Global fluctuations for 1D log-gas dynamics. (2) Covariance kernel and support

We consider the hydrodynamic limit in the macroscopic regime of the coupled system of stochastic differential equations, $ d\lambda_t^i=\frac{1}{\sqrt{N}} dW_t^i - V'(\lambda_t^i) dt+ \frac{\beta}{2N} \sum_{j\not=i} \frac{dt}{\lambda^i_t-\lambda^j_t}, \qquad i=1,\ldots,N, $ with $\beta>1$, sometimes called generalized Dyson's Brownian motion, describing the dissipative dynamics of a log-gas of $N$ equal charges with equilibrium measure corresponding to a $\beta$-ensemble, with sufficiently regular convex potential $V$. The limit $N\to\infty$ is known to satisfy a mean-field Mc Kean-Vlasov equation. Fluctuations around this limit have been shown by the author to define a Gaussian process solving some explicit martingale problem written in terms of a generalized transport equation. We prove a series of results concerning either the Mc Kean-Vlasov equation for the density $\rho_t$, notably regularity results and time-evolution of the support, or the associated hydrodynamic fluctuation process, whose space-time covariance kernel we compute explicitly.

with β > 1, sometimes called generalized Dyson's Brownian motion, describing the dissipative dynamics of a log-gas of N equal charges with equilibrium measure corresponding to a βensemble, with sufficiently regular convex potential V . The limit N → ∞ is known to satisfy a mean-field Mc Kean-Vlasov equation. Fluctuations around this limit have been shown [38] to define a Gaussian process solving some explicit martingale problem written in terms of a generalized transport equation.
1 Introduction and statement of main results

