A multi-layer extension of the stochastic heat equation
aa r X i v : . [ m a t h . P R ] J un A MULTI-LAYER EXTENSION OF THESTOCHASTIC HEAT EQUATION
NEIL O’CONNELL AND JON WARREN
Abstract.
Motivated by recent developments on solvable directed poly-mer models, we define a ‘multi-layer’ extension of the stochastic heat equa-tion involving non-intersecting Brownian motions. Introduction and summary
We consider the stochastic heat equation in one dimension(1) du = 12 ∂ y u dt + u dW ( t ) , with initial condition u (0 , x, y ) = δ ( x − y ), where W is a cylindrical Brown-ian motion on L ( R ) and the term u dW ( t ) is interpreted as an Itˆo integral.Formally, we can write this as(2) ∂ t u = 12 ∂ y u + u ˙ W , where ˙ W denotes space-time white noise associated with the Brownian motion W . This is a distribution-valued Gaussian field on [0 , ∞ ) × R with covariancefunction E ˙ W ( t, y ) ˙ W ( t ′ , y ′ ) = δ ( y − y ′ ) δ ( t − t ′ ) . For more background see, for example, [7, 23, 39, 51].We will write stochastic integrals R t H ( s ) dW ( s ) of a predictable L ( R )-valued process H as Z t H ( s, x ) W ( ds, dx ) . Then the solution u ( t, x, y ) to (1) is given by the chaos expansion u ( t, x, y ) = p ( t, x, y )+ ∞ X k =1 Z ∆ k ( t ) Z R k p ( t , x, x ) p ( t − t , x , x ) · · · p ( t − t k , x k , y )(3) × W ( dt , dx ) · · · W ( dt k , dx k ) , Mathematics Subject Classification.
Primary 60H15, 15A52. where ∆ k ( t ) = { < t < · · · < t k < t } and p ( t, x, y ) = 1 √ πt e − ( x − y ) / t . For each t > x, y ∈ R , the expansion (3) is convergent in L ( W ). Itsatisfies (1) in the sense that it satisfies the integral equation(4) u ( t, x, y ) = p ( t, x, y ) + Z t Z R p ( t − s, y ′ , y ) u ( s, x, y ′ ) W ( ds, dy ′ ) . For more details see, for example, [39]. The chaos series representation forthe solution can be viewed as an expansion of the Feynman-Kac formula. Toemphasize this interpretation, the solution (3) is often written as a generalisedWiener functional(5) u ( t, x, y ) = p ( t, x, y ) E exp ⋄ (cid:18)Z t dW ( s, X s ) (cid:19) , where the expectation is with respect to a Brownian bridge ( X s , s ≤ t ) whichstarts at x at time 0 and ends at y at time t [23]. The factor p ( t, x, y ) is arisesbecause of this use of the bridge in the representation, corresponding to the δ -function initial condition, rather than the more usual free Brownian motion.It is important to note that the integral R t dW ( s, X s ) ds , representing theintegral of the white noise ˙ W along the path s ( s, X s ), is not well-defined,and equation (5) is purely formal.This solution arises as a scaling limit of partition functions associated withlattice directed polymers, in the ‘intermediate disorder’ regime [1, 29]. In fact,as suggested by (5), it can be interpreted directly as a partition function forthe continuum random polymer [2]. On the other hand, it has been knownfor some time that h = log u arises as the scaling limit of the height profileof the weakly asymmetric simple exclusion process, at least for equilibriuminitial conditions [8], recently extended to include the present initial conditionin [3]. With this ‘surface growth’ interpretation, h is understood to be thephysically relevant solution (also known as the Cole-Hopf solution) to the KPZ(Kardar-Parisi-Zhang) equation [26] ∂ t h = 12 ∂ y h + 12 ( ∂ y h ) + ˙ W ( t, y ) , with ‘narrow wedge’ initial condition.In a remarkable recent development the exact distribution of the randomvariable u ( t, x, y ) has been determined. This has been acheived via twodistinct approaches. One of these [3, 41, 42, 43, 44] uses the asymmetricsimple exclusion process approximation together with recent work by Tracyand Widom [47, 48, 49, 50] in which exact formulas have been obtained forthat process using an approach based on the Bethe ansatz. Another ap-proach [11, 16, 17, 18] is based on replicas, where the moments of u ( t, x, y ) are MULTI-LAYER EXTENSION OF THE STOCHASTIC HEAT EQUATION 3 related to the attractive δ -Bose gas and computed via the Bethe ansatz. Thesedevelopments indicate that there is an underlying integrable structure behindthe KPZ and stochastic heat equations which is not yet fully understood.On the other hand, it has recently been found that there are exactly solvablediscrete (or semi-discrete) directed polymer models [15, 30, 32, 33, 34, 45, 46],yielding yet another approach. For these models, there is a direct connectionto integrable systems (specifically to the quantum Toda lattice) which onemight hope to understand in the continuum scaling limit. We will describehere one of the main results of the paper [32], which provided the motivationfor the present work.Define an ‘up/right path’ in R × Z to be an increasing path which eitherproceeds to the right or jumps up by one unit. For each sequence 0 < t < · · · < t N − < t we can associate an up/right path π from (0 ,
1) to ( t, N ) whichhas jumps between the points ( t i , i ) and ( t i , i + 1), for i = 1 , . . . , N −
1, andis continuous otherwise. Let B ( t ) = ( B ( t ) , . . . , B N ( t )), t ≥
0, be a standardBrownian motion in R N and define Z N ( t ) = Z e E ( π ) dπ, where E ( π ) = B ( t ) + B ( t ) − B ( t ) + · · · + B N ( t ) − B N ( t N − )and the integral is with respect to Lebesgue measure on the Euclidean set(6) { ( t , . . . , t N − ) ∈ R N − : 0 < t < · · · < t N − < t } of all such paths. This is the partition function for the model, which wasintroduced in [33]. In [32] an explicit integral formula is obtained for theLaplace transform of the distribution of Z N ( t ), via the following ‘multi-layer’construction. For n = 1 , , . . . , N , define(7) Z Nn ( t ) = Z e E ( π )+ ··· + E ( π n ) dπ . . . dπ n , where the integral is with respect to Lebesgue measure on the set of n -tuplesof non-intersecting (disjoint) up/right paths with respective initial points(0 , , . . . , (0 , n ) and respective end points ( t, N − n + 1) , . . . , ( t, N ). Here,the notion of Lebesgue measure arises by identifying this set of paths as asuitable subset of R n ( N − n ) , as in (6). Define X N ( t ) = log Z N ( t ) and, for n ≥ X Nn ( t ) = log[ Z Nn ( t ) /Z Nn − ( t )]. The relevance of this construction isanalogous to the role of the RSK correspondence in the study of last passagepercolation and longest increasing subsequence problems; in this setting it isbased on a geometric variant of the RSK correspondence. The main resultin [32] is the following. NEIL O’CONNELL AND JON WARREN
Theorem 1.1.
The process X N ( t ) = ( X N ( t ) , . . . , X NN ( t )) , t > , is a diffu-sion process in R N with infinitesimal generator given by (8) L = 12 ∆ + ∇ log ψ · ∇ where ψ is a (particular) ground state eigenfunction of the quantum Todalattice Hamiltonian H = − ∆ + 2 N − X i =1 e x i +1 − x i . The function ψ is a (class-one) GL ( n, R )-Whittaker function. The diffu-sion process with generator L is the analogue of Dyson’s Brownian motion inthis setting. The law of the logarithmic partition function X N ( t ) = log Z N ( t ),which can now be seen as the analogue of the top line (or largest eigenvalue) inthe Dyson process, is determined as a corollary. Similar results have been ob-tained in [15, 34] for a lattice directed polymer model with log-gamma weightswhich was introduced by Sepp¨al¨ainen [45]. In that setting, the eigenfunctionsof the quantum Toda lattice (also known as Whittaker functions) continue toplay a central role, as does the geometric lifting of the RSK correspondencewhich was introduced and studied in the papers [28, 31].Motivated by these developments, in this paper we introduce continuumversions of the partition functions Z Nn ( t ), which we expect will play an impor-tant role in our understanding of the integrable structure which appears tolie behind the KPZ and stochastic heat equations. The continuum partitionfunctions are defined as follows. For n = 1 , , . . . , t ≥ x, y ∈ R , define Z n ( t, x, y ) = p ( t, x, y ) n (cid:16) ∞ X k =1 Z ∆ k ( t ) Z R k R ( n ) k (( t , x ) , . . . , ( t k , x k ))(9) × W ( dt , dx ) · · · W ( dt k , dx k ) (cid:17) , where R ( n ) k is the k -point correlation function for a collection of n non-intersectingBrownian bridges which all start at x at time 0 and all end at y at time t , asdefined in section 2 below. Note that Z = u is the solution of the stochasticheat equation defined by (3) above. To explain the above definition we notethat, just as in (5), we can formally write (9) as(10) Z n ( t, x, y ) = p ( t, x, y ) n E exp ⋄ n X i =1 Z t dW ( s, X is ) ds ! , where ( X s , . . . , X ns , ≤ s ≤ t ) denote the trajectories of n non-intersectingBrownian bridges which all start at x at time 0 and all end at y at time t .These should be compared with the partition functions (7). The first mainresult of this paper is that the continuum partition functions Z n ( t, x, y ) arewell-defined. MULTI-LAYER EXTENSION OF THE STOCHASTIC HEAT EQUATION 5
Theorem 1.2.
