FREE ENERGY IN A MEAN FIELD OF BROWNIAN PARTICLES

We compute the limit of the free energy 1 NtN logE exp { 1 N ∑ 1≤j<k≤N ∫ tN 0 γ ( Bj(s)−Bk(s) ) ds } (N →∞) of the mean field generated by the independent Brownian particles {Bj(s)} interacting through the non-negative definite function γ(·). Our main theorem is relevant to the high moment asymptotics for the parabolic Anderson models with Gaussian noise that is white in time, white or colored in space. Our approach makes a novel connection to the celebrated Donsker-Varadhan’s large deviation principle for the i.i.d. random variables in infinite dimensional spaces. As an application of our main theorem, we provide a probabilistic treatment to the Hartree’s theory on the asymptotics for the ground state energy of bosonic quantum system.

(Communicated by Jaime San Martin) Abstract. We compute the limit of the free energy of the mean field generated by the independent Brownian particles {B j (s)} interacting through the non-negative definite function γ(·). Our main theorem is relevant to the high moment asymptotics for the parabolic Anderson models with Gaussian noise that is white in time, white or colored in space. Our approach makes a novel connection to the celebrated Donsker-Varadhan's large deviation principle for the i.i.d. random variables in infinite dimensional spaces.
As an application of our main theorem, we provide a probabilistic treatment to the Hartree's theory on the asymptotics for the ground state energy of bosonic quantum system.

1.
Introduction. Mean field theory considers the behavior of the stochastic system consisting of large number of small particles interacting to each other. In this paper, the independent d-dimensional Brownian motions B 1 (s) · · · , B N (s) represent the locations of these particles at the time s, and the function N −1 γ(x − y) measures the pairwise interactions among the Brownian particles. The long term behavior of the system is the result of the balance between two typical phenomena in the mean field regime: increasing number of the particles (i.e., N → ∞) and uniform negligibility of individual contribution (indicated by the multiple 1/N in the interaction function). The quantity stands for the integral potential of the system due to the interaction of the Brownian particles up to the time t N . In this work, t N → ∞ as N → ∞. Our goal is to study the asymptotics for the partition function In the case when d = 2, a slight different quantity corresponds to the N -body system with Schrödinger Hamiltonian with the parameter β ∈ (0, 1] that appears in the investigation of Bose-Einstein condensation ( [15]). Our work is also motivated by the recent investigations ( [7], [8] and [4]) of the spatial asymptotics max |x|≤R u(t, x) R → ∞ for the parabolic Anderson equation where V (t, x) is a Gaussian noise which is white in time, white or colored in space, i.e., Cov V (s, x), V (t, y) = δ 0 (s − t)γ(x − y) (s, x), (t, y) ∈ R + × R d .
In these works, the most substantial step is the investigation of the high moment asymptotics for Eu(t, x) N as N → ∞. Under proper positive homogeneity assumption on γ(·), the problem is relevant to the investigation proposed in this paper due to the moment representation (Theorem 5.3, [13] and Theorem 3.1, [6]) In this work we consider a more general phenomena beyond the setting of positive homogeneity. In addition, we shall work with a larger (than those considered in [4]) class of the covariance functions γ(x) which do not have to be, for instance, pointwise defined, non-negative or vanishing at infinity. Indeed, the issues such as the singularity of γ(x) arising from some practical needs pose new challenges. In addition, we point out that the approach given in [4] is no-longer working in the case when γ(·) switches signs. A practically interesting example is when where C H > 0 is a constant, which corresponds to the parabolic Anderson equation (4) with the Gaussian noise V (t, x) being white in time, fractional (with the Hurst parameter H) in space. Recently, it is pointed out ( [12]) that the parabolic Anderson equation (4) is solvable with the moment representation (5) as d = 1 and 1/4 < H < 1/2. On the other hand, the fact that 1 − 2H > 0 indicates that γ(x) is not defined at any point x ∈ R. Later we shall show that γ(·) is sign-switching in a suitable sense. This example provides a way to measure our capability in dealing with the issue of singularity and sign-alternativity.