Introduction
Let β ≥ 1 be a fixed parameter, and N ≥ 1 an integer. We consider the following system of coupled stochastic differential equations driven by N independent standard Brownian motions (W 1 t , . . . , W N t ) t≥0 , 3) is that of a β-log gas with confining potential V .
Let us start with a historical overview of the subject as a motivation for our study. This system of equations was originally considered in a particular case by Dyson [10] who wanted to describe the Markov evolution of a Hermitian matrix M t with i.i.d. increments dG t taken from the Gaussian unitary ensemble (GUE). In Dyson's idea, this matrix-valued process was to be a matrix analogue of Brownian motion. The latter time-evolution being invariant through conjugation by unitary matrices, we may project it onto a time-evolution of the set of eigenvalues {λ 1 t , . . . , λ N t } of the matrix, and obtain (1.1) with β = 2 and V ≡ 0. Keeping β = 2, it is easy to prove that (1.1) is equivalent to a generalized matrix Markov evolution, dM t = dG t − V ′ (M t )dt. The Gibbs measure can then be proved to be an equilibrium measure. Such measures, together with their projection onto the eigenvalue set, µ N eq ({λ 1 , . . . , λ N }), are the main object of random matrix theory, see e.g. [26], [2], [30]. The equilibrium eigenvalue distribution can be studied by various means, in particular using orthogonal polynomials with respect to the weight e −N V (λ) . The scaling in N (called macroscopic scaling in random matrix theory) ensures the convergence of the random point measure X N := 1 N N i=1 δ λ i to a deterministic measure µ V with compact support and density ρ when N → ∞ (see e.g. [19], Theorem 2.1). One finds e.g. the well-known semi-circle law, ρ(x) = 1 π √ 2 − x 2 , when V (x) = x 2 /2. Looking more closely at the limit of the point measure, one finds for arbitrary polynomial V (Johansson [19]) Gaussian fluctuations of order O(1/N ), contrasting with the O(1/ √ N) scaling of fluctuations for the means of N independent random variables, typical of the central limit theorem. Assuming that the support of the measure is connected (this essential "one-cut" condition holding in particular for V convex), Johansson proves that the covariance of the limiting law depends on V only through the support of the measure -it is thus universal up to a scaling coefficient -, while the means is equal to ρ, plus an apparently non-universal correction in O(1/N ).
Following Rogers and Shi [32], Li, Li and Xie [23] proved the following two facts: (i) two arbitrary eigenvalues never collide, which implies the non-explosion of (1.1); (ii) the random point process X N t := 1 N N i=1 δ λ i t satisfies in the limit N → ∞ a deterministic hydrodynamic equation of Mc Kean Vlasov type, namely, the asymptotic density ρ t ≡ X t := w−lim N →∞ X N t (1.4) satisfies the PDE in a weak (i.e. distribution) sense, where p.v. dy x−y ρ t (y) is a principal value integral.
The equilibrium measure ρ eq , defined as the solution of the integral equation (traditionally called: cut equation) β 2 p.v. dy x − y ρ eq (y) = V ′ (x), (1.6) cancels the right-hand side of (1.5), as is readily checked.
A complex Burgers-like PDE for the Stieltjes transform of X t U t (z) : is easily derived [32,18] from (1.5), In our recent article [38], in large part based on a previous paper by Israelsson [18] which dealt with the specific example of a harmonic potential, we introduced a process Y = (Y t ) t≥0 interpreted as asymptotic fluctuation process. Let Y N t := N (X N t − X t ) be the rescaled fluctuation process for finite N . Then it was proved that Y N t law → Y t when N → ∞, where (Y t ) t≥0 is the solution of a martingale problem, as can be briefly seen as follows. First, Itô's formula implies that if the test functions (f t ) 0≤t≤T , f t : R → R solve the following linear PDE Substituting formally to X N its deterministic limit X in the r.-h.s. of (1.11), one gets an equation which is the asymptotic limit of (1.11) in the limit N → ∞, namely, The main task in [38] consists in proving that eq. (1.11, 1.12) is akin to a transport equation on the cut complex plane C \ R. In the harmonic case (i.e. when V is quadratic), then the solution of, say, (1.12) Thus the solution of (1.12) may be represented formally as along the above characteristics, or equivalently, by solving the associated transport equation generated by the time-dependent operator Considering instead some arbitrary terminal condition and potential V , a similar formula holds, where the time-evolution is given up to a bounded perturbation by a transport operator whose characteristics are as (1.13) plus some extra term depending on V ′′′ . Then (at least formally), Itô's formula (see [18], p. 29) makes it possible to find the Markov kernel in the limit N → ∞. Namely, if f t be the solution of (1.12) with terminal condition f T , and Eq. (1.17) was proved for general potentials in our previous article [38]. Now, letting . . , n vary in dense subspace of L 1 (R), this martingale problem is solved in Bender [3] in the case of a harmonic potential using an explicit computation of the characteristics (1.13). Such is the present state of the art.

