THE VARIATIONAL FORMULATION OF THE FOKKER PLANCK EQUATION

Transcription

1 SIAM J. M A T H.ANAL. c 998 Society for Industrial and Applied Matematics Vol. 29, No., pp. 7, January THE VARIATIONAL FORMULATION OF THE FOKKER PLANCK EQUATION RICHARD JORDAN, DAVID KINDERLEHRER, AND FELIX OTTO In memory of Ricard Duffin Abstract. Te Fokker Planck equation, or forward Kolmogorov equation, describes te evolution of te probability density for a stocastic process associated wit an Ito stocastic differential equation. It pertains to a wide variety of time-dependent systems in wic randomness plays a role. In tis paper, we are concerned wit Fokker Planck equations for wic te drift term is given by te gradient of a potential. For a broad class of potentials, we construct a time discrete, iterative variational sceme wose solutions converge to te solution of te Fokker Planck equation. Te major novelty of tis iterative sceme is tat te time-step is governed by te Wasserstein metric on probability measures. Tis formulation enables us to reveal an appealing, and previously unexplored, relationsip between te Fokker Planck equation and te associated free energy functional. Namely, we demonstrate tat te dynamics may be regarded as a gradient flux, or a steepest descent, for te free energy wit respect to te Wasserstein metric. Key words. Fokker Planck equation, steepest descent, free energy, Wasserstein metric AMS subject classifications. 35A5, 35K5, 35Q99, 60J60 PII. S Introduction and overview. Te Fokker Planck equation plays a central role in statistical pysics and in te study of fluctuations in pysical and biological systems [7, 22, 23]. It is intimately connected wit te teory of stocastic differential equations: a (normalized) solution to a given Fokker Planck equation represents te probability density for te position (or velocity) of a particle wose motion is described by a corresponding Ito stocastic differential equation (or Langevin equation). We sall restrict our attention in tis paper to te case were te drift coefficient is a gradient. Te simplest relevant pysical setting is tat of a particle undergoing diffusion in a potential field [23]. It is known tat, under certain conditions on te drift and diffusion coefficients, te stationary solution of a Fokker Planck equation of te type tat we consider ere satisfies a variational principle. It minimizes a certain convex free energy functional over an appropriate admissible class of probability densities [2]. Tis free energy functional satisfies an H-teorem: it decreases in time for any solution of te Fokker Planck equation [22]. In tis work, we sall establis a deeper, and apparently previously unexplored, connection between te free energy functional and te Fokker Planck dynamics. Specifically, we sall demonstrate tat te solution of te Received by te editors May 3, 996; accepted for publication (in revised form) December 9, 996. Te researc of all tree autors is partially supported by te ARO and te NSF troug grants to te Center for Nonlinear Analysis. In addition, te tird autor is partially supported by te Deutsce Forscungsgemeinscaft (German Science Foundation), and te second autor is partially supported by grants NSF/DMS and DAAL ttp:// Center for Nonlinear Analysis, Carnegie Mellon University. Present address: Department of Matematics, University of Micigan (jordan@mat.lsa.umic.edu). Center for Nonlinear Analysis, Carnegie Mellon University (davidk@andrew.cmu.edu). Center for Nonlinear Analysis, Carnegie Mellon University and Department of Applied Matematics, University of Bonn. Present address: Courant Institute of Matematical Sciences (otto@cims.nyu.edu).

2 2 R. JORDAN, D. KINDERLEHRER, AND F. OTTO Fokker Planck equation follows, at eac instant in time, te direction of steepest descent of te associated free energy functional. Te notion of a steepest descent, or a gradient flux, makes sense only in context wit an appropriate metric. We sall sow tat te required metric in te case of te Fokker Planck equation is te Wasserstein metric (defined in section 3) on probability densities. As far as we know, te Wasserstein metric cannot be written as an induced metric for a metric tensor (te space of probability measures wit te Wasserstein metric is not a Riemannian manifold). Tus, in order to give meaning to te assertion tat te Fokker Planck equation may be regarded as a steepest descent, or gradient flux, of te free energy functional wit respect to tis metric, we switc to a discrete time formulation. We develop a discrete, iterative variational sceme wose solutions converge, in a sense to be made precise below, to te solution of te Fokker Planck equation. Te time-step in tis iterative sceme is associated wit te Wasserstein metric. For a different view on te use of implicit scemes for measures, see [6, 6]. For te purpose of comparison, let us consider te classical diffusion (or eat) equation ρ(t, x) t = ρ(t, x), t (0, ), x, wic is te Fokker Planck equation associated wit a standard n-dimensional Brownian motion. It is well known (see, for example, [5, 24]) tat tis equation is te gradient flux of te Diriclet integral 2 ρ 2 dx wit respect to te L 2 ( ) metric. Te classical discretization is given by te sceme Determine ρ (k) tat minimizes 2 ρ(k ) ρ 2 L 2 ( ) + 2 ρ 2 dx over an appropriate class of densities ρ. Here, is te time step size. On te oter and, we derive as a special case of our results below tat te sceme Determine ρ (k) tat minimizes () 2 d(ρ(k ), ρ) 2 + ρ log ρ dx over all ρ K, were K is te set of all probability densities on aving finite second moments, is also a discretization of te diffusion equation wen d is te Wasserstein metric. In particular, tis allows us to regard te diffusion equation as a steepest descent of te functional ρ log ρ dx wit respect to te Wasserstein metric. Tis reveals a novel link between te diffusion equation and te Gibbs Boltzmann entropy ( ρ log ρ dx) of te density ρ. Furtermore, tis formulation allows us to attac a precise interpretation to te conventional notion tat diffusion arises from te tendency of te system to maximize entropy. Te connection between te Wasserstein metric and dynamical problems involving dissipation or diffusion (suc as strongly overdamped fluid flow or nonlinear diffusion equations) seems to ave first been recognized by Otto in [9]. Te results in [9] togeter wit our recent researc on variational principles of entropy and free energy type for measures [2,, 5] provide te impetus for te present investigation. Te work in [2] was motivated by te desire to model and caracterize metastability

