Stochastic Lagrangian Transport and Generalized Relative Entropies Peter Constantin Department of Mathematics, The University of Chicago 5734 S. University Avenue, Chicago, Illinois 6637 Gautam Iyer Department of Mathematics, Stanford University Bldg. 38, 45 Serra Mall, Stanford, CA 9434 August 31, 26 Abstract We discuss stochastic representations of advection diffusion equations with variable diffusivity, stochastic integrals of motion and generalized relative entropies. Keywords: Relative entropies, stochastic integrals of motion, stochastically passive scalars, stochastic Lagrangian transport. AMS - MSC numbers: 35K45, 6H3. 1 Introduction Recently, Michel, Mischler and Perthame [8] discovered a remarkable property of certain unstable linear equations, in which decay of relative entropies takes place. Their observation was applied to population dynamics models, but the list of applications is growing. Of course, relative entropies have been used for a long time in kinetic theory and conservation laws. However, 1
the decay of relative entropies, was known before only in stable, self-adjoint situations in which a global attracting steady solution exists and no flow advection is present [9]. The property of decay of relative entropies was slightly generalized to variable diffusion coefficients and applied to Smoluchowski systems in [1]. A stochastic interpretation and proof in the case of constant diffusion coefficients was given in [2]. Here we provide a stochastic interpretation and proof in the case of variable diffusion coefficients. The method of proof and concepts are of more general interest [3, 4]. We consider a linear operator in R n, where Dρ = ν i (a ij j ρ) div x (Uρ) + V ρ (1) U(x, t) = (U j (x, t)) j=1,...n (2) is a smooth (C 2 ) function, V = V (x, t) is a continuous and bounded scalar potential and a ij (x, t) = σ ip (x, t)σ jp (x, t) (3) with the matrix σ(x, t) = (σ ij (x, t)) ij (4) a given smooth (C 2 ) matrix. We assume that σ is bounded and U and x σ decay at infinity. We use the shorthand notation A(D) for the operator and use also the non-divergence form where and A(D)ρ = a ij i j ρ (5) Dρ = νa(d)ρ u x ρ + P ρ (6) u j (x, t) = U j (x, t) ν i (a ij (x, t)) (7) The formal adjoint of the operator D in L 2 (R n ) is P = V div x (U). (8) D φ = ν i (a ij j φ) + U x φ + V φ. (9) The following is the result of Michel, Mischler and Perthame: 2
Theorem 1 [7, 8] Let f be a solution of t f = Df (1) and let ρ > be a positive solution of the same equation, t ρ = Dρ. (11) Let H be a smooth convex function of one variable and let φ be a non-negative function obeying pointwise t φ + D φ =. (12) Then d dt H ( ) f φρdx. (13) ρ 2 Stochastic Lagrangian Flow In order to represent solutions of equations like (1) we consider the drift v j (x, t) = u j + 2ν( k σ jp )σ kp = U j ν( k σ kp )σ jp + ν( k σ jp )σ kp. (14) Let X(a, t) be the strong solution of the stochastic differential system with initial data dx j (t) = v j (X, t)dt + 2νσ jp (X, t)dw p (15) X(a, ) = a. (16) Here W is a standard Brownian process in R n starting at time zero from the origin. This process will be fixed throughout the paper and all measurability issues will be with respect to the filtration associated to it and all almost sure statements will be with respect to the probability measure on the standard Wiener space. We will need the following result: Theorem 2 The inverse of the flow map a X(a, t), the stochastic map exists almost surely and satisfies its defining relations x A(x, t) (17) X(A(x, t), t) = x, x R n, A(X(a, t), t) = a, a R n, t, a.s. 3