To include the cases like (6), γ(x) is allowed to be generalized function defined as a linear functional γ: S(R d ) −→ R symbolically given as where S(R d ) is the Schwartz space of rapidly decreasing and infinitely smooth functions.
The quadratic form is defined as In connection to the parabolic Anderson model given in (4), the covariance function is non-negative definite. Throughout, we assume non-negative definite condition (7) on γ(·). According to Bochner representation, there is a positive and symmetric measure µ(dλ) on R d such that where is the Fourier transform of ϕ(·). Further, µ(dξ) is tempered in the sense that for some p > 0. Bochner representation can be written symbolically as Noticing that the difference of two independent Brownian motions is a constant multiple of a Brownian motion. To make sense of the exponential moment given in (1), it is required that the time integral t 0 γ B(s) ds be properly defined and exponentially integrable. When µ(dξ) is a finite measure, γ(x) is pointwise-defined, bounded and continuous. The above time integral is nothing more than an ordinary Riemann integral and the exponential integrability follows from the boundedness of γ(x).
The problem is highly non-trivial in the general setting. Given > 0, the function is non-negative definite with the finite spectral measure (see (10)) We assume that for every t > 0 One can see that the function γ(x) in (13) is replicable by γ(ax) for any constant a > 0 as In particular, the Brownian motion B(s) in (13) is replicable by B j (s) − B k (s). Finally, the exponential moment written in (1) is well-defined and finite under the assumption (13).
Theorem 1.1. Under non-negative definite condition (7) and the assumption (13) on γ(·), where which is well-defined and finite with We now comment on the condition (13) by considering the following three classes of γ(x) that are encompassed by (13).
The first class consists of the constant multiples of all characteristic functions γ(x) on R d that correspond to symmetric probability distributions µ(dξ). As mentioned, (13) becomes automatic when µ(dξ) is finite. In particular, Theorem 1.1 holds for every characteristic function γ(x) on R d . In view of the example we see that γ(x) is allowed to pick positive and negative values. In addition, one can make γ(x) periodic (in particular, γ(x) does not vanishing at ∞) by considering the distribution µ(dξ) supported on the lattice Z d . The second class consists of all non-negative and non-negative definite γ(x). In the case when γ(·) is not defined point-wise, "γ(·) ≥ 0" means γ (·) ≥ 0 for sufficiently small > 0. It is well-known ( [9]) that for any non-negative definite γ(·) ≥ 0, the assumption (13) is equivalent to the Dalang's condition This class includes some practically interesting cases such as γ(x) = δ 0 (x) for d = 1 where the time integral is The third class contains the non-negative definite functions γ(·) that have infinite spectral measure and are allowed to take negative values. Dalang's condition is no longer sufficient without assuming γ(·) ≥ 0. A good example is given in (6) where ξ(dξ) = |ξ| 1−2H dξ. It has been pointed out recently ( [12]) that (13) holds for 1/4 < H < 1/2. On the other hand, it is easy to see that the Dalang's condition (17) holds for any 0 < H < 1. Therefore, (17) alone is not sufficient if "H > 1/4" is necessary for (6).
To show the necessity of "H > 1/4", we start from an easy-to-check identity where τ is an independent exponential time with Eτ = 1. The integral on the right hand side is bounded from below by which is finite only when H > 1/4. On the other hand, by Brownian scaling Summarizing our argument, "H > 1/4" is necessary for (6).
Finding a condition for (13) that is "uniformly right" for the third class appears to be a hard problem beyond the scope of the current paper.
We now discuss the links to the high moment asymptotics of the parabolic Anderson equation and to the model of the Bose-Einstein condensation given in (2). By Brownian scaling, (14) can be rewritten as for any t > 0 In the special case when γ(·) satisfies the homogeneity γ(Cx) = |C| −α γ(x) (x ∈ R d and C ∈ R) for some 0 < α < 2, taking t N = N for the parabolic Anderson equation (4). In the special case of (6), α = 2 − 2H. By variable rescaling (6) with C H = 1. We therefore have Corollary 1. When 1/4 < H < 1/2 in the setting of (6), for any t > 0 Here we point out that (19) is achieved in [4] under the extra assumptions that α < d (The equality is allowed in the case d = 1 and γ(·) = δ 0 (·)) and γ(·) ≥ 0. These extra assumptions are not required by Theorem 1.1. Corollary 1 provides a concrete example of the improvement where γ(·) switches sign (as analyzed above) and α = 2 − 2H > 1 = d.