3 VARIATIONAL FORMULATION OF THE FOKKER PLANCK EQUATION 3 and ysteresis in pysical systems. We plan to explore in subsequent researc te relevance of te developments in te present paper to te study of suc penomena. Some preliminary results in tis direction may be found in [3, 4]. Te paper is organized as follows. In section 2, we first introduce te Fokker Planck equation and briefly discuss its relationsip to stocastic differential equations. We ten give te precise form of te associated stationary solution and of te free energy functional tat tis density minimizes. In section 3, te Wasserstein metric is defined, and a brief review of its properties and interpretations is given. Te iterative variational sceme is formulated in section 4, and te existence and uniqueness of its solutions are establised. Te main result of tis paper namely, te convergence of solutions of tis sceme (after interpolation) to te solution of te Fokker Planck equation is te topic of section 5. Tere, we state and prove te relevant convergence teorem. 2. Te Fokker Planck equation, stationary solutions, and te free energy functional. We are concerned wit Fokker Planck equations aving te form (2) ρ t = div ( Ψ(x)ρ)+β ρ, ρ(x, 0) = ρ 0 (x), were te potential Ψ(x) : [0, ) is a smoot function, β > 0 is a given constant, and ρ 0 (x) is a probability density on. Te solution ρ(t, x) of (2) must, terefore, be a probability density on for almost every fixed time t. Tat is, ρ(t, x) 0 for almost every (t, x) (0, ), and ρ(t, x) dx = for almost every t (0, ). It is well known tat te Fokker Planck equation (2) is inerently related to te Ito stocastic differential equation [7, 22, 23] (3) dx(t) = Ψ(X(t))dt + 2β dw (t), X(0) = X 0. Here, W (t) is a standard n-dimensional Wiener process, and X 0 is an n-dimensional random vector wit probability density ρ 0. Equation (3) is a model for te motion of a particle undergoing diffusion in te potential field Ψ. X(t) ten represents te position of te particle, and te positive parameter β is proportional to te inverse temperature. Tis stocastic differential equation arises, for example, as te Smolucowski Kramers approximation to te Langevin equation for te motion of a cemically bound particle [23, 4, 7]. In tat case, te function Ψ describes te cemical bonding forces, and te term 2β dw (t) represents wite noise forces resulting from molecular collisions [23]. Te solution ρ(t, x) of te Fokker Planck equation (2) furnises te probability density at time t for finding te particle at position x. If te potential Ψ satisfies appropriate growt conditions, ten tere is a unique stationary solution ρ s (x) of te Fokker Planck equation, and it takes te form of te Gibbs distribution [7, 22] (4) ρ s (x) =Z exp( βψ(x)), were te partition function Z is given by te expression Z = exp( βψ(x)) dx. Note tat, in order for equation (4) to make sense, Ψ must grow rapidly enoug to ensure tat Z is finite. Te probability measure ρ s (x) dx, wen it exists, is te unique

4 4 R. JORDAN, D. KINDERLEHRER, AND F. OTTO invariant measure for te Markov process X(t) defined by te stocastic differential equation (3). It is readily verified (see, for example, [2]) tat te Gibbs distribution ρ s satisfies a variational principle it minimizes over all probability densities on te free energy functional (5) F (ρ) =E(ρ)+β S(ρ), were (6) E(ρ) := Ψρ dx plays te role of an energy functional, and (7) S(ρ) := ρ log ρ dx is te negative of te Gibbs Boltzmann entropy functional. Even wen te Gibbs measure is not defined, te free energy (5) of a density ρ(t, x) satisfying te Fokker Planck equation (2) may be defined, provided tat F (ρ 0 ) is finite. Tis free energy functional ten serves as a Lyapunov function for te Fokker Planck equation: if ρ(t, x) satisfies (2), ten F (ρ(t, x)) can only decrease wit time [22, 4]. Tus, te free energy functional is an H-function for te dynamics. Te developments tat follow will enable us to regard te Fokker Planck dynamics as a gradient flux, or a steepest descent, of te free energy wit respect to a particular metric on an appropriate class of probability measures. Te requisite metric is te Wasserstein metric on te set of probability measures aving finite second moments. We now proceed to define tis metric. 3. Te Wasserstein metric. Te Wasserstein distance of order two, d(µ,µ 2 ), between two (Borel) probability measures µ and µ 2 on is defined by te formula (8) d(µ,µ 2 ) 2 = inf x y 2 p(dxdy), p P(µ,µ 2 ) were P(µ,µ 2 ) is te set of all probability measures on wit first marginal µ and second marginal µ 2, and te symbol denotes te usual Euclidean norm on. More precisely, a probability measure p is in P(µ,µ 2 ) if and only if for eac Borel subset A tere olds p(a )=µ (A), p( A) =µ 2 (A). Wasserstein distances of order q wit q different from 2 may be analogously defined [0]. Since no confusion sould arise in doing so, we sall refer to d in wat follows as simply te Wasserstein distance. It is well known tat d defines a metric on te set of probability measures µ on aving finite second moments: x 2 µ(dx) < [0, 2]. In particular, d satisfies te triangle inequality on tis set. Tat is, if µ,µ 2, and µ 3 are probability measures on wit finite second moments, ten (9) d(µ,µ 3 ) d(µ,µ 2 )+d(µ 2,µ 3 ). We sall make use of tis property at several points later on.