The map X is smooth and the determinant obeys the SDE with D(a, t) = det ( a X(a, t)) (18) d(det ( a X(a, t)) = [det( a X(a, t))] {[(div x v)(x, t) + 2νE(x, t)] x=x(a,t) dt + 2ν( k (σ kp ))(x, t) x=x(a,t) dw p } E(x, t) = i<j (19) det( i σ jp ) ij. (2) The map A(x, t) satisfies the stochastic partial differential system p da j + (u x A j νa(d)a j ) dt + 2ν( k A j )σ kp dw p = (21) with initial data A(x, ) =. Remark. In the statement above, det( i σ jp ) ij refers to the determinant of the two-by-two matrix ( r σ kp ) with r, k {i, j} for fixed i < j and p. Theorem 2 was originally proved in [3] for constant coefficients and in [4] for variable coefficients. For completeness, we reproduce the proof (with variable coefficients, as stated above) in Appendix A. 3 Stochastically Passive Scalars and Feynman-Kac Formula We consider first deterministic smooth time-independent functions f and note that the functions θ = θ f (x, t) = f (A(x, t)) are stochastically passive in the sense that they obey the equation with initial data dθ + (u x θ νa(d)θ) dt + 2ν k θσ kp dw p = (22) θ(x, ) = f(x). (23) Solutions of the SPDE (22) form an algebra; in particular, products of solutions are solutions, a nontrivial fact due to the presence of the stochastic 4
term. The expected values of these scalars obey advection-diffusion equations and do not form an algebra in general, if ν >. We consider now the function { } I(a, t) = exp P (X(a, s), s)ds (24) where P (x, t) is given in (8) and consider the function We have ψ = ψ f (x, t) = θ f (x, t)i(a(x, t), t) (25) Theorem 3 The process ψ = ψ f given by { } ψ(x, t) = f (A(x, t)) exp P (X(a, s), s)ds a=a(x,t) solves with initial datum ψ(x, ) = f (x). (26) dψ (Dψ) dt + 2ν x ψσdw = (27) The proof of this result follows using stochastic calculus [5], [6]. Indeed, the function I(a, t) obeys t I(a, t) = P (X(a, t), t)i(a, t) (28) pathwise (almost surely). Then, a calculation using (21) (see [3], [4]) shows that the function J(x, t) = I(A(x, t), t) (29) solves dj + (u x J P J νa(d)j)dt + 2ν x JσdW =. (3) The function ψ f is the product and therefore, from Itô s formula ψ f = θ f J, dψ f = Jdθ + θdj + d J, θ 5
and the equations obeyed by J, θ, we have dψ f = ( u x ψ f + P ψ f + νja(d)θ + νθa(d)j + 2ν( k J)σ kp ( j θ)σ jp )dt 2ν x ψ f σdw. This means dψ = ( u x ψ + P ψ + νa(d)ψ)dt 2ν x ψ f σ dw. Because of (6) we have (27). 4 Stochastic Integrals of Motion. Proposition 1 Consider a deterministic function φ that solves (12). Then the function { } M(a, t) = φ(x(a, t), t) det ( a X(a, t)) exp P (X(a, s), s)ds (31) is a martingale. Proof. We start by writing with M(a, t) = Φ(a, t)i(a, t)d(a, t) Φ(a, t) = φ(x(a, t), t), I given above in (24) and D given in (18). Next, we compute the equation obeyed by ΦI. In view of (28) and using Itô s formula we have d(φi) = I {( t φ(x(a, t), t) + P φ(x(a, t), t)) dt + + x φ X(a,t) dx + 1 2 i j φ X(a,t) d X i, X j }, which gives, in view of (15) d(φi) = = { t φ + P φ + v x φ + νa(d)φ} X(a,t) dt + 2νI(( i φ)σ ip ) X(a,t) dw p. 6
Using (12) and (14) we have d(φi) = I { 2ν( k (σ kp ))σ jp ( j φ) (div x U)φ} X(a,t) dt+ + 2νI(( j φ)σ jp ) X(a,t) dw p. (32) Now, by Itô, In view of (19) and (32) we have dm = Dd(ΦI) + ΦID + d D, ΦI. d D, ΦI = 2νDI {( k σ kp )σ jp ( j φ)} X(a,t) dt and consequently the terms ±2νDI( k σ kp )σ jp ( j φ)dt cancel and we obtain dm = ID { (div x U)φ + (div x v + 2νE)} X(a,t) dt+ + 2νID {σ jp ( j φ) + ( k σ kp )} X(a,t) dw p. Now, in view of (14) we have that (div x v) (div x U) = ν j [( k σ jp )σ kp ] ν j [( k σ kp )σ jp ] and therefore the coefficient of dt in dm is Now DI {2νE + ν j [( k σ jp )σ kp ] ν j [( k σ kp )σ jp ]} DI {ν j [( k σ jp )σ kp ] ν j [( k σ kp )σ jp ]} = DI {ν( k σ jp )( j σ kp ) ν( k σ kp )( j σ jp )} = = DI2 k<j p {ν( kσ jp )( j σ kp ) ν( k σ kp )( j σ jp )} = = 2νE and therefore the coefficient of dt in dm vanishes. We obtained that is, M is the martingale dm = 2νID {σ jp ( j φ) + ( k σ kp )} X(a,t) dw p, (33) M(a, t) = φ(a, ) + 2ν I(a, s)d(a, s) {σ jp( j φ) + ( k σ kp )} X(a,s) dw p (s) 7