In case d = 2, substituting t N = N 2β into (18) for some β > 0, we obtain the following asymptotic law for the model of the Bose-Einstein condensation given in (2): The proof of Theorem 1.1 consists two steps: The first step is carried out in Section 2 where we prove Theorem 1.1 in the special case when the spectral measure µ(dξ) is finite. The main idea in this step is linearization and tangent approximation. A fascinating feature of our treatment is its relevance to the famous Donsker-Varadhan large deviation principle for the i.i.d. random variables with values in infinite dimensional spaces. The general setting is treated in step 2 that is given in Section 3, where γ(·) is approximated by γ (·) defined in (11). After completing this work mathematically, we became aware of the literature on bosonic quantum system, the very recent development [16] on Hartree's theory and their relevance to the main topic of our paper. We therefore add Section 4 to address this link.
2. When the measure µ(dξ) is finite. In this case, everything stated in Theorem 1.1 can be directly defined. In particular, the fact that γ(x) is uniformly bounded Theorem 1.1 can be restated as 2.1. Lower bound for (22). By integral substitution . Then H is a real Hilbert space. Here we point out that in order for H to be a real Hilbert space, the functions in H do not have to be real valued. What matters is that for any real number c 1 , c 2 and h 1 , h 2 ∈ H, c 1 h 1 + c 2 h 2 ∈ H and that the linear functional takes real values. All those hold thanks to the symmetry of µ(dξ).
Let f ∈ H be a fixed bounded function. By the fact that By Proposition 3.1 in [5], Thus, Notice that the relation h 2 ≥ − f 2 + 2 f, h becomes an equality when h = f . Hence, for any dense sub-space We call the identity approximation by tangent planes. Let H 0 be the space of the bounded functions in H. Taking supremum over f ∈ H 0 on the right hand side of (24), it becomes Summarizing our estimates, we obtain the lower bound for (22): 2.2. Upper bound for (22). Let t > 0 be fixed but large. For the sake of simplification we assume that t N /t always remains to be an integer. By Markov property, where the supremum is taken overb = We now claim that for anyb = (b 1 , · · · , b N ) ∈ (R d ) N , and integer n ≥ 1, Indeed, where C(j 1 , · · · , j n ) are deterministic complex numbers with norm 1, and α l ∈ R d are deterministic such that N j1,··· ,jn=1 Therefore, where the inequality follows from the fact that Thus, we have proved (28). From (28), and by Taylor expansion, for anyb = Then, in view of (27), we have Recall the following Donsker-Varadhan's large deviation principle (Theorem 5.3, [11]): Let E be a real separable Banach space with E * as its topological dual. Let X = {X k } k≥1 be a sequence of i.i.d. random variables taking values in E such that Then for any close set F ⊂ E, and for any open set G ⊂ E, where the rate function is given as the convex conjugate This result appears to be an infinite dimensional extension of Cramer's large deviation (Theorem 2.2.3, p.27, [10]). We refer an interested reader also to [1] for an elegant proof of the Donsker-Varadhan large deviation principle. ξ) a.e.. Also, note that H is a real Hilbert space. To apply Donsker-Varadhan large deviation principle to the space E = H, we define the i.i.d.