5 (0) VARIATIONAL FORMULATION OF THE FOKKER PLANCK EQUATION 5 We note tat te Wasserstein metric may be equivalently defined by [2] d(µ,µ 2 ) 2 = inf E X Y 2, were E(U) denotes te expectation of te random variable U, and te infimum is taken over all random variables X and Y suc tat X as distribution µ and Y as distribution µ 2. In oter words, te infimum is over all possible couplings of te random variables X and Y. Convergence in te metric d is equivalent to te usual weak convergence plus convergence of second moments. Tis latter assertion may be demonstrated by appealing to te representation (0) and applying te well-known Skorood teorem from probability teory (see Teorem 29.6 of []). We omit te details. Te variational problem (8) is an example of a Monge Kantorovic mass transference problem wit te particular cost function c(x, y) = x y 2 [2]. In tat context, an infimizer p P(µ,µ 2 ) is referred to as an optimal transference plan. Wen µ and µ 2 ave finite second moments, te existence of suc a p for (8) is readily verified by a simple adaptation of our arguments in section 4. For a probabilistic proof tat te infimum in (8) is attained wen µ and µ 2 ave finite second moments, see [0]. Brenier [2] as establised te existence of a one-to-one optimal transference plan in te case tat te measures µ and µ 2 ave bounded support and are absolutely continuous wit respect to Lebesgue measure. Caffarelli [3] and Gangbo and McCann [8, 9] ave recently extended Brenier s results to more general cost functions c and to cases in wic te measures do not ave bounded support. If te measures µ and µ 2 are absolutely continuous wit respect to te Lebesgue measure, wit densities ρ and ρ 2, respectively, we will write P(ρ, ρ 2 ) for te set of probability measures aving first marginal µ and second marginal µ 2. Correspondingly, we will denote by d(ρ, ρ 2 ) te Wasserstein distance between µ and µ 2. Tis is te situation tat we will be concerned wit in wat follows. 4. Te discrete sceme. We sall now construct a time-discrete sceme tat is designed to converge in an appropriate sense (to be made precise in te next section) to a solution of te Fokker Planck equation. Te sceme tat we sall describe was motivated by a similar sceme developed by Otto in an investigation of pattern formation in magnetic fluids [9]. We sall make te following assumptions concerning te potential Ψ introduced in section 2: () (2) Ψ C ( ); Ψ(x) 0 for all x ; Ψ(x) C (Ψ(x) + ) for all x for some constant C<. Notice tat our assumptions on Ψ allow for cases in wic exp( βψ) dx is not defined, so te stationary density ρ s given by (4) does not exist. Tese assumptions allow us to treat a wide class of Fokker Planck equations. In particular, te classical diffusion equation ρ t = β ρ, for wic Ψ const., falls into tis category. We also introduce te set K of admissible probability densities: K := ρ: [0, ) measurable ρ(x) dx =,M(ρ) <, were M(ρ) = x 2 ρ(x) dx.

6 6 R. JORDAN, D. KINDERLEHRER, AND F. OTTO (3) Wit tese conventions in and, we now formulate te iterative discrete sceme: Determine ρ (k) tat minimizes 2 d(ρ(k ), ρ) 2 + F(ρ) over all ρ K. Here we use te notation ρ (0) = ρ 0. Te sceme (3) is te obvious generalization of te sceme () set fort in te Introduction for te diffusion equation. We sall now establis existence and uniqueness of te solution to (3). PROPOSITION 4.. Given ρ 0 K, tere exists a unique solution of te sceme (3). Proof. Let us first demonstrate tat S is well defined as a functional on K wit values in (, + ] and tat, in addition, tere exist α < and C< depending only on n suc tat (4) S(ρ) C (M(ρ) + ) α for all ρ K. Actually, we sall sow tat (4) is valid for any α ( n n+2, ). For future reference, we prove a somewat finer estimate. Namely, we demonstrate tat tere exists a C<, depending only on n and α, suc tat for all R 0, and for eac ρ K, tere olds (5) B R min{ρ log ρ, 0} dx C ( 2 + n) α n 2 R 2 + (M(ρ) + ) α, were B R denotes te ball of radius R centered at te origin in. Indeed, for α < tere olds Hence by Hölder s inequality, we obtain On te oter and, for min{z log z,0} Cz α for all z 0. min{ρ log ρ, 0} dx B R C ρ α dx B R α α α C dx x 2 (M(ρ) + ) α. + B R B R α α > n 2, we ave α α α α n 2 dx C x 2 + R 2 + Let us now prove tat for given ρ (k ) K, tere exists a minimizer ρ K of te functional. (6) K ρ 2 d(ρ(k ), ρ) 2 + F(ρ).