Theorem 4 Let h and ρ be smooth time independent deterministic functions. Consider the stochastically passive scalar θ h (x, t) = h (A(x, t)) and the process ψ ρ of (26) with initial datum ρ. Consider also φ(x, t), a deterministic solution of (12). Then the random variable E(t) = φ(x, t)ψ ρ (x, t)θ h (x, t)dx R n (34) is a martingale. In particular E(E(t)) = φ(a, )ρ (a)h (a)da R n (35) holds. Proof. In view of the change of variables formula and the definition of ψ ρ we have that E(t) = M(a, t)ρ (a)h (a)da (36) R n with M given in (31). The result follows then from the previous proposition. More precisely de = 2ν exp { R { s P (X(a, τ), τ)dτ n a=a(x,s)} {σjp ( j φ) + ( k σ kp )} dx } dw p (37) gives explicitly the SDE obeyed by E. 5 Generalized Relative Entropies We take now a smooth deterministic, time independent function H of one variable, a deterministic solution of (12), two smooth deterministic, time independent functions f and ρ, of which ρ is strictly positive. We form the processes ψ ρ and ψ f given by the expressions (26). Then it t follows that ( ) ψf (x,t) ψ ρ (x, t)φ(x, t)h ψ ρ = (x,t) ( ) ψ ρ (x, t)φ(x, t)h f (A(x,t)) ρ (A(x,t)) 8
( ψf holds. Thus, the quantity of interest, ψ ρ φh ψ ρ ), is the product of a stochastically passive scalar, ψ ρ and φ. By the previous theorem we have that ( ) ψf (x, t) E(t) = ψ ρ (x, t)h φ(x, t)dx (38) ψ ρ (x, t) is a martingale. The expected value is then constant in time: { ( ) } d dt E ψf ψ ρ H φdx =. (39) If we denote and ψ ρ f(x, t) = Eψ f (x, t) (4) ρ(x, t) = Eψ ρ (x, t) (41) we have from (27) that f solves (1), ρ > solves (11). We prove that we have (13). The starting point is (39). In view of (4) and (41), the statement that needs to be proved is E (ψ ρ ) H ( ) { E(ψf ) φdx E E(ψ ρ ) ψ ρ H ( ψf ψ ρ ) } φdx (42) The conservation (39) works for any H, but we expect (42) to hold only for convex H. Indeed, (42) can be reduced to a Jensen inequality. We claim more, that for all x, t we have E (ψ ρ ) H Considering the functions and we see that (43) becomes ( E(ψf ) E(ψ ρ ) ) H (E(v)) E g = ψ ρ E(ψ ρ ) v = ψ f E(ψ ρ ) 9 { E ψ ρ H { gh ( ψf ψ ρ )} (43) (44) (45) ( )} v. (46) g
This, however, is nothing but Jensen s inequality for the probability measure H ( P P h = E(gh), ( )) v P H g A Proof of Theorem 2 ( ) v. g We devote this appendix to proving Theorem 2. The original proof can be found in [3] for constant coefficients, and in [4] for variable coefficients. Lemma 1 Let X be the stochastic flow defined by (15), (16). Then the map X is spatially smooth (almost surely), and the determinant D = det( X) satisfies the equation dd = D [( v + 2νE) dt + ] 2ν k σ kp dw p where E = 1 2 [ iσ ip j σ jp j σ ip i σ jp ]. Proof. Differentiating (15) we have d( a X j ) = k v j a X k dt + 2ν k σ jp a X k dw p. (47) Let S n be the permutation group on n symbols, and ɛ τ denote the signature of the permutation τ S n. By Itô s formula, dd = [ ɛ τ c X τc d ( b X τb ) + ] d X τd d b X τb, c X τc τ S n c b c<b d b,c b=1...n = [ ɛ τ k v τb b X k c X τc dt + 2ν k σ τb,p b X k c X τc dw p + τ S n c b c b b=1...n + ν ] b X l l σ τb,p c X m m σ τc,p d X τd dt (48) c b d b,c 1
We compute each of the terms above individually: ɛ τ k v τb b X k ɛ τ τb v τb b X τb c X τc + τ S n b=1...n c b c X τc = τ S n b=1...n + τ S n b=1...n c b ɛ τ k v τb b X k τ 1 k k τ b = ( v) det( X)+ + b=1...n k b X k τ S n ɛ τ τk v τb b X τk k X τk = ( v) det( X) + c b,τ 1 k c b,k c X τc c X τc The second term above is zero because replacing replacing τ with τ (b k) in the inner sum produces a negative sign. Similarly we have ɛ τ k σ τb,p b X k c X τc dw p = k σ k,p det( X) dw p. τ S n b=1...n c b For the last term in (48), the only difference is that we have a few extra cases to consider: When l = τ(b), m = τ(c), we will get det( X) i σ ip j σ jp. When l = τ(c) and m = τ(b), we will get det( X) j σ ip i σ jp. In all other cases we get. This concludes proof