The series (9) is convergent in L ( W ) . The proof will given in Section 4. Define u = u and, for n ≥ u n = Z n /Z n − . From Theorem 1.1 we expect that, for each fixed t >
0, the process u t ( x ) = { u n ( t, , x ) , n ∈ N } , indexed by x ∈ R , is a diffusion process in R N which is a scaling limit at the edge of the diffusion with generator (8), justas the multi-layer Airy process introduced in [36] is a scaling limit at theedge of Dyson’s Brownian motion. Note that it is not at all clear a priorithat this process should have the Markov property: in fact, this is quite aremarkable property and can be regarded as the analogue, in this setting, ofPitman’s celebrated ‘2 M − X ’ theorem. Moreover, for large t , the process u t ( x ) , x ∈ R should rescale (after taking logarithms) to the multi-layer Airyprocess. At present, we only know this to be the case for the one-dimensionaldistributions of the first layer: it has been shown in the papers [3, 44] that thedistribution of log[ u ( t, , x ) /p ( t, , x )] (which is independent of x ) converges ina suitable scaling limit to the Tracy-Widom distribution. This result on thefirst layer has been tentatively extended to the finite-dimenisonal distributionsby Prolhac and Spohn [37, 38]. As such, it is natural to regard the process u t ( x ) , x ∈ R as the analogue of the multi-layer Airy process in the settingof the KPZ and stochastic heat equations. Similarly, for fixed t >
0, therandom sequence { u n ( t, . , n ∈ N } can be regarded as the analogue of theAiry point process. There are many things to understand in these directions,especially the law of the process ( u t ( x ) , x ∈ R ). For further recent progressin this direction, see [10, 14].It is also natural ask about the evolution of u t as t varies: after all, thisis an extension of the process ( u ( t, , · ) , t > u t (which involvesnon-intersecting Brownian bridges over the time interval [0 , t ] with the samestarting and ending points ) there is really no reason to expect the extendedprocess ( u t , t >
0) to have the Markov property. Nevertheless, we will presenta number of results which strongly indicate that it does indeed, somewhatremarkably, have the Markov property. In fact, we believe that for each n ,the process { ( u ( t, , x ) , . . . , u n ( t, , x )) , t ≥ } is Markov and give a proof of this claim in the case n = 2 (see Corollary 6.4).In order to develop a better understanding of the continuum partition func-tions Z n ( t, x, y ), we consider their analogues when the space-time white-noisepotential ˙ W ( t, x ) is replaced by a smooth time-varying potential φ ( t, x ). Forexample, we can take φ to be in the Schwartz space E of rapidly decreasingsmooth ( C ∞ ) functions on R + × R . For each n = 1 , , . . . , t > x, y ∈ R , NEIL O’CONNELL AND JON WARREN define(11) Z φn ( t, x, y ) = p ( t, x, y ) n E exp n X i =1 Z t φ ( s, X is ) ds ! , where once again ( X s , . . . , X ns , ≤ s ≤ t ) denote the trajectories of n non-intersecting Brownian bridges which all start at x at time 0 and all end at y at time t . On one hand, these are the analogues of the partition functions Z n introduced above with the white noise ˙ W replaced by the smooth potential φ .On the other, they are directly related to the Z n by the formula(12) Z φn ( t, x, y ) = E [ Z n ( t, x, y ) exp ⋄ ( W ( φ ))] , where W ( φ ) = Z ∞ Z R φ ( s, x ) W ( ds, dx )and exp ⋄ ( W ( φ )) is the Wick exponential of W ( φ ) defined byexp ⋄ ( W ( φ )) = exp (cid:18) W ( φ ) − Z ∞ Z R φ ( s, x ) dxds (cid:19) . In other words, as a function of φ , Z φn ( t, x, y ) is the S -transform of the whitenoise functional Z n ( t, x, y ) (see, for example, [23]). In the following, we dropthe superscript φ and write Z n = Z φn . Now, as above, we define u = u = Z and, for n ≥ u n = Z n /Z n − . Note that, by Feynman-Kac, u ( t, x, y ) satisfiesthe heat equation(13) ∂ t u = 12 ∂ y u + φ ( t, y ) u with initial condition u (0 , x, y ) = δ ( x − y ). In this setting we will show, usinga generalisation of the Karlin-McGregor formula (see Propositions 3.1 and 3.2below), that, for n ≥ Z n ( t, x, y ) = c n,t det (cid:2) ∂ ix ∂ jy u ( t, x, y ) (cid:3) n − i,j =0 , where c n,t = t n ( n − / ( Q n − j =1 j !) − . But this determinant is the Wronskianassociated with the solutions u, ∂ x u, . . . , ∂ n − x u of the heat equation (13). Itfollows (see for example [4]) that the functions u n are in fact Darboux trans-formations and satisfy the coupled system of heat equations(15) ∂ t u n = 12 ∂ y u n + [ φ ( t, y ) + ∂ y log (cid:0) Z n − /p n − (cid:1) ] u n with initial conditions u n (0 , x, y ) = δ ( x − y ). These equations are not imme-diately meaningful if we replace the smooth potential φ by space-time whitenoise. However they do suggest that, for each n , the multi-layer process (inthe white noise setting)( Z ( t, x, · ) , . . . , Z n ( t, x, · ) , t ≥ MULTI-LAYER EXTENSION OF THE STOCHASTIC HEAT EQUATION 7 has a Markov evolution. In order to confirm this, there are two possibledirections one could take. The first, and most obvious one, is to try to makesense of these evolution equations with white-noise replacing φ . To this end,it is helpful to consider the following change of variables which, as it happens,also reveal some deeper structure as we shall see. Set τ = 1 and, for n ≥ τ n = det (cid:2) ∂ ix ∂ jy u ( t, x, y ) (cid:3) n − i,j =0 , a n = τ n − τ n +1 τ n . We will show (see Propositon 3.7) that the evolution of the a n is given by(16) ∂ t a n = 12 ∂ y a n + ∂ y [ a n ∂ y log u n ] . It seems to be the case that these evolution equations will make sense asstochastic partial differential equations in the white-noise setting [21].We remark here in passing on an interesting connection to the 2D Todaequations. We will show (see Lemma 3.6) that a n = ∂ xy log τ n , for each n .Thus, if we define, for n ≥ q n = log( τ n /τ n − ) = log u n − log[ t n − ( n − , then a n = e q n +1 − q n and the q n satisfy the 2D Toda equations ∂ xy q n = e q n +1 − q n − e q n − q n − , n ≥ , with the convention that q = + ∞ . In this notation, the time-evolution ofthe a n is given by ∂ t a n = 12 ∂ y a n + ∂ y [ a n ∂ y q n ] . We will also show that there is in fact a second, and quite different, approachone can take in order to understand (and prove) the Markov property of themulti-layer process which exploits two quite remarkable formulas, presentedin Theorems 3.4 and 5.3 below, concerning a natural extension of the partitionfunctions Z n to collections of non-intersecting Brownian paths which start atdistinct points and end at distinct points. These are defined as follows. Set(17) Λ n = { x = ( x , . . . , x n ) ∈ R n : x ≥ · · · ≥ x n } , and denote the interior of Λ n by Λ ◦ n . For each t > x = ( x , . . . , x n )and y = ( y , . . . , y n ) in Λ ◦ n , define(18) K n ( t, x , y ) = p ∗ n ( t, x , y ) E exp n X i =1 Z t φ ( s, X is ) ds ! , where ( X s , . . . , X ns , ≤ s ≤ t ) denote the trajectories of a collection of non-intersecting Brownian bridges which start at positions x = ( x , . . . , x n ) andend at positions y = ( y , . . . , y n ) at time t , and p ∗ n ( t, x , y ) is the transition NEIL O’CONNELL AND JON WARREN density of a Brownian motion in Λ n killed when it first hits the boundary,given by the Karlin-McGregor formula [27], p ∗ n ( t, x , y ) = X σ ∈ a n sgn( σ ) n Y i =1 p ( t, x i , y σ ( i ) ) . According to the Feynman-Kac formula, K n satisfies the equation(19) ∂ t K n = 12 ∆ y K n + X i φ ( t, y i ) K n with Dirichlet boundary conditions on ∂ Λ n and initial condition K n (0 , x , y ) = Q i δ ( x i − y i ). In Proposition 3.2 below we show that the following generali-sation of the Karlin-McGregor formula holds:(20) K n ( t, x , y ) = det[ u ( t, x i , y j )] ni,j =1 . Now, define, for t > x , y ∈ Λ n ,(21) M n ( t, x , y ) = K n ( t, x , y )∆( x )∆( y ) . This extends continuously to the boundary of Λ n × Λ n ; by Proposition 3.2,for x ∈ R ,(22) M n ( t, x , y ) = ∆( y ) − det (cid:2) ∂ i − x u ( t, x, y j ) (cid:3) ni,j =1 . Rather surprisingly, we will show that the apparently richer object M n ( t, x , · )is, for a fixed x ∈ R and t >