7 VARIATIONAL FORMULATION OF THE FOKKER PLANCK EQUATION 7 Observe tat S is not bounded below on K and ence F is not bounded below on K eiter. Neverteless, using te inequality (7) M(ρ ) 2 M(ρ 0 )+2d(ρ 0, ρ ) 2 for all ρ 0, ρ K (wic immediately follows from te inequality y 2 2 x 2 +2 x y 2 and from te definition of d) togeter wit (4) we obtain (8) 2 d(ρ(k ), ρ) 2 + F(ρ) (7) 4 M(ρ) 2 M(ρ(k ) )+S(ρ) (4) 4 M(ρ) C (M(ρ) + )α 2 M(ρ(k ) ) for all ρ K, wic ensures tat (6) is bounded below. Now, let {ρ ν } be a minimizing sequence for (6). Obviously, we ave tat (9) {S(ρ ν )} ν is bounded above, and according to (8), (20) {M(ρ ν )} ν is bounded. Te latter result, togeter wit (5), implies tat min{ρ ν log ρ ν, 0} dx wic combined wit (9) yields tat max{ρ ν log ρ ν, 0} dx ν ν is bounded, is bounded. As z max{z log z,0},z [0, ), as superlinear growt, tis result, in conjunction wit (20), guarantees te existence of a ρ (k) K suc tat (at least for a subsequence) (2) ρ ν w ρ (k) in L ( ). Let us now sow tat (22) S(ρ (k) ) lim inf ν S(ρ ν). As [0, ) z z log z is convex and [0, ) z max{z log z,0} is convex and nonnegative, (2) implies tat for any R<, (23) ρ (k) log ρ (k) dx lim inf ρ ν log ρ ν dx, B ν R B R (24) max{ρ (k) log ρ (k), 0} dx lim inf max{ρ ν log ρ ν, 0} dx. B ν R B R On te oter and we ave according to (5) and (20) (25) lim sup min{ρ ν log ρ ν, 0} dx = 0. R ν N B R

8 8 R. JORDAN, D. KINDERLEHRER, AND F. OTTO Now observe tat for any R<, tere olds S(ρ (k) ) ρ (k) log ρ (k) dx + B R max{ρ (k) log ρ (k), 0} dx, B R wic togeter wit (23), (24), and (25) yields (22). It remains for us to sow tat (26) (27) E(ρ (k) ) lim inf ν E(ρ ν), d(ρ (k ), ρ (k) ) 2 lim inf ν d(ρ(k ), ρ ν ) 2. Equation (26) follows immediately from (2) and Fatou s lemma. coose p ν P(ρ (k ), ρ ν ) satisfying x y 2 p ν (dx dy) d(ρ (k ), ρ ν ) 2 + ν. As for (27), we By (20) te sequence of probability measures {ρ ν dx} ν is tigt, or relatively compact wit respect to te usual weak convergence in te space of probability measures on (i.e., convergence tested against bounded continuous functions) []. Tis, togeter wit te fact tat te density ρ (k ) as finite second moment, guarantees tat te sequence {p ν } ν of probability measures on is tigt. Hence, tere is a subsequence of {p ν } ν tat converges weakly to some probability measure p. From (2) we deduce tat p P(ρ (k ), ρ (k) ). We now could invoke te Skorood teorem [] and Fatou s lemma to infer (27) from tis weak convergence, but we prefer ere to give a more analytic proof. For R< let us select a continuous function η R : [0, ] suc tat η R = inside of B R and η R = 0 outside of B 2R. We ten ave (28) η R (x) η R (y) x y 2 p(dx dy) = lim η R (x) η R (y) x y 2 p ν (dx dy) ν lim inf ν d(ρ(k ), ρ ν ) 2 for eac fixed R<. On te oter and, using te monotone convergence teorem, we deduce tat d(ρ (k ), ρ (k) ) 2 x y 2 p(dx dy) R n = lim η R (x) η R (y) x y 2 p(dx dy), R wic combined wit (28) yields (27). To conclude te proof of te proposition we establis tat te functional (6) as at most one minimizer. Tis follows from te convexity of K and te strict convexity of (6). Te strict convexity of (6) follows from te strict convexity of S, te linearity of E, and te (obvious) convexity over K of te functional ρ d(ρ (k ), ρ) 2.