By the fact that X ≤ tµ(R d ), the condition of Donsker-Varadhan large deviation principle holds. Further, by Varadhan integral lemma (Theorem 4.3.1, [10]), Some delicate steps are needed in handling the variation on the right hand side. To this end we first claim that Λ * (h) = ∞ for any h ∈ H with h ∞ > 1, where Indeed, applying Hahn-Banach theorem there is a > 0 and f ∈ L( and f, h > 1 + . In addition, we may make f ∈ H. In particular, Hence, for any C > 0 Let An obvious modification of (25) leads to Hence, By duality lemma (Lemma 4.5.8, p.152, [10]) Therefore, it follows that Combining (29), (30) and (31), we obtain, Thus, all we need is to prove that lim sup Define τ t = inf{s ≥ 0; |B(s)| ≥ t 2 }. Then, Notice that |f (s, x)| ≤ µ(R d ). The second term on the right hand side is bounded by exp tµ(R d ) P max s≤1 |B(s)| ≥ t 3/2 which is negligible. As for the first term, first notice that wherep 1 (x) is the density of the measure ν(A) = P{B(1) ∈ A; τ t ≥ 1}. Let p 1 (x) be the density of B(1) and notice thatp 1 (x) ≤ p 1 (x) ≤ (2π) −d/2 . Hence, the right hand side is no greater than and the last step follows from Lemma 7.1 in [4]. Further, notice that where A d (t) = g(s, x); g(s, ·) ∈ F d for every 0 ≤ s ≤ t . Summarizing our computation, we obtain the bound By the fact that we therefore reach the bound (32).

XIA CHEN AND TUOC PHAN
3. When the measure µ(dξ) is infinite. The first thing we need to show is that E H is well defined and finite under our assumption. We specifically point out that the integrals have to be properly defined as g 2 (x) is not necessarily in S(R d ).
With the method used in proving (29), one can show that for any t 1 , t 2 > 0 Consequently, the limit exists and finite.
Recall that γ (x) is the non-negative definite function defined in (12). Given > > 0, notice that γ (·) − γ (·) is non-negative definite with the spectral measure For any given x ∈ R d , t > 0 and the integer n ≥ 1, similar to (28) By (13), this implies the moment convergence Further, the moment comparison similar to (28) also leads to for every x ∈ R d and θ > 0. By Theorem 4.1.6, p.99, [3], for any > 0 where p (x) is the density of B( ). Consequently, Noticing that for any g ∈ F d Consequently, for any θ > 0 and > 0 where the third step follows from translation invariance. Thus, by the relation γ (x − y)g 2 (x)g 2 (y)dxdy and monotonic convergence, the integral is well-defined and finite for every g ∈ F d . Further, In particular, E H is well-defined and finite.

XIA CHEN AND TUOC PHAN
For any integer n ≥ 1 By the fact that Therefore, it follows from Taylor expansion that Letting → 0 + on the right hand side leads to On the other hand, set ζ (x) = γ(x) − γ (x). We claim that for any θ > 0 By Jensen's inequality where the last step follows from (39) with n = 1. Thus, all we need is to show the upper bound of (41). Write By Hölder's inequality, For any n ≥ 1, by independence between B 1 and {B 2 , · · · , B N }

XIA CHEN AND TUOC PHAN
where the inequality follows from the fact that Hence, by Taylor expansion where the last step follows from the independence of the Brownian motions.
To prove (41), we need only to show that To simplify our notation, we may assume t N goes to infinity along the integers. By Markov property, On the other hand, letting → 0 + in (35), for every x ∈ R d . Therefore, Thus, (42) follows from the obvious fact that which is clearly a consequence of the assumption (13). We now prove the upper bound for (14). Let p, q > 1 are conjugate numbers. By Hölder inequality By (38) (with γ (·) being replaced by pγ (·)), (41) (with θ = q) and the fact that Letting p → 1 + on the right hand side leads to the upper bound Finally, Theorem 1.1 with its full generality follows from (40) and (43).