0, given as a function of( Z ( t, x, · ) , . . . , Z n ( t, x, · )) . For z ∈ Λ n − and y ∈ Λ n , write z ≺ y if y ≥ z > y ≥ · · · > y n − ≥ z n − >y n . For y ∈ Λ ◦ n , denote by GT ( y ) the Gelfand-Tsetlin polytope { ( y , y , . . . , y n − ) ∈ Λ × Λ × · · · × Λ n − : y ≺ y ≺ · · · ≺ y n − ≺ y } . Then (see Theorem 3.4) for t > x ∈ R and y ∈ Λ ◦ n ,(23) M n ( t, x , y ) = ∆( y ) − n Y i =1 u ( t, x, y i ) Z GT ( y ) n − Y k =1 n − k Y i =1 a k ( t, x, y n − ki ) dy n − ki . In the case φ = 0, this reduces to the fact that the volume of GT ( y ) isproportional to ∆( y ). Now, if the analogous formula holds in the white-noisesetting, with K n ( t, x , y ) defined by the chaos series (45) below, then this implythe Markov property of the multi-layer process( Z ( t, x, · ) , . . . , Z n ( t, x, · ) , t ≥ . This is because the process for each x ∈ R and for each n , the process( M n ( t, x , · ) , t ≥
0) clearly has the Markov property. However, an importantingredient in this argument is the Karlin-McGregor formula (20). A priori,
MULTI-LAYER EXTENSION OF THE STOCHASTIC HEAT EQUATION 9 there is no reason to expect this to hold in the white-noise setting. In fact,the natural analogue of this formula in the white-noise setting is the formula K n ( t, x , y ) = det ⋄ [ u ( t, x i , y j )] ni,j =1 , where det ⋄ indicates that the products in the expansion of the determinantare Wick products . Nevertheless, quite remarkably, we will show in Section 5that the formula (20) does in fact hold in the white-noise setting. In Section 6we argue that, modulo technical considerations, this indicates that the ana-logue of the integral formula (23) should hold in the white-noise setting. Wegive a proof in the case n = 2, just to demonstrate that it can be done. Themain technical issue concerns the continuity of M n ( t, x , y ) at the boundary ofΛ n × Λ n . A proof of the existence of an almost surely continuous extensionto the boundary based on Kolmogorov’s criterion would be long and technicaland beyond the scope of the present work. Here we satisfy ourselves with acontinuous extension in L , which allows us to establish the analogue of theintegral formula (23), and hence the Markov property of the multi-layer pro-cess, in the special case n = 2. We remark that an interesting consequenceof our proof of the L -continuity property is that the ratio of two solutionsto the stochastic heat equation is in H . In fact, such ratios have recentlybeen shown (in a slightly different setting, for smooth initial data and periodicboundary conditions) by Hairer [21] to be in C / − ǫ . The index 3 / Z n ( t, x, y )are locally Brownian in the space variable y .We conclude this section with some remarks on the RSK interpretation. Asremarked above, the multi-layer construction presented in this paper is basedon a geometric lifting of the RSK correspondence, so it is natural to considersuch an interpretation in the continuum setting. The analogue of RSK in thecontext of smooth potentials is the mapping φ (cid:12)(cid:12) [0 ,t ] × R
7→ { u n ( t, , · ) , n ≥ } . In the language of RSK, { u n ( t, , x ) , n ≥ x ≥ } is the P -tableau, { u n ( t, , − x ) , n ≥ x ≥ } is the Q -tableau, and their common ‘shape’ is thesequence { u n ( t, , , n ≥ } . We note the following symmetry, which corre-sponds to a well known symmetry property of the RSK correspondence. Writ-ing f = φ (cid:12)(cid:12) [0 ,t ] × R , P ( f ) = { u n ( t, , x ) , n ≥ x ≥ } , Q ( f ) = { u n ( t, , − x ) , n ≥ x ≥ } and f † ( s, x ) = f ( s, − x ), we have: P ( f † ) = Q ( f ) and Q ( f † ) = P ( f ).Similarly, in the white noise setting, we define P ( W [0 ,t ] ) = { u n ( t, , x ) , n ≥ x ≥ } and Q ( W [0 ,t ] ) = { u n ( t, , − x ) , n ≥ x ≥ } where W [0 ,t ] denotes the restriction of W to [0 , t ] × R . As explained above,we expect that, for each t > P ( W [0 ,t ] ) and Q ( W [0 ,t ] ) are diffusion processes in R N (indexed by x ≥
0) which are conditionally independent given theirstarting position { u n ( t, , , n ≥ } . This would be the analogue, in thissetting, of Pitman’s ‘2 M − X ’ theorem. Since the first draft of the present pa-per appeared, substantial progress has been made in this direction by Corwinand Hammond [14], where a natural candidate for this infinite-dimensionaldiffusion process has been constructed.The outline of the paper is as follows. In the next section we providesome background on non-intersecting Brownian motions and their bridges.Following this we study the analogue of the partition functions when thespace-time white noise is replaced by a smooth time-varying potential. In thissetting we establish a connection with Darboux transformations of solutionsto the heat equation, which give rise to evolution equations for the multi-layerprocess of partition functions. These equations are not directly meaningfulin the white noise setting, but suggest that the multi-layer process has aMarkovian evolution. We also give proofs of the integral formula (23) above,the evolution equations (16) and remark on the connection with the 2D Todaequations. In Section 4 we present the proof of Theorem 1.2 on the existenceof the continuum partition functions in the white-noise setting. In Section 5,we show that the Karlin-McGregor formula (5.3) holds in the white-noisesetting. In Section 6 we consider the evolution of the multi-layer process inthe white-noise setting and give a proof of the Markov property for n = 2. Acknowledgements.
Thanks to Ivan Corwin, Martin Hairer, Jeremy Quas-tel, Gregorio Moreno-Flores and Roger Tribe for helpful discussions. Thisresearch was supported in part by EPSRC grant EP/I014829/1.2.
Non-intersecting Brownian motions
Non-intersecting Brownian motions play a large role in this paper, and werecord here definitions and facts concerning them that will be useful to us.Recall that(24) Λ n = { x = ( x , . . . , x n ) ∈ R n : x ≥ · · · ≥ x n } , and denote the interior of Λ n by Λ ◦ n . Standard n -dimensional Brownian motionkilled on exiting Λ ◦ n has transition densities given by the Karlin-McGregorforumla [27], p ∗ n ( t, x , y ) = X σ ∈ a n sgn( σ ) n Y i =1 p ( t, x i , y σ ( i ) ) . Dyson Brownian motion is obtained as a Doob- h transform of this killed pro-cess by the harmonic function ∆( x ) = Q i For each t > , the transition density q n ( t, x , y ) extends con-tinuously to a uniformly bounded strictly positive function on Λ n × Λ n .Proof. By the Harish-Chandra/Itzykson-Zuber formula, we can write(26) q n ( t, x, y ) = (2 π ) − n/ t − n / c n Z U ( n ) e − tr ( X − UY U ∗ ) / t dU, where 1 /c n = Q n − j =1 j !, X and Y are diagonal matrices with entries given bythe vectors x and y , and the integral is with respect to normalised Haar mea-sure on the group of n × n unitary matrices. By bounded convergence, the RHSdefines a continuous function on Λ n × Λ n , and is bounded by (2 π ) − n/ t − n / c n .The strict positivity follows from the strict positivity of the integrand. (cid:3) The semigroup property q n ( s + t, x , z ) = Z Λ n q n ( s, x , y ) q n ( t, y , z )∆( y ) d y , thus also extends to the boundary, by continuity and dominated convergence.Moreover, it follows from the representation (26) that q n ( t, x , y )∆( y ) d y de-fines a probability measure on Λ n for every t > x ∈ Λ n . Consequently,Dyson Brownian motion can be started from any point on the boundary ofΛ n . In fact, as was shown by C´epa and L´epingle [12], it almost surely neversubsequently returns to the boundary.Let ( H t , t ≥ 0) be a standard Brownian motion in the space of n × n Hermit-ian matrices. Then the vector of ordered real-valued eigenvalues of H t evolvesas Dyson Brownian motion. This can be verified by deriving the transitiondensity for the eiqenvalues using the Harish-Chandra formula. Alternatively,following Dyson’s original approach, applying Itˆo’s formula shows that, de-noting the vector of eigenvalues by ( X t , X t , . . . , X nt ), the following stochasticdifferential equations are satisfied.