9 VARIATIONAL FORMULATION OF THE FOKKER PLANCK EQUATION 9 Remark. One of te referees as communicated to us te following simple estimate tat could be used in place of (4) (5) in te previous and subsequent analysis: for any Ω (in particular, for Ω = B R ) and for all ρ K tere olds (29) Ω min{ρ log ρ, 0} dx C Ω e x 2 dx + M(ρ)+ 4 for any > 0. To obtain te inequality (29), select C>0suc tat for all z [0, ], we ave z log z C z. Ten, defining te sets Ω 0 = Ω {ρ exp( x )} and Ω = Ω {exp( x ) < ρ }, we ave min{ρ log ρ, 0} dx = Ω ρ (log ρ) dx + Ω 0 ρ (log ρ) dx Ω C Ω e x 2 dx + Ω x ρ dx. Te desired result (29) ten follows from te inequality x x 2 +/(4) for > Convergence to te solution of te Fokker Planck equation. We come now to our main result. We sall demonstrate tat an appropriate interpolation of te solution to te sceme (3) converges to te unique solution of te Fokker Planck equation. Specifically, te convergence result tat we will prove ere is as follows. THEOREM 5.. Let ρ 0 K satisfy F (ρ 0 ) <, and for given >0, let {ρ (k) } k N be te solution of (3). Define te interpolation ρ : (0, ) [0, ) by Ten as 0, ρ (t) = ρ (k) for t [k, (k + ) ) and k N {0}. Ω ρ dx (30) ρ (t) ρ(t) weakly in L ( ) for all t (0, ), were ρ C ((0, ) ) is te unique solution of (3) wit initial condition ρ t = div(ρ Ψ) +β ρ, (32) ρ(t) ρ 0 strongly in L ( ) for t 0 and (33) M(ρ), E(ρ) L ((0,T)) for all T<. Remark. A finer analysis reveals tat ρ ρ strongly in L ((0,T) ) for all T<. Proof. Te proof basically follows along te lines of [9, Proposition 2, Teorem 3]. Te crucial step is to recognize tat te first variation of (6) wit respect to te independent variables indeed yields a time-discrete sceme for (3), as will now be demonstrated. For notational convenience only, we sall set β from ere on in. As will be evident from te ensuing arguments, our proof works for any positive β. In

10 0 R. JORDAN, D. KINDERLEHRER, AND F. OTTO fact, it is not difficult to see tat, wit appropriate modifications to te sceme (3), we can establis an analogous convergence result for time-dependent β. Let a smoot vector field wit bounded support, ξ C 0 (, ), be given, and define te corresponding flux {Φ τ } τ R by τ Φ τ = ξ Φ τ for all τ R and Φ 0 = id. For any τ R, let te measure ρ τ (y) dy be te pus forward of ρ (k) (y) dy under Φ τ. Tis means tat (34) ρ τ (y) ζ(y) dy = ρ (k) (y) ζ(φ τ (y)) dy for all ζ C0(R 0 n ). As Φ τ is invertible, (34) is equivalent to te following relation for te densities: (35) det Φ τ ρ τ Φ τ = ρ (k). By (6), we ave for eac τ > 0 (36) τ 2 d(ρ(k ), ρ τ ) 2 + F(ρ τ ) 2 d(ρ(k ), ρ (k) ) 2 + F(ρ (k) ) 0, wic we now investigate in te limit τ 0. Because Ψ is nonnegative, equation (34) also olds for ζ = Ψ, i.e., ρ τ (y) Ψ(y) dy = ρ (k) (y) Ψ(Φ τ (y)) dy. Tis yields τ E(ρ τ ) E(ρ (k) ) = τ (Ψ(Φ τ (y)) Ψ(y)) ρ (k) (y) dy. Observe tat te difference quotient under te integral converges uniformly to Ψ(y) ξ(y), ence implying tat d (37) d τ [E(ρ τ )] τ=0 = Ψ(y) ξ(y) ρ (k) (y) dy. d Next, we calculate d τ [S(ρ τ )] τ=0. Invoking an appropriate approximation argument (for instance approximating log by some function tat is bounded below), we obtain ρ τ (y) log(ρ τ (y)) dy (34) = ρ (k) (y) log(ρ τ (Φ τ (y))) dy R n (35) ρ = ρ (k) (k) (y) (y) log dy. R det Φ n τ (y) Terefore, we ave τ S(ρ τ ) S(ρ (k) ) = ρ (k) (y) τ log(det Φ τ (y)) dy.

11 Now using VARIATIONAL FORMULATION OF THE FOKKER PLANCK EQUATION d dτ [det Φ τ (y)] τ=0 = divξ(y), togeter wit te fact tat Φ 0 = id, we see tat te difference quotient under te integral converges uniformly to divξ, ence implying tat d (38) d τ [S(ρ τ )] τ=0 = ρ (k) divξ dy. Now, let p be optimal in te definition of d(ρ (k ), ρ (k) ) 2 (see section 3). Te formula ζ(x, y) p τ (dx dy) = ζ(x, Φ τ (y)) p(dx dy), ζ C0(R 0 n ) ten defines a p τ P(ρ (k ), ρ τ ). Consequently, tere olds τ 2 d(ρ(k ), ρ τ ) 2 2 d(ρ(k ), ρ (k) ) 2 2 Φ τ (y) x 2 2 y x 2 p(dx dy), τ wic implies tat (39) lim sup τ 0 τ 2 d(ρ(k ), ρ τ ) 2 2 d(ρ(k ), ρ (k) ) 2 (y x) ξ(y) p(dx dy). We now infer from (36), (37), (38), and (39) (and te symmetry in ξ ξ) tat (y x) ξ(y) p(dx dy) + ( Ψ ξ divξ) ρ (k) dy = 0 (40) for all ξ C0 (, ). Observe tat because p P(ρ (k ), ρ (k) ), tere olds (ρ (k) ρ (k ) ) ζ dy (y x) ζ(y) p(dx dy) R n = (ζ(y) ζ(x) +(x y) ζ(y)) p(dx dy) 2 sup 2 ζ y x 2 p(dx dy) = 2 sup 2 ζ d(ρ (k ), ρ (k) ) 2 for all ζ C0 ( ). Coosing ξ = ζ in (40) ten gives (ρ(k) ρ (k ) ) ζ +( Ψ ζ ζ) ρ (k) dy (4) 2 sup 2 ζ d(ρ(k ), ρ (k) ) 2 for all ζ C0 ( ).