Main results
We prove in this article two types of results. It is safe to assume that V is polynomial and strictly convex, though the reader will also find weaker sets of hypotheses, depending on the paragraph.
(A) The first series of results regards the Mc Kean-Vlasov equation (1.5). Little is known about it in general; the arguments in Li-Li-Xie [23] (see in particular Theorem 1.3) simply prove that it admits a unique solution in C([0, T ], P(R)), which is constructed as weak limit of the sequence of stochastic processes t → Y N (t). Unicity is proved using decrease of Wasserstein distance between two arbitrary solutions. A classical large-deviation argument (reviewed here) implies under our hypotheses a bound on the support of the measure ρ t ; in particular, ρ t is compactly supported.
Our first result is a regularity result: assuming that the analytic function z → U 0 (z), z ∈ Π + := {Im z > 0} extends to a continuous function on the closure Π + ∪ R of the upper half-plane, we prove that the same property holds for U t , t ≥ 0; see Theorem 2.1. Hence in particular (by Plemelj's formula), the density ρ t (·) = 1 2iπ (U t (· + i0) − U t (· − i0)) is a continuous function for every t ≥ 0.
Our second result concerns the support. We explain how to obtain the "external support" .) The external support is characterized, see eq. (2.26) and (2.27), in terms of characteristics of the generalized complex Burgers equation (1.8) -not surprisingly closely related to (1.13) -which are half-explicit in general and can be obtained in closed form in various cases, including for equilibrium dynamics or when V is harmonic. On the other hand, we do not prove any formula for the support itself. In particular, though under our hypotheses (more specifically, because V is convex) the support of the equilibrium density is a connected interval, we cannot exclude, even if supp(ρ 0 ) is connected, that e.g. supp The second series of results regards the fluctuation process (Y t ) t≥0 . While the above characteristic equations can be solved explicitly only when V is harmonic (see Bender [3]), yielding the covariance of the Stieltjes transform (SY t )(z) := Y t , 1 ·−z of the fluctuation process, their "trace" on the boundary of the upper (or lower) half-planes can be solved for arbitrary V . Then the covariance kernel Cov Our most general result in this direction is Theorem 3.1. A more explicit formula relying on Theorem 3.1 is Theorem 3.2 or Corollary 3.2 for equilibrium dynamics, see (3.70) for the specific case of a quartic (Landau-Ginzburg type) potential. The reader should compare the above results to those obtained by M. Duits [9] in a stochastic setting for fluctuations of noncolliding processes, and by N. Allegra, P. Calabrese, J. Dubail, J.-M. Stéphan and J. Viti [1], [6] in a condensed-matter context for the (real-time) propagator of the density field ρ(t 1 , x 1 )ρ(t 2 , x 2 ) ≡ (ψ † ψ)(t 1 , x 1 )(ψ † ψ)(t 2 , x 2 ) of a one-dimensional Fermi gas submitted to a confining potential V . Despite the difference of language, and the fact that an analytic continuation in time is necessary to go from one situation to the other, both series of works come to a similar conclusion. Focusing on the quantum setting, and considering the lowlying spectrum of the underlying N -particle quantum Hamiltonian, the authors predict (and confirm by some numerical simulations) that (assuming the theory to be free, i.e. Gaussian at large scale) the time-evolution equation obtained for the Wigner function in the semiclassical limit is essentially correct in the large N limit. The time-evolution equation for the chiral part of the two-point function is then the same as ours (compare e.g. our equation (3.59) to eq. (6) in [6]), taking as input the equilibrium density ρ eq computed by local-density approximation, see e.g. discussion in section A. of [5] or articles cited above. Then, in both situations, the fluctuation/density field is interpreted as a 2d Gaussian free field in a curved space with metric tensor ds 2 = e 2σ dz dz, with coordinate transform z = z(x, y) and conformal weight σ = σ(x, y) chosen by requiring that e σ(x,y) dz = dx + iπρ eq (x)dy, which yields ( [6], eq. (20)): z(x, y) = 1 π (G(x) + iπy), where G(x) := dx ρeq(x) , in exact correspondence with our Theorem 3.2. Therefore its law may be obtained from that of flat 2d Gaussian free field through a conformal transformation. The connection of our results to those is however lost at that point, since the single-time covariance kernel Cov(Y t (x 1 ), Y t (x 2 )) is (up to a simple scaling) independent of the potential, hence of ρ eq . It would be interesting to obtain a deeper understanding of this difference.

The Mc Kean-Vlasov equation
We study in this section eq. (1.5) indirectly through the time-evolution of its Stieltjes transform As shown in [32], [18], U t satisfies following generalized complex Burgers equation, But in general, T t is an unknown time-dependent quantity for which an independent equation should be provided. For V polynomial, however, say, deg(V ) =: 2n, T t (z) is easily seen [19] to be some explicit polynomial in z of order ≤ 2n − 2, with coefficients in the linear span of the 2n − 2 first moments of the unknown density ρ t , namely, . Looking at the asymptotic expansion of U t at infinity, T t (z) may also be defined (up to an additive constant) as minus the part polynomial in z of V ′ (z)U t (z), so that ∂Ut(z) when z → ∞, in coherence with the leading term of the expansion, −U t (z) ∼ z→∞ 1/z. Projecting (2.2) onto the linear subspace ⊕ k≥0 Cz −k−1 yields an infinite system of coupled ODEs for the moments x k ρ t (x) dx k≥0 , which in principle can be solved numerically on short time-intervals.
We make in this section the following Assumptions.

