Γ-convergence for a class of action functionals induced by gradients of convex functions
aa r X i v : . [ m a t h . O C ] J a n Γ -convergence for a class of action functionals induced bygradients of convex functions Luigi
Ambrosio ∗ Aymeric
Baradat † Yann
Brenier ‡ January 20, 2021
Abstract
Given a real function f , the rate function for the large deviations of the diffusion processof drift ∇ f given by the Freidlin-Wentzell theorem coincides with the time integral of theenergy dissipation for the gradient flow associated with f . This paper is concerned with thestability in the hilbertian framework of this common action functional when f varies. Moreprecisely, we show that if ( f h ) h is uniformly λ -convex for some λ ∈ R and converges towards f in the sense of Mosco convergence, then the related functionals Γ -converge in the strongtopology of curves. Action functionals of the form I f ( γ ) := Z n | ˙ γ ( t ) | + |∇ f | ( γ ( t )) o d t, and the closely related ones (since they differ by a null lagrangian, the term f ( γ (1)) − f ( γ (0)) ) Z | ˙ γ ( t ) − ∇ f ( γ ( t )) | d t, (1)appear in many areas of Mathematics, for instance in the Freidlin-Wentzell theory of largedeviations for the SDE d X ǫt = ∇ f ( X ǫt )d t + √ ǫ d B t (see for instance [9]) or in the variationaltheory of gradient flows pioneered by De Giorgi, where they correspond to the integral form of theenergy dissipation (see [4]). In this paper, we investigate the stability of the action functionals I f with respect to Γ -convergence of the functions f (actually with respect to the stronger notionof Mosco convergence, see below). More precisely, we are concerned with the case when thefunctions under consideration are λ -convex and defined in a Hilbert space H . In this case, thefunctional I f is well defined if we understand ∇ f ( x ) as the element with minimal norm in thesubdifferential ∂f ( x ) : this choice, very natural in the theory of gradient flows, grants the jointlower semicontinuity property of ( x, f )
7→ |∇ f | ( x ) that turns out to be very useful when provingstability of gradient flows, see [12], [5] and the more recent papers [10], [11] where emphasis is puton the convergence of the dissipation functionals. In more abstract terms, we are dealing withautonomous Lagrangians L ( x, p ) = | p | + |∇ f | ( x ) that are unbounded and very discontinuous ∗ Scuola Normale Superiore, Pisa. E-mail: [email protected] † Institut Camille Jordan, Lyon. E-mail: [email protected] ‡ École Normale Supérieure, Paris. E-mail: [email protected] x , and this is a source of difficulty in the construction of recovery sequences, inthe proof of the Γ -limsup inequality.Our interest in this problem comes from [3], where we dealt with the derivation of thediscrete Monge-Ampère equation from the stochastic model of a Brownian point cloud, usinglarge deviations and Freidlin-Wentzell theory, along the lines of [6]. In that case H = R Nd wasfinite dimensional, f ( x ) := max σ ∈ S N h x, A σ i , (with A = ( A , . . . , A N ) ∈ R Nd given and A σ = ( A σ (1) , . . . , A σ ( N ) ) for all σ ∈ S N , the set of allpermutations of J , N K ), and the approximating functions f ǫ were given by f ǫ ( t, x ) = ǫt log (cid:20) N ! X σ ∈ Σ N exp (cid:18) h x, A σ i εt (cid:19)(cid:21) . In that case, our proof used some simplifications due to finite dimensionality, and a uniformLipschitz condition. In this paper, building upon some ideas in [3], we provide the conver-gence result in a more general and natural context. For the sake of simplicity, unlike [3], weconsider only the autonomous case. However it should be possible to adapt our proof to thecase when time-dependent λ -convex functions f ( t, · ) are considered, under additional regularityassumptions with respect to t , as in [3].In the infinite-dimensional case, Mosco convergence (see Definition 4.1) is stronger and moreappropriate than Γ -convergence, since it ensures convergence of the resolvent operators (un-der equi-coercitivity assumptions, the two notions are equivalent). Also, since in the infinite-dimensional case, the finiteness domains of the functions can be pretty different, the addition ofthe