of Lemma 1. Lemma 2 For any time t, the map X t has a (spatially) smooth inverse. Proof. Define λ by [ ( λ = exp v + 2νE ν( k σ kp ) 2) dt + 2ν ] k σ kp dw s (p) The Itô s formula immediately shows that λ satisfies equation (19). Since (19) is a linear SDE with smooth coefficients, uniqueness of the solution guarantees D = exp(λ) almost surely, and hence D > almost surely. The spatial invertibility of X now follows as X t is locally orientation preserving and has degree 1 (because X t is properly homotopic to X, the identity map). The (spatial) smoothness of the inverse is guaranteed by the inverse function theorem. 11
The above lemma shows existence of a spatial inverse of X. As before, we let A denote the spatial inverse of X. We now derive a stochastic evolution equation of A [equation (21)]. Lemma 3 Let Y be a C 1 stochastic flow of semi-martingales adapted to F t, the filtration of W t. If for all a R n, t > we have then Y (X s (a), ds) = b(x s (a), s) ds + Y t (a) = Y (a) + b(a, t) dt + σ (X s (a), s)dw s σ (a, t) dw t. Proof. Let Y be the process defined by Y t (a) = Y (a) + b(a, t) dt + σ (a, t) dw t, and set δ = Y Y. Since δ is adapted to F t, there exists a non-negative predictable function a such that a(x, y, s) ds = δ(x), δ(y) t. Now, by definition of the generalized Itô integral we have δ(x s, s) almost surely, and hence a(x s, X s, s) ds almost surely. Since X is a flow of homeomorphisms (diffeomorphisms actually), we must have t, a(x, x, t) almost surely. Thus δ = Y Y is of bounded variation. Since we have shown above that δ has bounded variation, δ(x s, ds) = t δ Xs,s ds and hence t, t δ t. At time, δ by definition, and hence δ t almost surely for all t, concluding the proof. Lemma 4 There exists a process B of bounded variation such that A t = B t 2ν 12 ( A s )σ dw s (49)
Proof. Applying the generalized Itô formula to A X we have = = A(X s, ds) + A dx Xs,s s + 1 2 ija 2 d Xs,s X (i), X (j) + t t t s + i A(X s, ds), X (i) t X (i) t t [ A(X s, ds) + A v + νa Xs,s ij ija 2 Xs,s t t + 2ν A σ dw Xs,s s + t t ] ds+ (5) i A(X s, ds), X (i) t X (i) t. Notice that the second and fourth terms on the right are of bounded variation. Applying Lemma 3 we conclude the proof. Lemma 5 The process A satisfies the equation da t +(v )A t dt νa ij 2 ija t dt 2ν j A t ( i σ jk )σ ik dt+( A t )σ dw t = (51) Proof. Since the joint quadratic variation term in (5) depends only on the martingale part of i A, we can compute it explicitly by ( ) i A(X s, ds), X (i) = 2ν 2 ij A s σ jk σ ik + j A s ( i σ jk )σ ik Xs ds t = 2ν t t t ( aij 2 ija s + j A s ( i σ jk )σ ik ) Xs ds. Substituting (52) in (5) and applying Lemma 3 we conclude the proof. Acknowledgment. PC partially supported by NSF grant DMS-54213. (52) References [1] P. Constantin, Smoluchowski Navier-Stokes Systems, Proc. AMS conference (25), to appear. [2] P. Constantin, Generalized relative entropies and stochastic representation, IMRN (26) (to appear.) 13
[3] P. Constantin, G. Iyer, A stochastic Lagrangian representation of the 3d incompressible Navier-Stokes equations, Commun. Pure Applied Math, to appear (26). [4] G. Iyer, Ph.D Thesis, The University of Chicago, (26). [5] I. Karatzas, S. E. Shreve, Brownian motion and Stochastic Calculus, Graduate Texts in Mathematics 113 (1991), Springer-Verlag, New York. [6] H. Kunita, Stochastic flows and stochastic differential equations, Cambridge studies in advanced mathematics, 24 (199), Cambridge University Press, Cambridge. [7] B. Perthame, Talk at Northwestern University, June 25. [8] P. Michel, S. Mischler, B. Perthame, General entropy equations for structured population models and scattering, C.R. Acad. Sci. Paris, Ser. I 338 (24), 697-72. [9] C. Villani Topics in Optimal Transportation Graduate studies in Mathematics 58 (23), American Mathematical Society, Providence, Rhode Island. 14