An example: scaling solution in the Hermite case
In this paragraph, we assume that β = 2 and V (x) = x 2 2 , and look for simple solution of (2.2) other than the constant solution ρ eq . By reference to the underlying equilibrium unitary ensemble, we call this case the Hermite case.
Explicit formulas. The equilibrium density corresponds to the semi-circle law, ρ eq (x) ≡ and Stieltjes transform U eq (z) ≡ −z + √ z 2 − 2 continuously extending to the real line, and U eq (z) = U eq (z), hence (by Schwarz's extension lemma) U eq extends to a holomorphic function (still called U eq ) on the cut plane C \ [− √ 2, √ 2]. Note that U ′ eq is singular in the neighbourhood of the ends of the support, ± √ 2; namely, U ′ eq (±( Scaling solution. Assume that ρ 0 (x) := 1 s ρ eq (x/s) (s > 0), or equivalently, U 0 (z) := 1 s U eq (z/s). Then we use the following Ansatz, for some unknown scaling function t → s(t), corresponding to a time-dependent support Hence our Ansatz is correct provided we choose s(t) to be the solution of the odeṡ = 1 s − s, namely, Equivalently, s 2 (t)−1 s 2 (0)−1 = e −2t , which means that the "radius" b t := √ 2 s(t) converges exponentially fast and monotonously to its equilibrium value, √ 2.

Regularity
As proved in our previous article [38] -extending uniform-in-time moment bounds proved in [2] in the harmonic case -, there exists R = R(T ) and c, C > 0 such that, for all N ≥ 1, Using Borel-Cantelli's lemma, one immediately deduces the following: for any test function f : for every t ≤ T . In particular, for every n = 0, 1, . . ., the function t → x n ρ t (x) dx (0 ≤ t ≤ T ) is bounded and continuous; which implies in turn that t → T ′ t (z) is a polynomial in z depending continuously on t.
Our main result in this subsection is Theorem 2.1. Under the Assumptions of section 2, U t Π + extends to a contiuous function onΠ + for every t ≥ 0. In particular, x → ρ t (x) is a continuous function for every t ≥ 0.

A. (Case of a harmonic potential).
Then d dz T t (z) ≡ 0 and so (2.2) is a closed equation for U t which can be solved on C \ R, where it is analytic, using the method of characteristics. We shall use this to derive the evolution of the support.
Characteristics. For definiteness we choose V (x) = x 2 2 . Let Z t (z 0 ) be the solution at time t ≥ 0 of the following differential equation, decreases and the characteristics may eventually cross the real axis, after which the characteristic method makes no sense because of the discontinuity. So we decide to kill characteristics as sooon as they cross the real axis. Let t max (z 0 ) := inf{t > 0 | Z t (z 0 ) ∈ R} ∈ (0, +∞]; for every T < t max (z 0 ), there exists a neighbourhood B(z 0 ) of z 0 in Π + that is mapped inside Π + . Hence characteristics (2.8) started from B(z 0 ) are well-defined up to time T , and define for every t ≤ T a one-to-one mapping into a time-dependent region Solving instead backwards in time, one gets Since Im U t (z) ≥ 0, it is apparent from (2.12) that φ t : Π + → Π + , with Im φ t (z) ≥ Im z; this can be deduced, even without knowing the explicit formula (2.12), from (2.8), since − dz dt ∈ Π + as long as z(t) ∈ Π + . Let Thus (see Rudin [33], Theorem 14.19) the map φ t extends to a homeomorphismΠ + →Π t , while the boundary ∂Π t is a Jordan curve. Hence U t : z → e t U 0 (φ t (z)) extends to a continuous function onΠ + .
This makes it plain enough that (somewhat counter-intuitively) characteristics do not follow the time-evolution of the support or the singularities of U t on the real axis (see next subsection for more).

B. General case
The general case is similar, except that the time evolution of the (2n − 2) first moments of the density must be determined independently. Namely, instead of (2.8), we consider the generalized characteristics Z t (z 0 ), solution of the o.d.e.
, solved as Differentiating (2.14) yieldsz with initial condition Solving for T t by some independent means (e.g. numerically), (2.18) can be solved numerically for short time knowing U 0 (and even by quadrature when T t is constant, e.g. for equilibrium dynamics). However (due to the multi-valuedness of the square-root function on C), eq. (2.18) stops making sense in general when the function inside the square-root vanishes. On the other hand, an unambiguous definition may be given in terms of the second-order differential equation (2.16), in its matrix form .