4. Link to bosonic quantum system. After completion of this project, we became aware of the subject on bosonic quantum system or more specifically, a very recent development [16] on Hartree's theory. Limited to our setting and in our notation, Hartree's theory supports the statements such that where E H is given in (15) and is known as Hartree energy in the literature of quantum mechanics, and (x = (x 1 , · · · , x N )) is called the ground state energy which appears as the principal eigenvalue of the N -body problem ( [14]) with the Shrödinger Hamiltonian Here F N d is given in (16) with d being replaced by N d. In quantum mechanics, H N formulates N -body problem (see, e.g, [14]). The statement (44) claims that the ground state of the non-linear Shrödinger operator (also called Hartree operator) H = ∆ + (γ * g 2 )g is approximated by the ground states of linear Shrödinger operators given in (45). The reason behind (44) is simple: Replacing F N d by the sub-class of the functions of the formg This leads to the lower bound for (44): The above analysis shows that (44) rests on the fact that the maximizer for the variation in (45) is approximated in a suitable sense by the functions in the form given in (47) as N → ∞.
Most of the earlier results (to which we refer the references cited in [16]) were for the ground state energy where In the case when γ(·) ≡ 0, it is easy to see that the functions given in (47) maximize the variation E V H (N ), which partially explains why we have (50). On the other hand, (50) does not give a clear picture on the role played by the function γ(·) in this "factorization" dynamics. One might take V ≡ 0 for observing the behavior of γ(·). In the case when lim |x|→∞ γ(x) = 0, E 0 H = 0 and E 0 (N ) = 0 for all N ≥ 1, which is not necessarily the consequence of factorization, as one can make E 0 (N ) = 0 by choosing "flat" functions g(x). Therefore, it makes sense even only for the sake of understanding how the pair interaction function γ(·) response to factorization, to switch the sign of γ(·) in the variation E 0 (N ) so that it becomes the problem given in (45) where the two terms in the variation compete against each other and Hartree's theory takes the form given in (44). On the other hand, note that, as remarked by Lewin, Nam and Rougerie in [16, p. 579], "The validity of Hartree's theory in this simple case 1 is already a nontrivial problem and does not seem to have been proven before". It appears that the paper [16] is the first work where Hartree's theory mathematically includes the form given (44).
As an application of Theorem 1.1, we provide a probabilistic treatment to Hartree's theory. It should be pointed out that results obtained by Lewin, Nam and Rougerie [16] takes a form more general than (44). As for the assumptions, both the condition (13) and the condition posted in [16] are sufficient but "nearly necessary" for which ensures the finiteness of E(N ). Our goal here is not to establish the most general form of Hartree's theory but to show a probabilistic relevance to Hartree's theory.
Proof of Theorem 4.1. In view of (48), all we need is to establish the upper bound lim sup The idea is Feynman-Kac formula. To this end we work on the (N d)-dimensional Brownian motion B(s) = B 1 (s), · · · , B N (s) s ≥ 0.
We introduce the notationx = (x 1 , · · · , x N ) for x 1 , · · · , x N ∈ R d . We use "Ex" for the expectation associated with the Brownian motion B(s) with B(0) =x. For fixed N , the transform T t g(x) = Ex exp 1 N 1≤j<k≤N t 0 γ B j (s) − B k (s) ds g B(t) ,x ∈ R N d defines a semi-group of continuous linear operators on L 2 (R N d ) in the sense that T s+t = T t • T s . Further, the semi-group {T t ; t ≥ 0} takes the Schrödinger operator H N (given in (46)) as its infinitesimal generator and is formally written as T t = e tH N . Here we mention the fact that H N is initially a symmetric linear operator and can be extended into a self-adjoined operator by Friedrichs's extension. Let > 0 be fixed and notice the fact that E(N ) = sup g∈F d g, H N g . For any N ≥ 1, one can find g N ∈ F N d such that g N is locally supported, and g N , H N g N > E(N ) − N . Let R N > 0 be the radius of the (N d)-dimensional ball which supports g N and let t N → ∞ (N → ∞) with sufficient increasing rate so that lim N →∞ where ω N d is the volume of the (N d)-dimensional unit ball.
By the moment comparison similar to (28), one can show that for anyx ∈ R N d Ex exp Consequently, By the spectral representation where the measure µ g N (dλ) satisfies µ g N (R) = g N 2 2 = 1. In other words, µ g N is a probability measure on R. By Jensen's inequality Combining this with (52) and (53), In view of Theorem 1.1, letting → 0 + on the right hand side leads to (51).