12 2 R. JORDAN, D. KINDERLEHRER, AND F. OTTO We wis now to pass to te limit 0. In order to do so we will first establis te following a priori estimates: for any T<, tere exists a constant C< suc tat for all N N and all [0, ] wit N T, tere olds (42) (43) max{ρ (N) (44) N (45) k= M(ρ (N) ) C, log ρ (N), 0} dx C, d(ρ (k ) E(ρ (N) ) C,, ρ (k) )2 C. Let us verify tat te estimate (42) olds. Since ρ (k ) is admissible in te variational principle (3), we ave tat 2 d(ρ(k ), ρ (k) )2 + F(ρ (k) ) F(ρ(k ) ), wic may be summed over k to give N (46), ρ (k) )2 + F (ρ (N) ) F (ρ0 ). 2 d(ρ(k ) k= As in Proposition 4., we must confront te tecnical difficulty tat F is not bounded below. Te inequality (42) is establised via te following calculations: M(ρ (N) ) (7) 2 d(ρ 0, ρ (N) )2 +2M(ρ 0 ) N 2 N d(ρ (k ), ρ (k) )2 +2M(ρ 0 ) (46) 4 N (4) 4 T k= F (ρ 0 ) F (ρ (N) ) +2M(ρ 0 ) F (ρ 0 )+C (M(ρ (N) ) + )α +2M(ρ 0 ), wic clearly gives (42). To obtain te second line of te above display, we ave made use of te triangle inequality for te Wasserstein metric (see equation (9)) and te Caucy Scwarz inequality. Te estimates (43), (44), and (45) now follow readily from te bounds (4) and (5), te estimate (42), and te inequality (46), as follows: log ρ (N), 0} dx S(ρ(N) )+ log ρ (N), 0} dx max{ρ (N) E(ρ (N) (5) min{ρ (N) S(ρ (N) )+C(M(ρ(N) ) + )α F (ρ (N) )+C(M(ρ(N) ) + )α (46) F (ρ 0 )+C(M(ρ (N) ) + )α ; ) = F (ρ(n)) S(ρ(N) (4) ) F (ρ (N) )+C(M(ρ(N) ) + )α (46) F (ρ 0 )+C(M(ρ (N) ) + )α ;

13 VARIATIONAL FORMULATION OF THE FOKKER PLANCK EQUATION 3 N k= d(ρ (k ), ρ (k) (46) )2 2 F (ρ 0 ) F (ρ (N) ) (4) 2 F (ρ 0 )+C(M(ρ (N) ) + )α. Now, owing to te estimates (42) and (43), we may conclude tat tere exists a measurable ρ(t, x) suc tat, after extraction of a subsequence, (47) ρ ρ weakly in L ((0,T) ) for all T<. A straigtforward analysis reveals tat (42), (43), and (44) guarantee tat (48) ρ(t) K for a.e. t (0, ), M(ρ), E(ρ) L ((0,T)) for all T<. Let us now improve upon te convergence in (47) by sowing tat (30) olds. For a given finite time orizon T<, tere exists a constant C< suc tat for all N,N N and all [0, ] wit N T, and N T, we ave d(ρ (N ), ρ (N) )2 C N N. Tis result is obtained from (45) by use of te triangle inequality (9) for d and te Caucy Scwarz inequality. Furtermore, for all ρ, ρ K,p P(ρ, ρ ), and ζ C0 ( ), tere olds ζρ dx ζρdx = (ζ(x) ζ(y)) p(dx dy) R n sup ζ sup ζ x y p(dx dy) x y 2 p(dx dy) so from te definition of d we obtain ζρ dx ζρdx sup ζ d(ρ, ρ ) for ρ, ρ K and ζ C0 ( ). Hence, it follows tat ζρ (t ) dx ζρ (t) dx C sup ζ ( t t + ) 2 (49) for all t, t (0,T), and ζ C0 ( ). Let t (0,T) and ζ C0 ( ) be given, and notice tat for any δ > 0, we ave ζρ (t) dx ζρ(t) dx t+δ ζρ (t) dx 2 δ ζρ (τ) dx dτ t δ R n t+δ t+δ + 2 δ ζρ (τ) dx dτ 2 δ ζρ(τ) dx dτ t δ t δ R n t+δ + 2 δ ζρ(τ) dx dτ ζρ(t) dx R. n t δ 2,