(2.19)
Writing V ′ (z) ∼ z→∞ c n z 2n−1 + . . ., we get for 0 < b < 1: Im V ′ (a + ib) ∼ a→∞ (2n − 1)c n a 2n−2 b, whence there exists a max ≥ 0 such that: On the other hand, since V is strictly convex, there exists b max ∈ (0, 1) such that Thus (see (2.14)) − dz dt ∈ Π + as in the harmonic case, providing one restricts to the strip Im z ∈ (0, b max ). The rest of the argument proceeds as in the previous subsection if one restricts to characteristics , extends to a continuous function onΠ + , proving Theorem 2.1 in whole generality.

Support
In this paragraph we study the time evolution of the external support Using the characteristics introduced in the previous subsection, we shall be able to give a defining formula for a t , b t (t ≥ 0).
Exactly as in the example developed in §2.1, and for the same reasons, the function U 0 has a maximal analytic extension to the cut plane C\[a 0 , b 0 ], which is real-valued and real-analytic on R \ [a 0 , b 0 ]. Thus the characteristics t → Z t (x 0 ) issued from x 0 > b 0 , as defined by (2.14), is well-defined and real-valued for t small enough. As long as the characteristics (z s ) 0≤s≤t , , the sign is unambiguously a minus sign, z ≈ −V ′ (z), and characteristics may not cross: for t ≤ T fixed and b max > b 0 large enough, is an increasing, real-analytic diffeomorphism on its image. On the other hand, taking the derivative of (2.19) with respect to the initial [ · ] 1 =1st component, a complicated formula from which no general rule to guess the possible vanishing of Z ′ t (x 0 ) can be expected. Let us illustrate this on the simple Hermite case where β = 2 and V (x) = x 2 2 , and characteristics are explicit (see A. of last subsection). When −∞ for all t > 0, which does happen e.g. when U 0 (z) = 1 s U eq (z/s) is a rescaling of the equilibrium solution U eq .
Eq. (2.23) excludes the possibility that the latter quantity vanish, so actually d Conversely, if (by absurd) U t were analytic at b * 0 (t), then (2.14) would imply that We now claim that the function t → b t is càdlàg, i.e. right-continuous with left limits. Furthermore, it doesn't have any positive jumps, i.e. b t ≤ lim t ′ →t,t ′ <t b t ′ . (On the other hand, we cannot exclude negative jumps, with ρ t ′ [bt,b t − ] → t ′ →t,t ′ <t 0 pointwise). Namely, is a sequence such that t n → t, b tn → lim inf t ′ →t b t ′ and b tn < b, which is incompatible with the fact that the measure ρ s (x)dx depends continuously on s; (ii) lim sup t ′ →t,t ′ >t b t ′ ≤ b t , as follows from the characteristic method developed above; Then there exist characteristics moving by an amount b ′ − b in arbitrary small time, which is contradictory with previous arguments.
Let us illustrate this with the example of the scaling solution of §2.1. We find from (2.10) (2.29) Easy but tedious computations yield as expected.

Kernel of the fluctuation process
We give in this section formulas for the distribution-valued covariance kernel of the asymptotic fluctuation process (Y t ) t≥0 . The proof is indirect. First we obtain an evolution equation for the Stieltjes transformed covariance kernels which are the boundary values of the kernel Λ : All these formulas are to be understood in a distribution sense.  (see Introduction) is an explicit transport equation, which is the key to the PDE we obtain for the kernel g ±,± ; see Theorem 3.1. This PDE can be solved in terms of the characteristics (see (3.52)). In the stationary case one gets a more explicit formula (see Theorem 3.2 and Corollary 3.8).
We end this section with the interesting case of a quartic potential, V (x) = 1 4 t 4 + c 2 t 2 + d (c > 0), for which computations can be made totally explicit (see eq. (3.70)).

General framework
We collect here those notations and results proved in our previous article [38] which are necessary for the present study.