(27) X it = X i + β it + X j = i Z t dsX is − X js , where β i , i = 1 , , . . . , n are a collection of independent standard one-dimensionalBrownian motions. Note that these equations hold even if the intitial value H of the Hermitian Brownian motion has repeated eigenvalues, in which case theDyson Brownian motion is starting from the boundary of Λ n . This can beenseen by the following argument. Applying Itˆo’s formula from some strictly positive time ǫ onwards ( recalling the process of eigenvalues does not visitthe boundary) we obtain for t ≥ ǫ ,(28) X it = X iǫ + β i,ǫt + X j = i Z tǫ dsX is − X js , where β i,ǫ are Brownian motions. Now for ǫ < ǫ ′ the increments of β i,ǫǫ ′ − ǫ + · and β i,ǫ ′ agree, and by virtue of this consistency there exist Brownian motions β i starting from 0 such that β i,ǫt = β iǫ + t − β iǫ for all ǫ > 0. Now returning to (28),writing it using β i , and letting ǫ tend down to 0 gives (27) as desired even inthe case when the Dyson Brownian motion starts from the boundary.We can construct bridges for Dyson Brownian motion using the standardMarkovian framework, see for example Proposition 1 of [19]. Specifically givenpoints x and y belonging to Λ n , we define the bridge from x at time 0 endingat y at time t , to be a process ( X s , ≤ s ≤ t ) whose law over [0 , s ], for any s < t , is absolutely continuous with respect to that of Dyson Brownian motionstarting from x , with a density(29) q n ( t − s, X s , y ) q n ( t, x , y ) . Note that this is well-defined as the denominator is strictly positive, by Lemma 2.1above. We will also refer to this bridge, somewhat informally, as a collectionof non-intersecting Brownian bridges. In the special case x = x , y = y ,where x, y ∈ R and we denote by ∈ R n the vector with all coordinates equalto 1, the process is also often referred to as a watermelon. The correlationfunction R ( n ) k (( t , x ) , . . . , ( t k , x k )) appearing in the definition (9) is definedto be the sum over i , i , . . . , i k of the (continuous) probability densities of( X i t , . . . , X i k t k ) with respect to Lebesgue measure evaluated at ( x , x , . . . , x k ).Correlation functions for non-intersecting Brownian bridges with arbitrarystarting and ending positions, which appear in Section 5 below, are definedanalogously. 3. Darboux transformations In this section we replace the white noise potential ˙ W by a smooth potential φ , which we assume for convenience to be in the Schwartz space E of rapidlydecreasing smooth ( C ∞ ) functions on R + × R .For each n = 1 , , . . . , t > x, y ∈ R , define(30) Z φn ( t, x, y ) = p ( t, x, y ) n E exp n X i =1 Z t φ ( s, X is ) ds ! , where ( X s , . . . , X ns ) , ≤ s ≤ t, denote the trajectories of n non-intersectingBrownian bridges which all start at x at time 0 and all end at y at time t . Onone hand, these are the analogues of the partition functions Z n introduced in MULTI-LAYER EXTENSION OF THE STOCHASTIC HEAT EQUATION 13 the previous section with the white noise ˙ W replaced by a smooth potential φ . On the other, they are directly related to the Z n by the formula(31) Z φn ( t, x, y ) = E [ Z n ( t, x, y ) exp ⋄ ( W ( φ ))] , where W ( φ ) = Z ∞ Z R φ ( s, x ) W ( ds, dx )and exp ⋄ ( W ( φ )) is the Wick exponential of W ( φ ) defined byexp ⋄ ( W ( φ )) = exp (cid:18) W ( φ ) − Z ∞ Z R φ ( s, x ) dxds (cid:19) . In other words, as a function of φ , Z φn ( t, x, y ) is the S -transform of the whitenoise functional Z n ( t, x, y ) (see, for example, [23]). To see that (31) holds, onthe RHS replace Z n ( t, x, y ) by the series (9) and exp ⋄ ( W ( φ )) by its Wienerchaos expansion; computing the expectation of the product of these two serieswe obtain p ( t, x, y ) n ∞ X k =0 Z ∆ k ( t ) Z R n φ ( t , x ) . . . φ ( t k , x k ) × R ( n ) k (( t , x ) , . . . , ( t k , x n )) dx . . . dx k dt . . . dt k = p ( t, x, y ) n E exp n X i =1 Z t φ ( s, X is ) ds ! . For the remainder of this section we will only consider the case of smoothpotential φ . For notational convenience we will drop the superscript andsimply write Z n ( t, x, y ) = Z φn ( t, x, y ). By the Feynman-Kac formula, u := Z satisfies the heat equation(32) ∂ t u = 12 ∂ y u + φ ( t, y ) u with initial condition u (0 , x, y ) = δ ( x − y ). Proposition 3.1. For n ≥ , (33) Z n ( t, x, y ) = c n,t det (cid:2) ∂ ix ∂ jy u ( t, x, y ) (cid:3) n − i,j =0 , where c n,t = t n ( n − / c n and /c n = Q n − j =1 j ! . We will prove this via a generalisation of the Karlin-McGregor formula.For each t > x = ( x , . . . , x n ) and y = ( y , . . . , y n ) in Λ ◦ n , define(34) K n ( t, x , y ) = p ∗ n ( t, x , y ) E exp n X i =1 Z t φ ( s, X is ) ds ! , where ( X s , . . . , X ns , ≤ s ≤ t ) , denote the trajectories of a collection ofnon-intersecting Brownian bridges which start at positions x , . . . , x n and endat positions y , . . . , y n at time t , and p ∗ n ( t, x , y ) is the transition density of a Brownian motion in Λ n killed when it first hits the boundary, given by theKarlin-McGregor formula. Proposition 3.2. (35) K n ( t, x , y ) = det[ u ( t, x i , y j )] ni,j =1 . Proof. According to the Feynman-Kac formula, K n satisfies the equation(36) ∂ t K n = 12 ∆ y K n + X i φ ( t, y i ) K n with Dirichlet boundary conditions on ∂ Λ n and initial condition K n (0 , x , y ) = Q i δ ( x i − y i ). Moreover it is the unique solution to this initial-boundary valueproblem which vanishes as | y | → ∞ uniformly for t in compact intervals.This follows from a variant of the maximum principle (see, for example, [20,Chapter 2, Theorem 2]), which applies in this setting since φ is bounded andcontinuous.On the other hand, det[ Z ( t, x i , y j )] ni,j =1 satisfies the same initial-boundaryvalue problem and vanishes as | y | → ∞ uniformly for t in compact intervals.So the identity follows by uniqueness. (cid:3) We remark that the same argument can be applied to more general ex-pressions than (34) in which the potential ( s, x ) P φ ( s, x i ) is replaced bya bounded continuous potential ψ ( t, x ) which is a symmetric function of thecoordinates of x ; we will make use of this fact in the proof of Theorem 5.3below. Proof of Proposition 3.1. It is immediate from the definitions that Z n ( t, a, b ) p ( t, a, b ) n = lim x → a , y → b K n ( t, x , y ) p ∗ n ( t, x , y ) . Now lim x → a , y → b p ∗ n ( t, x , y )∆( x )∆( y ) = p ( t, a, b ) n t − n ( n − / c n , where ∆( x ) = Q i 0) and ǫ → (cid:3) Define u n ( t, x, y ) recursively by Z n = u u · · · u n . MULTI-LAYER EXTENSION OF THE STOCHASTIC HEAT EQUATION 15 Proposition 3.3. The functions u n satisfy the coupled system of heat equa-tions (37) ∂ t u n = 12 ∂ y u n + [ φ ( t, y ) + ∂ y log (cid:0) Z n − /p n − (cid:1) ] u n with initial conditions u n (0 , x, y ) = δ ( x − y ) and the convention Z = 1 . The equations (37) follow from Proposition 3.1 together with known prop-erties of Darboux transformations of solutions to one-dimensional heat equa-tions with time-varying potentials, see for example [4, 5]. For completeness,we will include a direct proof of Proposition 3.3 just after the statement ofProposition 3.7 below.The coupled heat equations of Proposition 3.3 are not immediately mean-ingful if we replace the smooth potential φ by space-time white noise. Howeverthey do suggest that the multi-layer process (in the white noise setting)( Z ( t, x, · ) , . . . , Z n ( t, x, · ) , t ≥ Z n which will play an important role in our understanding of the Markovproperty when we return to the white noise setting.Define, for t > x , y ∈ Λ n ,(38) M n ( t, x , y ) = K n ( t, x , y )∆( x )∆( y ) . This extends continuously to the boundary of Λ n × Λ n ; by Proposition 3.2,for x ∈ R ,(39) M n ( t, x , y ) = ∆( y ) − det (cid:2) ∂ i − x u ( t, x, y j ) (cid:3) ni,j =1 . Rather surprisingly, we will now show that the apparently richer object M n ( t, x , · )is, for a fixed x ∈ R and t > 0, given as a function of( Z ( t, x, · ) , . . . , Z n ( t, x, · )) . Recall from Proposition 3.1 that, for n ≥ Z n ( t, x, y ) = c n,t det (cid:2) ∂ ix ∂ jy u ( t, x, y ) (cid:3) n − i,j =0 , where c n,t = t n ( n − / c n . Let us write(40) τ n ( t, x, y ) = det (cid:2) ∂ ix ∂ jy u ( t, x, y ) (cid:3) n − i,j =0 . For notational convenience, set τ = Z = 1 and Λ = R . For n ≥ 1, define(41) a n = τ n − τ n +1 τ n = nt Z n − Z n +1 Z n . Here we are using the fact that c n − ,t c n +1 ,t /c n,t = t/n . For z ∈ Λ n − and y ∈ Λ n , write z ≺ y if y ≥ z > y ≥ · · · > y n − ≥ z n − > y n . For y ∈ Λ ◦ n , denote by GT ( y ) the Gelfand-Tsetlin polytope { ( y , y , . . . , y n − ) ∈ Λ × Λ × · · · × Λ n − : y ≺ y ≺ · · · ≺ y n − ≺ y } . Theorem 3.4. For t > , x ∈ R and y ∈ Λ ◦ n , M n ( t, x , y ) = ∆( y ) − n Y i =1 u ( t, x, y i ) Z GT ( y ) n − Y k =1 n − k Y i =1 a k ( t, x, y n − ki ) dy n − ki . In the case φ = 0, this reduces to the fact that the volume of GT ( y ) isproportional to ∆( y ). By (39) and Proposition 3.1, this theorem can be seenas a consequence of the next two lemmas. Lemma 3.5. If f , f , . . . is a sequence of continuously differentiable functionson R with f ≡ then det[ f i ( y j )] ni,j =1 = Z z ≺ y det[ f ′ i +1 ( z j )] n − i,j =1 dz . . . dz n − . Proof. Using the formula1 z ≺ y = det (cid:2) y j +1 Let g ( x, y ) be a smooth function and define W = 1 , W = g and, for n ≥ , W n = det h ∂ ix ∂ jy g ( x, y ) i n − i,j =0 . Suppose that W n is strictlypositive for all n ≥ and define T n = W n − W n +1 /W n . Then the followingidentities hold: T n = ∂ xy log W n = ∂ y ( ∂ y ( · · · ∂ y ( ∂ nx g/g ) /T ) /T ) · · · ) /T n − ) , det[ ∂ i − x g ( x, y j )] ni,j =1 = n Y i =1 g ( x, y i ) Z GT ( y ) n − Y k =1 n − k Y i =1 T k ( x, y n − ki ) dy n − ki . MULTI-LAYER EXTENSION OF THE STOCHASTIC HEAT EQUATION 17 Proof. From well-known properties of Wronskian determinants, ∂ x W n , ∂ y W n and ∂ xy W n can be expressed as determinants, namely ∂ x W n = det (cid:2) ∂ ix ∂ jy g ( x, y ) (cid:3) i =0 , ,...,n − ,n ; j =0 ,...,n − ,∂ y W n = det (cid:2) ∂ ix ∂ jy g ( x, y ) (cid:3) i =0 ,...,n − j =0 , ,...,n − ,n ,∂ xy W n = det (cid:2) ∂ ix ∂ jy g ( x, y ) (cid:3) i =0 , ,...,n − ,n ; j =0 , ,...,n − ,n . It follows from Sylvester’s determinant identity [22, p22] that W n ∂ xy W n − ( ∂ x W n )( ∂ y W n ) = W n − W n +1 , proving the first identity. Essentially the same argument shows that, for any k ≥ W n ∂ kx ∂ y W n − ( ∂ kx W n )( ∂ y W n ) = W n − ∂ k − x W n +1 . This implies that ( ∂ y ( ∂ kx W n /W n )) /T n = ( ∂ k − x W n +1 ) /W n +1 . In particular, ( ∂ y ( ∂ nx g/g )) /T = ( ∂ n − x W ) /W , ( ∂ y ( ∂ n − x W /W )) /T = ( ∂ n − x W ) /W , and so on, yielding the second identity.Now, by Lemma 3.5,det (cid:20) ∂ i − x g ( x, y j ) g ( x, y j ) (cid:21) ni,j =1 = Z y n − ≺ y det " ∂ y n − j ∂ ix g ( x, y n − j ) g ( x, y n − j ) n − i,j =1 n − Y i =1 dy n − i = Z y n − ≺ y det ∂ y n − j ∂ ix g ( x,y n − j ) g ( x,y n − j ) T ( x, y n − j ) n − i,j =1 n − Y i =1 T ( x, y n − i ) dy n − i . Applying Lemma 3.5 again, using T = ∂ y ( ∂ x g/g ), we obtaindet ∂ y n − j ∂ ix g ( x,y n − j ) g ( x,y n − j ) T ( x, y n − j ) n − i,j =1 = Z y n − ≺ y n − det ∂ y n − j ∂ y n − j ∂ i +1 x g ( x,y n − j ) g ( x,y n − j ) T ( x, y n − j ) n − i,j =1 n − Y i =1 dy n − i = Z y n − ≺ y n − det ∂ y n − j ∂ yn − j ∂i +1 x g ( x,yn − j ) g ( x,yn − j ) T ( x,y n − j ) T ( x, y n − j ) n − i,j =1 n − Y i =1 T ( x, y n − i ) dy n − i . Now apply Lemma 3.5 again, using T = ∂ y ( ∂ y ( ∂ x g/g ) /T ), and so on, toobtain the third identity. (cid:3) Note that, by Lemma 3.6,(42) a n = ∂ xy log τ n . Thus, if we define, for n ≥ q n = log( τ n /τ n − ) = log u n − log[ t n − ( n − , then a n = e q n +1 − q n and, by (42), the functions q n satisfy the 2D Toda equa-tions ∂ xy q n = e q n +1 − q n − e q n − q n − , n ≥ , with the convention that q = + ∞ .The time-evolution of the a n is given by the following proposition. Proposition 3.7. For n ≥ , (43) ∂ t a n = 12 ∂ y a n + ∂ y [ a n ∂ y log u n ] . Proof of Propositions 3.3 and 3.7. First we note that the initial condition u n (0 , x, y ) = δ ( x − y ) follows immediately from the definition (30) of Z n . We will verify theequations (37) and (43) simultaneously by induction over n . Write h n = log u n and note that (37) is equivalent to ∂ t h n = 12 ∂ y h n + 12 ( ∂ y h n ) + φ ( t, y ) + ∂ y log (cid:0) Z n − /p n − (cid:1) . For n = 1, the equation (37) holds by hypothesis. Thus ∂ t h = 12 ∂ y h n + 12 ( ∂ y h n ) + φ ( t, y ) . MULTI-LAYER EXTENSION OF THE STOCHASTIC HEAT EQUATION 19 It follows that a = ∂ xy h satisfies ∂ t a = 12 ∂ y a + ∂ y [ a ∂ y h ] , as required.Now assume the induction hypothesis (with two parts): ∂ t u n = 12 ∂ y u n + [ φ ( t, y ) + ∂ y log (cid:0) Z n − /p n − (cid:1) ] u n and ∂ t a n = 12 ∂ y a n + ∂ y [ a n ∂ y h n ] . Then, using u n +1 = nta n u n and ∂ y log(1 /p n ) = n/t , ∂ t u n +1 = na n u n + nt [ a n ∂ t u n + u n ∂ t a n ]= 1 t u n +1 + nta n [ 12 ∂ y u n [ φ ( t, y ) + ∂ y log (cid:0) Z n − /p n − (cid:1) ] u n ]+ ntu n [ 12 ∂ y a n + ∂ y [ a n ∂ y h n ]]= 12 ∂ y u n +1 + [ φ ( t, y ) + ∂ y log (cid:0) Z n − /p n − (cid:1) + ∂ y h n + 1 t ] u n +1 = 12 ∂ y u n +1 + [ φ ( t, y ) + ∂ y log ( Z n /p n )] u n +1 , as required. Note that this implies ∂ t h n +1 = 12 ∂ y h n +1 + 12 ( ∂ y h n +1 ) + φ ( t, y ) + ∂ y log ( Z n /p n ) . By (42) we can write a n +1 = a n + ∂ xy h n +1 . Thus, using the second part ofthe induction hypothesis again, ∂ t a n +1 = 12 ∂ y a n +1 + ∂ y [ a n ∂ y h n ] + ∂ y [( a n +1 − a n ) ∂ y h n +1 ] + ∂ y a n . But ∂ y a n = ∂ y [ a n ∂ y log a n ] = ∂ y [ a n ( ∂ y h n +1 − ∂ y h n )] , so we have ∂ t a n +1 = 12 ∂ y a n +1 + ∂ y [ a n +1 ∂ y h n +1 ] , as required. (cid:3) Proof of Theorem 1.2 We return now to the white noise setting, denoting by u ( t, x, y ) the solutionto the stochastic heat equation (1) with initial condition u (0 , x, y ) = δ ( x − y ). In this section we will show that for each n ≥ Z n ( t, x, y ) defined by (9) isconvergent in L ( W ) or, equivalently,(44) ∞ X k =0 Z ∆ k ( t ) Z R k R ( n ) k (( t , x ) , . . . , ( t k , x k )) dx · · · dx k dt · · · dt k < ∞ . The first step is to show that this is equivalent to E e L < ∞ , where L is thetotal intersection local time between two independent copies of the system of n non-intersecting Brownian bridges. Let ( X is , ≤ s ≤ t, i = 1 , . . . , n ) be acollection of non-intersecting bridges which all start at x at time 0 and all endat y at time t , and let and ( Y is , ≤ s ≤ t, i = 1 , . . . , n ) be an independentcopy of X . Define ( L ijs , ≤ s ≤ t ) to be the semimartingale local time processat 0 of ( X i − Y j ) / 2, as defined for example in [40, Chapter VI]. The totalintersection local time is defined by L = L t = P ni,j =1 L ijt . Lemma 4.1. In the above notation, E e L is given by (44) .Proof. We show, by induction on k , that the k th term of (44) is equal to E [ L kt ] /k !.First recall that R ( n )1 (( t , x )) is the sum over i of the marginal probabilitydensities for each X it evaluated at x . Consequently ( R ( n )1 (( t , x )) can beexpressed as the sum over all i and j of the joint density of ( X it , Y jt ) evaluatedat ( x , x ). Integrating over x and t gives an expression which agrees withthat for E [ L ] given by the occupation time formula ( see [40, Chapter 6]).Similarly R ( n ) k (( t , x ) , . . . , ( t k , x k )) is the sum over i , i , . . . , i k of the den-sities of the ( X i t , . . . , X i k t k ) evaluated at ( x , x , . . . , x k ). Consequently( R ( n ) k (( t , x ) , . . . , ( t k , x k ))) can be expressed as the sum over all i , . . . , i k and j , . . . j k of the joint densityof ( X i t , . . . , X i k t k , Y j t , . . . , Y j k t k ) evaluated at( x , x , . . . , x k , x , x , . . . , x k ) . This is integrated over all x i and t i to give an expression for the k th termof (44). On the other hand we can derive the same expression for E [ L kt ]by writing L kt = k R t L k − s dL s , and evaluating the expectation of this usingProposition 3 and Lemma 1 of [19], together with inductive hypothesis. (cid:3) Next will show that, in fact, all exponential moments of L t are finite. Firstnote that L t = A + B , where A is the intersection local time on the timeinterval [0 , t/ 2] and B is the remainder. Thus, by Cauchy-Schwartz, it sufficesto show that A and B each have finite exponential moments of all orders. Now,on the time interval [0 , t/ X = ( X , X , . . . X n )and Y = ( Y , Y , . . . , Y n ) is equivalent to that of two independent copies MULTI-LAYER EXTENSION OF THE STOCHASTIC HEAT EQUATION 21 of Dyson Brownian motion with Radon-Nikodym density a product of twofactors each given by (29) with s = t/ x = x , y = y . By Lemma 2.1, thisRadon-Nikodym density is a bounded random variable. A similar statementholds on [ t/ , t ] after time reversal. It therefore suffices to show that, for twoindependent Dyson Brownian motions the total intersection local time hasfinite exponential moments of all orders. This is established as a special caseof the following lemma which controls the intersection local time for arbitrarystarting points of the Dyson Brownian motions. Proposition 4.2. Let X and Y be independent Dyson Brownian motionsstarting from points X = u and Y = v belonging to Λ n . Their total in-tersection time L t has finite exponential moments of all orders for all t > .Moreover for any β > , and ǫ > one may choose t > small enough thatthat E (cid:2) e βL t (cid:3) < ǫ uniformly for all u , v ∈ Λ n .Proof. The processes X and Y satisfy a system of SDEs X it = u i + β it + X j = i Z t dsX is − X js , Y it = v i + γ it + X j = i Z t dsY is − Y js , where β i , γ i , i = 1 , , . . . , n are a collection of independent standard one-dimensional Brownian motions. Recall this holds even if u and v lie on theboundary of the Weyl chamber. By Cauchy-Schwartz, it suffices to show thatfor each distinct pair i, j , L ijt has finite exponential moments of all orders,each of which can be bounded arbitrarily close to 1, uniformly in u and v , bychoosing t small.Recall that L ij is the local time process of ( X i − Y j ) / L ijt = | X it − Y jt | − | u i − v j | − Z t sgn( X is − Y js ) d ( X is − Y js )= | X it − Y jt | − | u i − v j | − Z t sgn( X is − Y js ) d ( β is − γ js ) − Z t sgn( X is − Y js )( D is − E js ) ds ≤ | X it − u i | + | Y jt − v j | + (cid:12)(cid:12)(cid:12)(cid:12)Z t sgn( X is − Y js ) d ( β is − γ js ) (cid:12)(cid:12)(cid:12)(cid:12) + Z t | D is | ds + Z t | E js | ds, where D is = X j = i X is − X js , E js = X j = i Y is − Y js . Thus, it suffices to show that each of the random variables | X it − u i | , | Y jt − v j | , (cid:12)(cid:12)(cid:12)(cid:12)Z t sgn( X is − Y js ) d ( β is − γ js ) (cid:12)(cid:12)(cid:12)(cid:12) , Z t | D is | ds and Z t | E js | ds, have finite exponential moments of all orders, each of which can be boundedarbitrarily close to 1, uniformly in u and v , by choosing t small.The third of the above random variables is the absolute value of a Gaussianrandom variable with mean zero and variance 2 t and so the desired propertyholds straightforwardly. (To see this, note that the stochastic integral is acontinuous martingale with quadratic variation process 2 t , hence is a Brownianmotion by L´evy’s characterisation theorem [40, Chapter IV, Theorem 3.6].)To control the exponential moments of the first and second of the aboverandom variables, we recall that Dyson’s Brownian motion arises as the pro-cess of eigenvalues of a Brownian motion ( H t , t ≥ 0) in the space of n × n Hermitian matrices. Then ˜ H t = H t − H defines a Hermitian Brownian mo-tion starting from the zero matrix, and applying Weyl’s eigenvalue inqualitiesto the sum H + ˜ H t we deduce that | X it − u i | ≤ σ t where σ t is the spectral radius of ˜ H t . Since the Brownian motion ( ˜ H t , t ≥ u , and the spectral radius of ˜ H t has exponentialmoments that approach 1 as t tends 0, this gives the desired control for | X it − u i | . The same argument applies of course to | Y jt − v j | .The fourth and fifth random variables are essentially the same, so it remainsto show that, for each i , ξ i := Z t | D is | ds has finite exponential moments which can be made arbitrarily close to 1. Wewill prove this by induction over i . For i < j , define ξ ij = Z t X is − X js ds. First we note that ξ = Z t D is ds = X t − u − β t , has finite exponential moments which may be bounded as desired. Now, since ξ = ξ + · · · + ξ n and each term is non-negative, this implies that ξ j ≤ ξ and hence that ξ j has exponential moments satisfying the same bound for each j = 2 , . . . , n . MULTI-LAYER EXTENSION OF THE STOCHASTIC HEAT EQUATION 23 Now ξ = ξ + ξ + · · · + ξ n = Z t D s ds + 2 ξ = X t − u − β t + 2 ξ . Thus ξ and ξ , . . . , ξ n all have exponential moments of all orders which maybe bounded as desired. Similarly, ξ = ξ + ξ + ξ + · · · + ξ n = Z t D s ds +2 ξ +2 ξ = X t − u − β t +2 ξ +2 ξ , the fourth term is handled similarly, and so on. (cid:3) We note that, by the stationarity of space-time white noise, the law of Z n ( t, x, y ) /p ( t, x, y ) n does not depend on x, y and the above proposition im-plies that C t := E | Z n ( t, x, y ) /p ( t, x, y ) n | < ∞ .5. Karlin-McGregor type formula in the white noise setting For n = 1 , , . . . and x , y ∈ Λ ◦ n , define K n ( t, x , y ) = p ∗ n ( t, x , y ) (cid:16) ∞ X k =1 Z ∆ k ( t ) Z R k R ( x , y ) k (( t , x ) , . . . , ( t k , x k ))(45) × W ( dt , dx ) · · · W ( dt k , dx k ) (cid:17) , where R ( x , y ) k is the k -point correlation function for a collection of n non-intersecting Brownian bridges started at positions x = ( x , x , . . . x n ) andending at positions y = ( y , y , . . . , y n ) at time t . Proposition 5.1. The series (45) is convergent in L ( W ) .Proof. We need to show that E [ e L A ] < ∞ where L is the total intersectionlocal time between two independent sets of n independent Brownian bridgesstarted at positions x and ending at positions y at time t , and A is the eventthat each set is non-intersecting. So it suffices to show that E e L < ∞ . Byconsidering pairwise intersection local times and applying H¨older’s inequalityone obtains E e L ≤ E e n √ t/ R where R is the local time at zero of a standardBrownian bridge on [0 , P ( R > r ) = e − r / , r > (cid:3) In fact, Proposition 4.2 yields the following stronger statement. Proposition 5.2. For each t > , there is a constant d t < ∞ such that forall x , y ∈ Λ ◦ n , E | K n ( t, x , y ) | ≤ d t p ∗ n ( t, x , y )∆( x )∆( y ) . Proof. As in the proof of Theorem 1.2, let L be the total intersection local timebetween two independent copies of the system of n non-intersecting Brownianbridges started at positions x and ending at positions y at time t . Then (cf.Lemma 4.1) E | K n ( t, x , y ) | = p ∗ n ( t, x , y ) Ee L . Write L t = A + B , where A is the total intersection local time on the timeinterval [0 , t/ 2] and B is the remainder. By Cauchy-Schwartz, E e L ≤ (cid:0) E e A (cid:1) / (cid:0) E e B (cid:1) / . Now, as explained in Section 2, E e A = ˆ E (cid:20) q n ( t/ , X t/ , y ) q n ( t, x , y ) e A (cid:21) , where ˆ E denotes the expectation with respect to a Dyson Brownian mo-tion started at x . By Lemma 2.1 and Proposition 4.2 this is bounded by d t /q n ( t, x , y ) where d t is a constant independent of x , y . The second term istreated similarly, and the statement of the proposition follows. (cid:3) Before proceeding to the Karlin-McGregor formula, which is the main resultof this section, we recall the approach of Bertini and Cancrini to the stochasticheat equation. In [7] these authors make sense of the formal Feynman-Kacrepresentation (5) by means of smoothing the white noise. For κ > W κ defined by W κ ( t, x ) = Z t Z R δ κ ( x − y ) W ( ds, dy ) , where δ κ ( · ) is the centered Gaussian density of variance 1 /κ . Then the ana-logue of (5) is then meaningful and defines random variables u κ ( t, x, y ).Moreover, for each ( t, x, y ), and any p ≥ u κ ( t, x, y ) → u ( t, x, y ) in L p ( W ) as κ → ∞ . Theorem 5.3. For x , y ∈ Λ ◦ n , K n ( t, x , y ) = det[ u ( t, x i , y j )] ni,j =1 .Proof. Let φ ∈ E , multiply both sides by exp ⋄ W ( φ ) and take expectations.The lefthand side becomes K φn ( t, x , y ), defined earlier by the Feynman-Kacexpression (34). The righthand side becomes C n ( t, x, y ) := E (cid:2) det[ u ( t, x i , y j )] ni,j =1 exp ⋄ ( W ( φ )) (cid:3) , which will we now argue is also given by (34), and since φ ∈ E is arbitrarythe statement of the theorem will follow.Consider the quantity C κn ( t, x , y ) := E (cid:2) det[ u κ ( t, x i , y j )] ni,j =1 exp ⋄ ( W ( φ )) (cid:3) . MULTI-LAYER EXTENSION OF THE STOCHASTIC HEAT EQUATION 25 Replacing each u κ ( t, x i , y j ) by its Feynman-Kac representation, using Fubini,and integrating over W we obtain C κn ( t, x , y ) = X σ sgn( σ ) p n ( t, x , σ y ) E exp (cid:18)Z t ψ κ ( s, B σs ) ds (cid:19) where for each permutation σ , B σ is a bridge of standard n -dimensional Brow-nian motion starting from x and ending at σ y = ( y σ (1) , . . . , y σ ( n ) ), and ψ κ isgiven by ψ κ ( s, z ) = n X i =1 φ k ( s, z i ) + X i For each x , y ∈ Λ ◦ n , K n ( s + t, x , y ) = Z Λ n K n ( s, x , z ) K n ( t, z , y ; s ) d z , almost surely. Consequently, for each x ∈ Λ ◦ n , K n ( t, x , · ) , t > is a Markovprocess taking values in C (Λ ◦ n , R ) .Proof. This follows from Theorem 5.3 using the (generalised) Cauchy-Binetformula [24] together with the corresponding flow property for the solution ofthe stochastic heat equation, namely that for each x, y ∈ R , u ( s + t, x, y ) = Z R u ( s, x, z ) u ( t, z, y ; s ) dz almost surely. (cid:3) We conclude this section with the following. Proposition 5.5. For each x , y ∈ Λ ◦ n , K n ( t, x , y ) ≥ almost surely.Proof. In the above notation, we first claim that for each x , y ∈ Λ ◦ n and κ > u κ ( t, x i , y j )] ni,j =1 > . To see this, we use the Feyman-Kac representation, from [7, (2.17)], u κ ( t, x, y ) = p ( t, x, y ) E F κt ( b )where the expectation is with respect to a Brownian bridge b starting at x and ending at y at time t , and F κs ( b ) , ≤ s ≤ t is an almost surely continuous,strictly positive, multiplicative functional of b . It follows, by a standard path-switching argument (see, e.g., [25, Section 1.2]), that(48) det[ u κ ( t, x i , y j )] ni,j =1 = p ∗ n ( t, x , y ) E n Y i =1 F κt ( X i )where the expectation is with respect to a collection of n non-intersectingBrownian bridges ( X , . . . , X n ) started at positions x and ending at positions y at time t . Indeed, multiplying both sides of (48) by 1 A ( y ), where A is ameasurable subset of Λ n , and integrating with respect to y ∈ Λ n , gives(49) X σ ∈ S n ( − σ E (cid:2) n Y i =1 F κt ( B i ); B ( t ) ∈ σA (cid:3) = E (cid:2) n Y i =1 F κt ( B i ); B ( t ) ∈ A ; T > t (cid:3) , where B is a standard Brownian motion in R n started at x and T is the firstexit time of B from Λ n . To prove (48) it suffices to show that (49) holds forevery measurable A ⊂ Λ n . Let us writeΓ( B ) = n Y i =1 F κt ( B i )and note that (49) is equivalent to X σ ∈ S n ( − σ E (cid:2) Γ( B ); B ( t ) ∈ σA ; T ≤ t (cid:3) = 0 . MULTI-LAYER EXTENSION OF THE STOCHASTIC HEAT EQUATION 27 Now, X σ ∈ S n ( − σ E (cid:2) Γ( B ); B ( t ) ∈ σA ; T ≤ t (cid:3) = n − X i =1 X σ ∈ S n ( − σ E (cid:2) Γ( B ); B ( t ) ∈ σA ; T = T i ≤ t (cid:3) where T i = inf { t ≥ B i ( t ) = B i +1 ( t ) } , so it suffices to show that X σ ∈ S n ( − σ E (cid:2) Γ( B ); B ( t ) ∈ σA ; T = T i ≤ t (cid:3) = 0for each i = 1 , . . . , n − 1. Fix i and define˜ B ( t ) = ( B ( t ) t ≤ T i s i B ( t ) t ≥ T i where s i denotes the adjacent transposition ( i, i + 1). By the strong Markovproperty, ˜ B has the same law as B . Moreover, since F κt is a multiplicativefunctional, we also have Γ( ˜ B ) = Γ( B ). Hence, E (cid:2) Γ( B ); B ( t ) ∈ σA ; T = T i ≤ t (cid:3) = E (cid:2) Γ( ˜ B ); ˜ B ( t ) ∈ s i σA ; T = T i ≤ t (cid:3) = E (cid:2) Γ( B ); B ( t ) ∈ s i σA ; T = T i ≤ t (cid:3) and it follows that X σ ∈ S n ( − σ E (cid:2) Γ( B ); B ( t ) ∈ σA ; T = T i ≤ t (cid:3) = X σ ∈ S n ( − σ E (cid:2) Γ( B ); B ( t ) ∈ s i σA ; T = T i ≤ t (cid:3) = − X σ ∈ S n ( − s i σ E (cid:2) Γ( B ); B ( t ) ∈ s i σA ; T = T i ≤ t (cid:3) = − X σ ∈ S n ( − σ E (cid:2) Γ( B ); B ( t ) ∈ σA ; T = T i ≤ t (cid:3) , as required. The result now follows from (48), letting κ → ∞ . (cid:3) On the evolution of the Z n in the white noise setting In this section we discuss the analogue of Theorem 3.4 in the white noisesetting, and the implication that ( Z ( t, x, · ) , . . . , Z n ( t, x, · )) , t ≥ t > M n ( t, x , y ) = K n ( t, x , y )∆( x )∆( y ) has a version which almost surely extends continuously to a strictly positivefunction on Λ n × Λ n . In particular, for each t > 0, almost surely,(50) Z n ( t, a, b ) = c n,t lim x → a , y → b M n ( t, x , y ) , uniformly on compact intervals. Assuming this continuity it can be shownthat the analogue of Theorem 3.4 holds in the white-noise setting, that is, ifwe set Z = 1 and define, for n ≥ a n = nt Z n − Z n +1 Z n , then, for t > x ∈ R and y ∈ Λ ◦ n ,(51) M n ( t, x , y ) = ∆( y ) − n Y i =1 u ( t, x, y i ) Z GT ( y ) n − Y k =1 n − k Y i =1 a k ( t, x, y n − ki ) dy n − ki . It is not difficult to see (from the flow property described in Corollary 5.4 ofthe previous section) that for each x ∈ R and for each n , the process( M ( t, x , · ) , . . . , M n ( t, x , · )) , t ≥ Z ( t, x, · ) , . . . , Z n ( t, x, · )) , t ≥ M n ( t, x , y )based on Kolmogorov’s criterion would be long and technical. Here we willsatisfy ourselves with a continuous extension in L , which then allows us toprove (51), and hence the Markov property of the multi-layer process, in thespecial case n = 2. Lemma 6.1. For each t > , M n ( t, x , y ) = K n ( t, x , y )∆( x )∆( y ) extends continuously in L ( W ) to Λ n × Λ n . Moreover this extension satisfies Z n ( t, x, y ) = c n,t M n ( t, x , y ) . Proof. First we recall that we have the representation M n ( t, x , y ) = p ∗ n ( t, x , y )∆( x )∆( y ) (cid:16) ∞ X k =1 Z ∆ k ( t ) Z R k R ( x , y ) k (( t , x ) , . . . , ( t k , x k ))(52) × W ( dt , dx ) · · · W ( dt k , dx k ) (cid:17) , where R ( x , y ) k are the correlations functions of a collection of n non-intersectingBrownian bridges starting at x and ending at time t at y . Since p ∗ n ( t, x , y )∆( x )∆( y ) MULTI-LAYER EXTENSION OF THE STOCHASTIC HEAT EQUATION 29 extends continuously to Λ n × Λ n this representation naturally defines theextension of M n ( t, x , y ) to Λ n × Λ n . Our task is show continuity in L ( W ).For this it is enough to show that( x , y , x ′ , y ′ ) E (cid:2) M n ( t, x , y ) M n ( t, x ′ , y ′ ) (cid:3) is continuous. Now, as in the proof of Theorem 1.2 this expectation is equalto p ∗ n ( t, x , y ) p ∗ n ( t, x ′ , y ′ )∆( x )∆( y )∆( x ′ )∆( y ′ ) E [ e L ]where L is the total intersection local time of two independent sets of non-intersecting bridges, X and X ′ say, starting at positions x = ( x , . . . x n ) and x ′ = ( x ′ , x ′ , . . . , x ′ n ) and ending at y = ( y , . . . , y n ) and y ′ = ( y ′ , y ′ , . . . , y ′ n )at time t .Let us write L = L [0 ,δ ] + L [ δ,t − δ ] + L [ t − δ,t ] , where L [0 ,ǫ ] denotes the localtime accrued over the time periods [0 , δ ] and so on. By conditioning on theposition of the bridges at times δ and t − δ we have E (cid:2) exp( L [ δ,t − δ ] ) | ( X (0) , X ′ (0) , X ( t ) , X ′ ( t )) = ( x , x ′ , y , y ′ ) (cid:3) = Z p (( x , x ′ , y , y ′ ) , ( ξ , ξ ′ , η , η ′ )) × E (cid:2) exp( L [ δ,t − δ ] ) | ( X ( δ ) , X ′ ( δ ) , X ( t − δ ) , X ′ ( t − δ )) = ( ξ , ξ ′ , η , η ′ ) (cid:3) d ξ d ξ ′ d η d η ′ . where the kernel p ( · , · ) can be written as a product of transition densities fornon-intersecting Brownian motions, and is thus seen to be continuous. Fromthis it follows by a dominated convergence argument that E (cid:2) exp( L [ δ,t − δ ] ) | ( X (0) , X ′ (0) , X ( t ) , X ′ ( t )) = ( x , x ′ , y , y ′ ) (cid:3) depends continuously on ( x , x ′ , y , y ′ ) also.To deduce the continuity of z E (cid:2) exp( L ) | ( X (0) , X ′ (0) , X ( t ) , X ′ ( t )) = ( x , x ′ , y , y ′ ) (cid:3) we must show that the difference E (cid:2) exp( L ) | ( X (0) , X ′ (0) , X ( t ) , X ′ ( t )) = ( x , x ′ , y , y ′ ) (cid:3) − E (cid:2) exp( L [ δ,t − δ ] ) | ( X (0) , X ′ (0) , X ( t ) , X ′ ( t )) = ( x , x ′ , y , y ′ ) (cid:3) can be made uniformly small for z within compact sets by choosing δ smallenough. Applying the Cauchy-Schwartz inequality this amounts to showingthat E (cid:2) exp(4 L [0 ,δ ] ) | ( X (0) , X ′ (0) , X ( t ) , X ′ ( t )) = ( x , x ′ , y , y ′ ) (cid:3) and E (cid:2) exp(4 L [ t − δ,t ] ) | ( X (0) , X ′ (0) , X ( t ) , X ′ ( t )) = ( x , x ′ , y , y ′ (cid:3) can