(k)dB . T s r 2ω(k) · Z Z−T Z−T Or T Sˆ := Z(s,w) dB , T s · Z−T where s ϕˆ(k) 2 Z(s,w) = dB | | δ⊥(k)e−(s−r)ω(k)e−ik·(Br−Bs)d3k . r 2ω(k) Z−T (cid:18)Z (cid:19) 10 Double stochastic integrals Remark 2.6 In Proposition 2.5, the diagonal term T d3k|ϕˆ(k)|2 is absorbed in the 3 2ω(k) normalization constant, since it does not depend on the Brownian path B. Moreover R from Remark 2.4 it follows that T ϕˆ(k) 2 exp Sˆ exp d3k| | . T − ≤ 3 2ω(k) (cid:12) (cid:16) (cid:17)(cid:12) (cid:18) Z (cid:19) Thus SˆT may be writte(cid:12)(cid:12)n symbolica(cid:12)(cid:12)lly as 1 ˆ S = dB W(B B ,t s)dB . T s t s t 2 · − − Z[−T,T]×[−T,T]\{s=t} 3 The infinite volume limit 3.1 Tightness and the Pauli-Fierz model The idea of the proof of the infinite volume limit we are about to give is not straightforward. We will show that it follows from showing that the bottom of the spectrum of a self-adjoint operator is eigenvalue. Actually, in the case of pair potential W under consideration, associated self-adjoint operator is realized as the Pauli-Fierz Hamilto- nian H in the non-relativistic quantum electrodynamics. Fortunately it is established that H has the unique ground state for not only confining external potential V, e.g., V(x) = x 2, but also the Coulomb V(x) = 1/ x , which is the most important case. | | − | | Let us begin with defining the Pauli-Fierz Hamiltonian with form factor ϕˆ as a self- adjoint operator on some Hilbert space H and we will review the functional integral representation of the C semigroup e−tH. 0 Let F := ∞ [ nL2(R3 1,2 )] be the Boson Fock space. The state space of n=0 s ×{ } one electron minimally coupled with the photon (bose) field is given by L N H := L2(R3) F. (3.1) ⊗ We denote the formal kernels of the annihilation operator and the creation operator on F by a(k,j) and a∗(k,j), respectively, which satisfy the canonical commutation relations: [a(k,j),a∗(k′,j′)] = δ(k k′)δ , [a(k,j),a(k′,j′)] = 0 = [a∗(k,j),a∗(k′,j′)]. (3.2) jj′ − The free Hamiltonian in F is defined by H := ω(k)a∗(k,j)a(k,j)d3k. (3.3) f j=1,2Z X Here dispersion relation ω is given by (2.1). Let us fix a function ϕˆ satisfying Assump- tion (A) The quantized radiation field A = (A ,A ,A ) with form factor ϕˆ is defined 1 2 3 by A := ⊕A (x)d3x, where we used the isomorphism H = ⊕L2(R3) dx and µ R3 µ ∼ R3 R 1 ϕˆ(k) ϕˆ( Rk) A (x) := e (k,j) e−ikx a∗(k,j)+eikx − a(k,j) d3k. (3.4) µ µ √2 √ω(k) ω(k) ! j=1,2Z X p