14 4 R. JORDAN, D. KINDERLEHRER, AND F. OTTO According to (49), te first term on te rigt-and side of tis equation is bounded by C sup ζ (δ + ) 2, and owing to (47), te second term converges to zero as 0 for any fixed δ > 0. At tis point, let us remark tat from te result (47) we may deduce tat ρ is smoot on (0, ). Tis is te conclusion of assertion (a) below, wic will be proved later. From tis smootness property, we ascertain tat te final term on te rigt-and side of te above display converges to zero as δ 0. Terefore, we ave establised tat (50) ζρ (t) dx ζρ(t) dx for all ζ C0 ( ). However, te estimate (42) guarantees tat M(ρ (t)) is bounded for 0. Consequently, (50) olds for any ζ L ( ), and terefore, te convergence result (30) does indeed old. It now follows immediately from (4), (45), and (47) tat ρ satisfies ρ ( t ζ Ψ ζ + ζ) dx dt = ρ 0 ζ(0) dx, (5) (0, ) for all ζ C0 (R ). In addition, we know tat ρ satisfies (33). We now sow tat (a) any solution of (5) is smoot on (0, ) and satisfies equation (3); (b) any solution of (5) for wic (33) olds satisfies te initial condition (32); (c) tere is at most one smoot solution of (3) wic satisfies (32) and (33). Te corresponding arguments are, for te most part, fairly classical. Let us sketc te proof of te regularity part (a). First observe tat (5) implies ρ(t ) ζ(t ) dx ρ ( t ζ Ψ ζ + ζ) dx dt (t 0,t ) (52) = ρ(t 0 ) ζ(t 0 ) dx for all ζ C 0 (R ) and a.e. 0 t 0 <t. We fix a function η C0 ( ) to serve as a cutoff in te spatial variables. It ten follows directly from (52) tat for eac ζ C0 (R ) and for almost every 0 t 0 <t, tere olds ηρ(t ) ζ(t ) dx ηρ( t ζ + ζ) dx dt (t 0,t ) = ρ ( η Ψ η) ζ dx dt (53) (t 0,t ) + ρ (2 η η Ψ) ζ dx dt (t 0,t ) R n + ηρ(t 0 ) ζ(t 0 ) dx. Notice tat for fixed (t,x ) (0, ) and for eac δ > 0, te function ζ δ (t, x) = G(t + δ t, x x )

15 VARIATIONAL FORMULATION OF THE FOKKER PLANCK EQUATION 5 is an admissible test function in (53). Here G is te eat kernel (54) G(t, x) = t n 2 g(t 2 x) wit g(x) = (2π) n 2 exp( 2 x 2 ). Inserting ζ δ into (53) and taking te limit δ 0, we obtain te equation t (ρη)(t )= [ρ(t)( η Ψ η)] G(t t) dt t 0 (55) t + [ρ(t) (2 η η Ψ)] G(t t) dt t 0 +(ρη)(t 0 ) G(t t 0 ) for a.e. 0 t 0 <t, were denotes convolution in te x-variables. From (55), we extract te following estimate in te L p -norm: (ρη)(t ) L p = Now observe tat wic leads to (ρη)(t ) L p + t t 0 t t 0 ρ(t)( η Ψ η) L G(t t) L p dt ρ(t) (2 η η Ψ) L G(t t) L p dt + (ρη)(t 0 ) L G(t t 0 ) L p for a.e. 0 t 0 <t. G(t) L p = t ( p ) n 2 gl p, G(t) L p, = t n p 2 n + 2 g L p, = ess sup ρ(t)( η Ψ η) L t (t 0,t ) t t 0 t ( p ) n 2 0 gl p dt For p< n n + ess sup ρ(t) (2 η η Ψ) L t (t 0,t ) t t 0 + (ρη)(t 0 ) L G(t t 0 ) L p for a.e. 0 t 0 <t. 0 t n p 2 n + 2 g L p dt te t-integrals are finite, from wic we deduce tat ρ L p loc ((0, ) Rn ). We now appeal to te L p -estimates [8, section 3, (3.), and (3.2)] for te potentials in (55) to conclude by te usual bootstrap arguments tat any derivative of ρ is in L p loc ((0, ) Rn ), from wic we obtain te stated regularity condition (a). We now prove assertion (b). Using (55) wit t 0 = 0, and proceeding as above, we obtain (ρη)(t ) (ρ 0 η) G(t ) L = ess sup ρ(t)( η Ψ η) L t (0,t ) + ess sup ρ(t) (2 η η Ψ) L t (0,t ) t 0 t 0 g L dt t 2 gl dt for all t > 0