Assumptions
Our Assumptions in this section are of three different types.
Assumptions on the potential.
We assume that V is convex and C 11 .
The convexity assumption on V is essential for the convergence of the finite N -density to the solution ρ t of the Mc Kean-Vlasov equation, see [23], and for Johansson's universal formula for equilibrium fluctuations to apply [19], see §3.4 below. The extra regularity assumptions on V have been used in [38] for semi-group estimates and in some perturbation arguments. Later on (see end of §3.3, and §3.4), we shall further assume that V extends analytically to an entire function V : C → C in order to get more explicit formulas.
Assumptions on the initial measure.
Let µ N 0 = µ 0 ({λ i 0 } i ) be the initial measure of the stochastic process {λ i t } t≥0,i=1,...,N , and be the initial empirical measure. Since N varies, we find it useful here to add an extra upper index (λ N,i 0 ) i=1,...,N to denote the initial condition of the process for a given value of N . We assume that: (i) (large deviation estimate for the initial support) there exist some constants C 0 , c 0 , R 0 > 0 such that, for every N ≥ 1, x−z is the Stieltjes transform of ρ 0 .
As proved in [38], the initial large deviation estimate (i) implies a uniform-in-time large deviation estimate for the support of the random point measure: Proposition 3.1. (see [38], Lemma 5.1) Assume (i) holds for some constants R 0 , c 0 , C 0 > 0. Let T > 0. There exists some radius R = R(T ) and constant c, depending on V and R 0 , c 0 but uniform in N , such that Finally, as in section 2, we add a Regularity assumption on the initial density.
We assume that the Stieltjes transform U 0 Π + of the initial density ρ 0 on the upper-half plane extends to a continuous functionΠ + → C.
Though this Assumption is probably unnecessary, it is natural, holds true in all examples treated below, and allows stating convergence results in a stronger sense.

Summary of results
All results presented here come from our previous article [38]. The measure-valued process has been shown in [38] to converge in C([0, T ], H −14 ): Proposition 3.3 (Gaussianity of limit fluctuation process). (see [38], Main Theorem) Let Y N t be the finite N fluctuation process (3.8). Then: where (f s ) 0≤s≤T is the solution of the asymptotic equation (1.12).
The main point of the proof has been to rewrite the evolution equation for (f t ) 0≤t≤T in terms of a "quasi"-transport operator on functions on the upper half-plane. Let us briefly recapitulate how this is done. (ii) Let, for z ∈ C \ R, and be the Stieltjes transform of X N t , resp. X t .
Definition 3.6. Let, for p ∈ [1, +∞] and b max > 0, (3.13) The value of b max is unessential, so we fix some constant b max > 0 (e.g. b max = 1) and omit the b max -dependence in the estimates. (3.14) Thanks to the symmetry condition, h(z) = h(z), (3.14) may be rewritten in the form from which it is apparent that f is indeed real-valued.
Various Stieltjes decompositions, following Israelsson [18], have been constructed in [38]. The simplest one consists in defining h : . When κ is even, it is proved (see [38], (2.13)) that where |||K κ bmax ||| L 1 (R)→L 1 (R) , |||K κ bmax ||| L ∞ (R)→L ∞ (R) = O(1). (We shall only need to consider κ = 0 in the present article). Since in the sequel we want to focus on narrow strips around the real axis, one might think of taking the limit b max → 0. However, this introduces awkward boundary terms. Instead we fix b max > 0 and define h : (a, b) → e −b/ε K κ bmax,ε (ε > 0), where K κ bmax,ε is the Fourier multiplication operator by K κ bmax,ε (s) : . Similarly to (3.16), we get (specifically for κ = 0) . The Fourier multiplication operator in the r.h.s. of (3.17) is not a differential operator any more: where H is the Hilbert transform (see Appendix). Note that the most singular term in O(ε −2 ) is simply a constant.
In [38], we wrote down an explicit time-dependent operator H(t) such that the right-hand side of (1.12) for f t decomposed as Note that, since Stieltjes decompositions are not unique, the operator H(t) is very underdetermined. The essential features of the operator H(t) chosen in [38] are recapitulated in Appendix, see section 4; in particular, for κ ≥ 0, H(t) is the generator of a timeinhomogeneous semi-group of L p , p ≥ 1, which is a bounded perturbation of a transport operator. Moving around the operator H(t) to the function f z (x), one obtains an operator L(t) which is a "twisted adjoint" of H(t), [38], eq. (3.18) for details). For κ = −1 (at least formally), L(t) = H † (t), and h t may be directly interpreted as a density in C \ R exactly as in (1.14), so that L(t) is the direct generalization of (1.15) to an arbitrary potential.