be made uniformly close to 1. This follows from Proposition 4.2, not-ing that the joint law of X and X ′ over the time interval [0 , δ ] is absolutelycontinuous to that of a pair of independent Dyson Brownian motions, with a density, specified by equation (29) that, by virtue of Lemma 2.1, is boundeduniformly for ( x , x ′ , y , y ′ ) belonging to compact sets. (cid:3) For each t > 0, the continuity in L ( W ) of the mapping ( x , y ) M n ( t, x , y )implies the existence of a version of the stochastic process M n ( t, · , · ) which ismeasurable, see Cohn [13]. Henceforth we will always assume that we areusing this version, and likewise with regard to Z n ( t, · , · ).Recall that K n ( t, z , y ; s ) is defined via the chaos expansion (45) but withthe shifted white noise ˙ W ( s + · , · ), and define M n ( t, z , y ; s ) from it via M n ( t, x , y ; s ) = K n ( t, x , y ; s )∆( x )∆( y ) , Corollary 6.2. For each x , y ∈ Λ n , M n ( s + t, x , y ) = Z Λ n M n ( s, x , z ) M n ( t, z , y ; s )∆( z ) d z , almost surely.Proof. First note that, for x , y ∈ Λ ◦ n this is an immediate consequence of theflow property for K n given by Corollary 5.4.For y ∈ Λ ◦ n , we extend the result to an x ∈ Λ n \ Λ ◦ n by taking a sequence x n of points in Λ ◦ n converging to x . Then, by Lemma 6.1, M n ( s + t, x n , y ) → M n ( s + t, x , y ) , in L ( W ), and hence also in L ( W ). On the otherhand we have, E (cid:12)(cid:12)(cid:12)(cid:12)Z Λ n M n ( s, x n , z ) M n ( t, z , y ; s )∆( z ) d z − Z Λ n M n ( s, x , z ) M n ( t, z , y ; s )∆( z ) d z (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z Λ n E (cid:2) | M n ( s, x n , z ) − M n ( s, x , z ) | (cid:3) E (cid:2) M n ( t, z , y ; s ) (cid:3) ∆( z ) d z , where we have used the independence of M n ( s, x , z ) and M n ( t, z , y ; s ) and thepositivity of M n ( t, z , y ; s ) which follows from Proposition 5.5. Now Lemma6.1 certain implies that the integrand on the righthandside tends to 0 for every z . Moreover, E (cid:2) M n ( s, x n , z ) (cid:3) = p ∗ n ( s, x n , z )∆( x n )∆( z )is uniformly bounded by Lemma 2.1, as is E (cid:2) M n ( s, x , z ) (cid:3) , and Z Λ n E (cid:2) M n ( t, z , y ; s ) (cid:3) ∆( z ) d z = Z Λ n p ∗ n ( t, z , y ) ∆( z )∆( y ) d z = 1 . Consequently, by Dominated Convegence, the integral on the righthandside ofthe displayed inequalities converges to 0 and hence the flow property is provedto extend to x . MULTI-LAYER EXTENSION OF THE STOCHASTIC HEAT EQUATION 31 To further extend the flow property to y ∈ Λ n \ Λ ◦ n , we take y n ∈ Λ ◦ n converging to y , and apply essentially the same arguments again, making useof the result just proved. (cid:3) Theorem 6.3. For x ∈ R and y ∈ Λ ◦ , (53) M ( t, x , y ) u ( t, x, y ) u ( t, x, y ) = 1 y − y Z y y M ( t, x , z ) u ( t, x, z ) dz. Proof. It is known [7] that the solution to the stochastic equation u ( t, x, y )admits a version that is almost surely continuous in t and y and moreover isstrictly positive. We assume in the following that we are using this version. Inparticular, having fixed t , x and y > y we let A ǫ ( x ) be the event { u ( t, x, z ) >ǫ for all z ∈ [ y , y + 1] } . Then as ǫ ↓ P ( A ǫ ( x )) ↑ x , y ∈ Λ ◦ , M ( t, x , y ) = 1∆( x )∆( y ) [ u ( t, x , y ) u ( t, x , y ) − u ( t, x , y ) u ( t, x , y )] . Hence,(54) M ( t, x , y ) u ( t, x , y ) u ( t, x , y ) = 1∆( x )∆( y ) (cid:20) u ( t, x , y ) u ( t, x , y ) − u ( t, x , y ) u ( t, x , y ) (cid:21) . Writing y = ( z + h, z ) where h > M ( t, x , ( z + h, z )) u ( t, x , z + h ) u ( t, x , z ) = 1( x − x ) h (cid:20) u ( t, x , z + h ) u ( t, x , z + h ) − u ( t, x , z ) u ( t, x , z ) (cid:21) . Integrating this equation with respect to z over the interval [ y , y ] we obtain Z y y M ( t, x , ( z + h, z )) u ( t, x , z + h ) u ( t, x , z ) dz = 1( x − x ) h (cid:20)Z y + hy u ( t, x , z ) u ( t, x , z ) dz − Z y + hy u ( t, x , z ) u ( t, x , z ) dz (cid:21) . Now let h tend to zero. By the continuity of u ( t, x , · ) and u ( t, x , · ) the RHSconverges almost surely to1( x − x ) (cid:20) u ( t, x , y ) u ( t, x , y ) − u ( t, x , y ) u ( t, x , y ) (cid:21) . We want to identify the limit of the LHS. Consider E = (cid:12)(cid:12)(cid:12)(cid:12)Z y y M ( t, x , ( z + h, z )) u ( t, x , z + h ) u ( t, x , z ) dz − Z y y M ( t, x , z ) u ( t, x , z ) dz (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z y y | M ( t, x , ( z + h, z )) − M ( t, x , z ) | u ( t, x , z ) dz + Z y y | u ( t, x , z + h ) − u ( t, x , z ) | M ( t, x , ( z + h, z )) u ( t, x , z ) u ( t, x , z + h ) dz. We have(56) E (cid:2) E ; A ǫ ( x ) (cid:3) ≤ ǫ − Z y y E | M ( t, x , ( z + h, z )) − M ( t, x , z ) | dz + ǫ − Z y y (cid:0) E [ M ( t, x , ( z + h, z )) ] E [( u ( t, x , z + h ) − u ( t, x , z )) ]) / (cid:1) dz By virtue of the uniform continuity in L of the mappings ( z , z ) M ( t, x , ( z , z ))and z u ( t, x , z ) these integrals tend to zero as h ↓ 0, and consequently E tends to 0 in probability. Thus we have proven(57) 1( x − x ) (cid:20) u ( t, x , y ) u ( t, x , y ) − u ( t, x , y ) u ( t, x , y ) (cid:21) = Z y y M ( t, x , z )) u ( t, x , z ) dz. Next let x = ( x + h, x ) and let h ↓ 0. The LHS of (57) can be rewritten as( y − y ) M ( t, ( x + h, x ) , ( y , y )) u ( t, x, y ) u ( t, x, y ) ;as h ↓ y − y ) M ( t, x , ( y , y )) u ( t, x, y ) u ( t, x, y ) . On the other hand, if we consider F = (cid:12)(cid:12)(cid:12)(cid:12)Z y y M ( t, ( x + h, x ) , z )) u ( t, x, z ) dz − Z y y M ( t, x , z )) u ( t, x, z ) dz (cid:12)(cid:12)(cid:12)(cid:12) we have E (cid:2) F ; A ǫ ( x ) (cid:3) ≤ ǫ − Z y y E | M ( t, ( x + h, x ) , z ) − M ( t, x , z ) | dz which again by the L continuity of M converges to 0 as h ↓ 0. From this itfollows the RHS of (57) converges to Z y y M ( t, x , z )) u ( t, x, z ) dz, as required. (cid:3) We remark that the identity (57) shows that the ratio of two solutions tothe stochastic heat equation is in H ; in fact, such ratios have recently beenshown (in a slightly different setting) by Hairer [21] to be in C / − ǫ . Corollary 6.4. For each x ∈ R , the process ( Z ( t, x, · ) , Z ( t, x, · )) , t ≥ has the Markov property. MULTI-LAYER EXTENSION OF THE STOCHASTIC HEAT EQUATION 33 Proof. Fix times 0 ≤ s < t . Suppose that F = F ( Z ( t, x, · ) , Z ( t, x, · )) is abounded random variable (depending on the random fields at a finite numberof points). We wish to show that the conditional expectation given the whitenoise W [0 ,s ] of this random variable is measurable with respect to the randomfields ( Z ( s, x, · ) , Z ( s, x, · )). To see this, note firstly that by Lemma 6.1 thesame random variable F is a function of the fields ( M ( t, x , · ) , M ( t, x , · )).Now the flow property for M n obtained in Corollary 6.2 together with theindependence of M ( t, · , · ; s ) from W [0 ,s ] implies the conditional expectation E (cid:2) F | W [0 ,s ] (cid:3) has a version G of the form G = G ( M ( s, x , · ) , M ( s, x , · )).We have M ( s, x , · ) is proportional Z ( t, x, · ), and more profoundly, by thepreceeding Theorem, M ( s, x , · )) can be expressed in terms of Z ( t, x, · ) and Z ( t, x, · ). Thus we see that G is of the required form. (cid:3) It would be interesting to understand the evolution of the multi-layer pro-cess in terms of a system of stochastic partial differential equations. Motivatedby the evolution equations obtained in Section 2 in the the case of a smoothpotential, it is natural to consider and to try to make sense of the system ofequations(58) ∂ t a n = 12 ∂ y a n + ∂ y [ a n ∂ y log u n ] , where Z n = u · · · u n . For recent progress in this direction, in the case ofsmooth initial data and periodic boundary conditions, see [21]. References [1] T. Alberts, K. Khanin and J. Quastel. The intermediate disorder regime for directedpolymers in dimension 1 + 1. Phys. Rev. Lett. J.Stat. Phys. 154 (2014) 305–326.[3] G. Amir, I. Corwin and J. Quastel. Probability distribution of the free energy of thecontinuum directed random polymer in 1+1 dimensions. Comm. Pure Appl. Math. J. Phys. A 35 (2002) L389–L399.[5] D.J. Arrigo and F. Hickling. 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