16 6 R. JORDAN, D. KINDERLEHRER, AND F. OTTO and terefore, On te oter and, we ave wic leads to (ρη)(t) (ρ 0 η) G(t) 0 in L ( ) for t 0. (ρ 0 η) G(t) ρ 0 η in L ( ) for t 0, (ρη)(t) ρ 0 η in L ( ) for t 0. From tis result, togeter wit te boundedness of {M(ρ(t))} t 0, we infer tat (32) is satisfied. Finally, we prove te uniqueness result (c) using a well-known metod from te teory of elliptic parabolic equations (see, for instance, [20]). Let ρ, ρ 2 be solutions of (32) wic are smoot on (0, ) and satisfy (32), (33). Teir difference ρ satisfies te equation ρ div [ρ Ψ + ρ] = 0. t We multiply tis equation for ρ by φ δ (ρ), were te family {φ δ} δ 0 is a convex and smoot approximation to te modulus function. For example, we could take Tis procedure yields te inequality φ δ (z) = (z 2 + δ 2 ) 2. t [φ δ (ρ)] div [φ δ (ρ) Ψ + [φ δ (ρ)]] = φ δ (ρ) ρ 2 +(φ δ(ρ) ρ φ δ (ρ)) Ψ (φ δ(ρ) ρ φ δ (ρ)) Ψ, wic we ten multiply by a nonnegative spatial cutoff function η C0 ( ) and integrate over to obtain d φ δ (ρ(t)) η dx + φ δ (ρ(t)) ( Ψ η η) dx dt R n (φ δ(ρ) ρ φ δ (ρ)) Ψ η dx. Integrating over (0,t) for given t (0, ), we obtain wit elp of (32) φ δ (ρ(t)) η dx + φ δ (ρ(t)) ( Ψ η η) dx dt (0,t) R n (φ δ(ρ) ρ φ δ (ρ)) Ψ η dx dt. (0,t) Letting δ tend to zero yields (56) ρ(t) η dx + ρ(t) ( Ψ η η) dx dt 0. (0,t) According to (2) and (33), ρ and ρ Ψ are integrable on te entire. Hence, if we replace η in (56) by a function η R satisfying x η R (x) = η, were η (x) = for x, η (x) = 0 for x 2, R and let R tend to infinity, we obtain ρ(t) dx = 0. Tis produces te desired uniqueness result.

17 VARIATIONAL FORMULATION OF THE FOKKER PLANCK EQUATION 7 Acknowledgments. Te autors would like to tank te anonymous referees for valuable suggestions and comments. REFERENCES [] P. BILLINGSLEY, Probability and Measure, Jon Wiley, New York, 986. [2] Y. BRENIER, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Mat., 44 (99), pp [3] L. A. CAFFARELLI, Allocation maps wit general cost functions, in Partial Differential Equations and Applications, P. Marcellini, G. G. Talenti, and E. Vesintini, eds., Lecture Notes in Pure and Applied Matematics 77, Marcel Dekker, New York, NY, 996, pp [4] S. CHANDRASEKHAR, Stocastic problems in pysics and astronomy, Rev. Mod. Pys., 5 (942), pp. 89. [5] R. COURANT, K. FRIEDRICHS, AND H. LEWY, Über die partiellen Differenzgleicungen der matematiscen Pysik, Mat. Ann., 00 (928), pp. 74. [6] S. DEMOULINI, Young measure solutions for a nonlinear parabolic equation of forward backward type, SIAM J. Mat. Anal., 27 (996), pp [7] C. W. GARDINER, Handbook of stocastic metods, 2nd ed., Springer-Verlag, Berlin, Heidelberg, 985. [8] W. GANGBO AND R. J. MCCANN, Optimal maps in Monge s mass transport problems, C. R. Acad. Sci. Paris, 32 (995), pp [9] W. GANGBO AND R. J. MCCANN, Te geometry of optimal transportation, Acta Mat., 77 (996), pp [0] C. R. GIVENS AND R. M. SHORTT, A class of Wasserstein metrics for probability distributions, Micigan Mat. J., 3 (984), pp [] R. JORDAN, A statistical equilibrium model of coerent structures in magnetoydrodynamics, Nonlinearity, 8 (995), pp [2] R. JORDAN AND D. KINDERLEHRER, An extended variational principle, in Partial Differential Equations and Applications, P. Marcellini, G. G. Talenti, and E. Vesintini, eds., Lecture Notes in Pure and Applied Matematics 77, Marcel Dekker, New York, NY, 996, pp [3] R. JORDAN, D. KINDERLEHRER, AND F. OTTO, Free energy and te Fokker Planck equation, Pysica D., to appear. [4] R. JORDAN, D. KINDERLEHRER, AND F. OTTO, Te route to stability troug te Fokker Planck dynamics, Proc. First U.S. Cina Conference on Differential Equations and Applications, to appear. [5] R. JORDAN AND B. TURKINGTON, Ideal magnetofluid turbulence in two dimensions, J. Stat. Pys., 87 (997), pp [6] D. KINDERLEHRER AND P. PEDREGAL, Weak convergence of integrands and te Young measure representation, SIAM J. Mat. Anal., 23 (992), pp. 9. [7] H. A. KRAMERS, Brownian motion in a field of force and te diffusion model of cemical reactions, Pysica, 7 (940), pp [8] O. A. LADYŽENSKAJA, V. A. SOLONNIKOV, AND N. N. URAL CEVA, Linear and Quasi Linear Equations of Parabolic Type, American Matematical Society, Providence, RI, 968. [9] F. OTTO, Dynamics of labyrintine pattern formation in magnetic fluids: A mean field teory, Arcive Rat. Mec. Anal., to appear. [20] F. OTTO, L contraction and uniqueness for quasilinear elliptic parabolic equations, J. Differential Equations, 3 (996), pp [2] S. T. RACHEV, Probability metrics and te stability of stocastic models, Jon Wiley, New York, 99. [22] H. RISKEN, Te Fokker-Planck equation: Metods of solution and applications, 2nd ed., Springer-Verlag, Berlin, Heidelberg, 989. [23] Z. SCHUSS, Singular perturbation metods in stocastic differential equations of matematical pysics, SIAM Rev., 22 (980), pp [24] J. C. STRIKWERDA, Finite Difference Scemes and Partial Differential Equations, Wardswort & Brooks/Cole, New York, 989.