The stationary covariance kernel in the Hermite case
We rewrite in appropriate coordinates the formulas for the covariance found by Israelsson in the stationary regime, in the Hermite case, i.e. when V is harmonic (V (x) = x 2 /2) and β = 2.
It is very instructive to compute the short-distance asymptotics in a scaled limit, ∆t = εδt 12 → 0, x 1 − x 2 = εδx 12 → 0. Formula (3.25) implies in the angular coordinates independently of θ, from which Note that only the first term in the r.-h.s. of (3.25) contributes to (3.35).

PDE for the covariance kernel: the general case
We shall now derive a PDE for g +,± 1,2 in whole generality. (A PDE for g −,± 1,2 is then obtained by complex-conjugating the first space coordinate.) Theorem 3.1 (hydrodynamic fluctuation equation for general V ). The kernel g +,± 1,2 (t 1 , x 1 ; t 2 , x 2 ) satisfies the following PDE in a weak sense, that is, for any smooth, compactly supported test function ψ = ψ(x 1 ), Remark. The product U t 1 (x 1 + i0)g ±,± 1,2 (t 1 , x 1 ; ·) makes sense as a distribution because both x 1 → U t 1 (x 1 + i0) and x 1 → g ±,± 1,2 (t 1 , x 1 ; ·) are obtained by convolution with respect to the function x → 1 x+i0 , hence have Fourier support ⊂ supp F(x → 1 x+i0 ) = R + . Proof. A short but non rigorous proof goes as follows. Fix κ = −1. (4.2)), we consider the limit when N → ∞ and b 1 → 0 + of the characteristic equations associated to L transport := H † transport , see (3.21) and below, thus obtaining directly the solution of the evolution equation with terminal condition f t 1 = f z 1 , where z 1 ≡ a t 1 + ib t 1 . One finds (3.38) Explicit formulas (4.4,4.5,4.6) for v hor , v vert , τ −1 yield (as follows from easy explicit computation, of from [38], eq. (3.15),(3.41), (3.45) and (3.48), where one has set b ≡ 0 + ) Consider these to be the characteristics of a generalized transport operator L hol acting on a function f z 1 analytic on Π + , so that ∂ z 1 ≡ ∂ z 1 + ∂z 1 ≡ ∂ x 1 : then which is exactly the operator featuring in the r.-h.s. of (3.36), acting on the x 1 -variable. This makes it possible to keep b t ≡ 0 + during the time evolution.
where f s is the solution at time s ≤ t of (1.12) with terminal condition f t ≡ f z . Differentiating w.r. to t and Taylor expanding to order 1 in λ yields , where L ± = L + if z ∈ Π + , resp. L − if z ∈ Π − , a generalized transport equation on C \ R with the same characteristics as above.
How explicit can these formulas be made ? One may of course try to answer this question through case-by-case inspection. Let us point out at two specific but sufficiently general cases. The first one is the harmonic case, i.e. V (x) = x 2 2 , treated in an exhaustive way by Bender [3] (see in particular Theorem 2.3) for an arbitrary parameter β > 1 and an arbitrary initial condition. Though the mapping Φ t 1 t is explicit (see (2.10)), the inverse mapping, (Φ t 1 t ) −1 , of course, is not in general. It requires some skill to provide explicit formulas not relying on the use of (Φ t 1 t ) −1 , see e.g. the beautiful result using Schwartzian derivatives ( [3], Theorem 2.7) for Cov( Y t 1 , F 1 , Y t 2 , F 2 ) when F 1 , F 2 are bounded analytic functions on a neighbourhood of the real axis. The second one is the stationary case, where β and V are general but ρ t = ρ eq is assumed to be the equilibrium measure. This is the subject of the next subsection.

Solution of the PDE in the stationary case
We restrict to the stationary case in this subsection, and assume as stated before that V extends analytically to an entire function V : C → C. Let us first state two essential facts. First, the universality (up to simple scaling and translation) of Johansson's formula for equilibrium fluctuations implies, assuming that supp(ρ eq ) = [−A, A] (A > 0): where Λ (3.24) is as in (3.24), and A cos(θ j ) = x j , A sin(θ j ) = A 2 − x 2 j , j = 1, 2 is up to scaling the change of variables used in the Hermite case. Second, using (1.6) and (5.3), (5.4), which may also be interpreted as saying that ρ eq extends analytically to Π ± as 1 ±iπ (U eq (z) + 2 β V ′ (z)). Therefore, Theorem 3.1 may be restated in this simple form, which generalizes (3.31). Solving (3.58) for short time and x 1 → x 2 , with initial condition (t 1 = t 2 ) given by Johansson's equilibrium formula, one finds the same short-distance asymptotics as in (3.35), namely, (3.59) See discussion in the Introduction.

Appendix. Generator and semi-group estimates
A large part of the work in our previous article [38] has been to write down explicitly a time-dependent operator H κ (t) (called: generator) such that, assuming f T = C κ (h T ), the function f t = C κ (h t ) with h t solution for t ≤ T of the backwards evolution equation is solution of (1.12).
The transport operator H κ transport (t) can be exponentiated backward in time for κ ≥ 0 as results from the sign of Re τ κ : namely,

Appendix. Stieltjes and Hilbert transforms
We collect in this section some definitions and elementary properties concerning Stieltjes and Hilbert transforms, in the periodic and in the non-periodic cases.
2π 0 dt f (t) cot(θ − t), (5.11) making it plain that the periodic Hilbert transformation is a rational generalization of the Hilbert transform (5.3) on the real line.

On Ornstein-Uhlenbeck processes
An Ornstein-Uhlenbeck process is a (Hilbert-space valued) stochastic process Y (t) satisfying a linear stochastic differential equation of the forṁ where η is delta-correlated white noise, time-derivative of a Wiener process, and A, Σ are some operators; see [17], §5 for details. If Y : R + → R is one-dimensional, and Σ = √ T > 0, Y (t) modelizes either the velocity of a massive Brownian particle under the influence of friction, or the position of an infinitely massive Brownian particle submitted to friction and to a harmonic potential V (Y ) = 1 2 AY 2 ; in the first interpretation, A is the friction coefficient. In both cases T plays the rôle of a temperature, as appears in the Maxwell-like equilibrium distribution e −AY 2 /2T = e −V (Y )/T . In our context Y = Y (t, x) is the random fluctuation process, η = η(t, y) is space-time white noise, and (6.1) is a Langevin equation for Y . Under adequate assumptions, notably on the analytic properties and long-time behavior of the semi-group e −tA , t ≥ 0 generated by A, this equation has a unique stationary measure µ ∞ , and the law µ t of Y t converges to µ ∞ for any reasonable initial measure µ 0 . Furthermore, µ ∞ is Gaussian, with covariance kernel K ∞ = K ∞ (x, y) defined uniquely by Sym(K ∞ A † ) = 1 2 ΣΣ † (6.2) with Sym(B) := 1 2 (B + B † ) (see [17], Theorem 5.22). If Σ, A are self-adjoint and commute, and A ≥ 0, then (starting from any initial measure) Y (t) = e −tA Y (0) + e −tA t 0 e sA Ση(s), so K ∞ = lim t→∞ t 0 ds e −(t−s)A ΣΣ † e −(t−s)A = 1 2 Σ 2 /A, confirming (6.2). Assume conversely that some stationary Gaussian process Y (t) is given, with known two-time covariance kernel K ∞ (t 1 , x 1 ; t 2 , x 2 ) = K ∞ (t 1 − t 2 ; x 1 , x 2 ). Then Y is the solution of (6.1) with