endpoint condition is an additional source of difficulties, that we handle with an interpolationlemma which is very much related to the structure of monotone operators, see Lemma 3.1.Defining the functionals Θ f,x ,x : C ([0 , H ) → [0 , ∞ ] by Θ f,x ,x ( γ ) := ( I f ( γ ) if γ ∈ AC ([0 , H ) , γ (0) = x , γ (1) = x ; + ∞ otherwise, (2)our main result reads as follows: Theorem 1.1. If ( f h ) h is uniformly λ -convex for some λ ∈ R , if f h → f w.r.t. Mosco conver-gence, and if lim h →∞ x h,i = x i , sup h |∇ f h | ( x h,i ) < ∞ , i = 0 , , then Θ f h ,x h, ,x h, Γ -converge to Θ f,x ,x in the C ([0 , H ) topology. As a byproduct, under an additional equi-coercitivity assumption our theorem grants con-vergence of minimal values to minimal values and of minimizers to minimizers. Obviously thecondition x h,i → x i is necessary, and we believe that at least some (possibly more refined) boundson the gradients at the endpoints are necessary as well. If we ask also that x h,i are recoverysequences, i.e. f h ( x h,i ) → f ( x i ) , then the result can also be read in terms of the functionals (1).As a final comment, it would be interesting to investigate this type of convergence resultsalso in a non-Hilbertian context, as it happened for the theory of gradient flows. For instance,a natural context would be the space of probability measures with finite quadratic moment.Functionals of this form, where f is a constant multiple of the logarithmic entropy, appearin the so-called entropic regularization of the Wasserstein distance (see [8] and the referencestherein). 2 cknowledgements. We dedicate this paper to Edoardo Vesentini, a great mathematician anda former President of the Accademia dei Lincei. As Director of the Scuola Normale, he has beenthe pioneer of many projects that shaped the Scuola Normale for many years to come. The firstauthor acknowledges the support of the PRIN 2017 project “Gradient flows, Optimal Transportand Metric Measure Structures”. This work was prepared during the stay of the second authorat the Max Planck Institute for Mathematics in the Sciences in Leipzig, that he would like tothank for its hospitality.
Let H be a Hilbert space. For a function f : H → ( −∞ , ∞ ] we denote by D ( f ) the finitenessdomain of f . We say that f is λ -convex if x f ( x ) − λ | x | is convex. It is easily seen that λ -convex functions satisfy the perturbed convexity inequality f (cid:0) (1 − t ) x + ty (cid:1) ≤ (1 − t ) f ( x ) + tf ( y ) − λ t (1 − t ) | x − y | , t ∈ [0 , . We denote by ∂f ( x ) the Gateaux subdifferential of f at x ∈ D ( f ) , namely the set ∂f ( x ) := (cid:26) p ∈ H : lim inf t → + f ( x + th ) − f ( x ) t ≥ t h h, p i ∀ h ∈ H (cid:27) . It is a closed convex set, possibly empty. We denote by D ( ∂f ) the domain of the subdifferential.In the case when f is λ -convex, the monotonicity of difference quotients gives the equivalent,non asymptotic definition: ∂f ( x ) := (cid:26) p ∈ H : f ( y ) ≥ f ( x ) + h y − x, p i + λ | y − x | ∀ y ∈ H (cid:27) . (3)For any x ∈ D ( ∂f ) we consider the vector ∇ f ( x ) as the element with minimal norm of ∂f ( x ) .We agree that |∇ f ( x ) | = ∞ if either x / ∈ D ( f ) of x ∈ D ( f ) and ∂f ( x ) = ∅ . For λ -convexfunctions, relying on (3), it can be easily proved that ∂f ( x ) is not empty if and only if sup y = x (cid:2) f ( x ) − f ( y ) + λ | x − y | (cid:3) + | x − y | < ∞ (4)and that |∇ f | ( x ) is precisely equal to the supremum (see for instance Theorem 2.4.9 in [4]).For τ > we denote by f τ the regularized function f τ ( x ) := min y ∈ H f ( y ) + | y − x | τ (5)and we denote by J τ = (Id + τ ∂f ) − : H → D ( ∂f ) the so-called resolvent map associatingto x the minimizer y in (5). When f is proper, λ -convex and lower semicontinuous, existenceand uniqueness of J τ ( x ) follow by the strict convexity of y f ( y ) + | y − x | / (2 τ ) , as soon as τ < − /λ when λ < , and for all τ > otherwise (we shall call admissible these values of τ ).We also use the notation J f,τ to emphasize the dependence on f .Now we recall a few basic and well-known facts (see for instance [7], [4]), providing for thereader’s convenience sketchy proofs. Theorem 2.1.
Assume that f : H → ( −∞ , ∞ ] is proper, λ -convex and lower semicontinuous.For all admissible τ > one has: i) f τ is differentiable everywhere, and for all x ∈ H , ∇ f τ ( x ) = x − J τ ( x ) τ ∈ ∂f ( J τ ( x )) . (6) (ii) J τ is (1 + λτ ) − -Lipschitz, and f τ ∈ C , ( H ) with Lip( ∇ f τ ) ≤ /τ as soon as there holds (1 + τ λ ) − ≤ .(iii) For all x ∈ D ( ∂f ) , ∇ f τ ( x + τ ∇ f ( x )) = ∇ f ( x ) . (7) (iv) The following monotonicity properties hold for all x ∈ H : |∇ f | ( J τ ( x )) ≤ |∇ f τ | ( x ) = | x − J τ ( x ) | τ ≤
11 + λτ |∇ f | ( x ) . (8) Proof.
The inclusion in (6) follows from performing variations around J τ ( x ) in (5).Before proving the equality in (6), let us prove the Lipschitz property for J τ given in (ii).Recall that the convexity of g = f − λ | · | yields that ∂f is λ -monotone, namely h ξ − η, a − b i ≥ λ | a − b | ∀ ξ ∈ ∂f ( a ) , η ∈ ∂f ( b ) . Given x and y , we apply this property to a := J τ ( x ) , b := J τ ( y ) , ξ := ( x − J τ ( x )) /τ and η := ( y − J τ ( y )) /τ . (Thanks to the inclusion in (6), we have ξ ∈ ∂f ( a ) and η ∈ ∂f ( b ) .) Byrearranging the terms, we get h x − y, J τ ( x ) − J τ ( y ) i ≥ (1 + λτ ) | J τ ( x ) − J τ ( y ) | . Hence, by the Cauchy-Schwarz inequality, J τ is (1 + λτ ) − -Lipschitz.Let us go back to proving the equality in (6). For any x and z , one has (using y = J τ ( x ) asan admissible competitor in the definition of f τ ( x + z ) ) f τ ( x + z ) − f τ ( x ) ≤ | J τ ( x ) − ( x + z ) | τ − | J τ ( x ) − x | τ = (cid:28) z, x − J τ ( x ) τ (cid:29) + | z | τ and, reversing the roles of x and x + z , f τ ( x ) − f τ ( x + z ) ≤ (cid:28) − z, x + z − J τ ( x + z ) τ (cid:29) + | z | τ . These two identities together with the continuity of J τ imply that f τ is differentiable at x andprovides the equality in (6) and hence the one in (8). The Lipschitz property for ∇ f τ announcedfollows directly from this identity and the Lipschitz property for J τ .To get (7), it suffices to remark that for all x ∈ D ( ∂f ) , belongs to the subdifferential ofthe strictly convex function y f ( y ) + | x + τ ∇ f ( x ) − y | τ at y = x . Hence, x is the minimizer of this function, and J τ ( x + τ ∇ f ( x )) = x . Then, wededuce (6) from (7).The first inequality in (8) follows from the inclusion in (6). In order to prove the secondinequality, we perform a variation along the affine curve joining x to J τ ( x ) , namely, γ ( t ) :=(1 − t ) x + tJ τ ( x ) . Since f ( J τ ( x )) + 12 τ | x − J τ ( x ) | ≤ f ( γ ( t )) + 12 τ | x − γ ( t ) | ≤ (1 − t ) f ( x ) + tf ( J τ ( x )) + t τ (cid:0) t − λτ (1 − t ) (cid:1) | x − J τ ( x ) | t ∈ [0 , , taking the left derivative at t = 1 gives (cid:18) λ τ (cid:19) | x − J τ ( x ) | ≤ f ( x ) − f ( J τ ( x )) , so that the representation formula (4) for |∇ f | ( x ) gives (cid:18) λ τ (cid:19) | x − J τ ( x ) | ≤ |∇ f | ( x ) | x − J τ ( x ) | − λ | x − J τ ( x ) | . By rearranging the terms, this leads to the second inequality in (8).Another remarkable property of |∇ f | , for f λ -convex and lower semicontinuous, is the uppergradient property, namely, f ( γ (0)) , f ( γ ( δ )) < ∞ and | f ( γ ( δ )) − f ( γ (0)) | ≤ Z δ |∇ f | ( γ ( t )) | ˙ γ ( t ) | d t for any δ > and any absolutely continuous γ : [0 , δ ] → H (with the convention × ∞ = 0 ),whenever γ is not constant and the integral in the right hand side is finite (see for instanceCorollary 2.4.10 in [4] for the proof). For δ > and f : H → ( −∞ , ∞ ] proper, λ -convex and lower semicontinuous, we consider theautonomous functionals I δf : C ([0 , δ ]; H ) → [0 , ∞ ] defined by I δf ( γ ) := Z δ n | ˙ γ | + |∇ f | ( γ ) o d t, set to + ∞ on C ([0 , δ ]; H ) \ AC ([0 , δ ]; H ) . Notice also that I δf ( γ ) < ∞ implies γ ∈ D ( ∂f ) a.e.in (0 , δ ) .Identity (4) ensures the lower semicontinuity of |∇ f | ; hence, under a coercitivity assumptionof the form { f ≤ t } compact in H for all t ∈ R , the infimum Γ δ ( x , x δ ) := inf n I δf ( γ ) : γ (0) = x , γ ( δ ) = x δ o x , x δ ∈ H (9)is always attained whenever finite.Also, by the Young inequality and the upper gradient property of |∇ f | , one has that I δf ( γ ) < ∞ implies γ (0) , γ ( δ ) ∈ D ( f ) and | f ( γ ( δ )) − f ( γ (0)) | ≤ I δf ( γ ) . The same argument shows thatwe may add to I δf a null Lagrangian. Namely, as done in [3], we can consider the functionals Z δ | ˙ γ − ∇ f ( γ ) | d t which differ from I δf precisely by the term f ( γ ( δ )) − f ( γ (0)) , whenever γ is admissible in (9)with I δf ( γ ) < ∞ .Because of the lack of continuity of x
7→ ∇ f ( x ) , very little is known in general about theregularity of minimizers in (9), even when H is finite-dimensional. However, one may use thefact that I f is autonomous to perform variations of type γ γ ◦ (Id + ǫφ ) , φ ∈ C ∞ c (0 , δ ) , toobtain the Dubois-Reymond equation (see for instance [2]) dd t (cid:2) | ˙ γ | − |∇ f | ( γ )) (cid:3) = 0 in the sense of distributions in (0 , δ ) .
5t implies Lipschitz regularity of the minimizers when, for instance, |∇ f | is bounded on boundedsets (an assumption satisfied in [3], but obviously too strong for some applications in infinitedimension).We will need the following lemma, estimating Γ δ from above, to adjust the values of thecurves at the endpoints. The heuristic idea is to interpolate on the graph of f τ and then readback this interpolation in the original variables. This is related to Minty’s trick (see [1] for anextensive use of this idea): a rotation of π/ maps the graph of the subdifferential onto thegraph of an entire -Lipschitz function; here we use only slightly tilted variables, of order τ . Lemma 3.1 (Interpolation) . Let f : H → ( −∞ , ∞ ] be a proper, λ -convex and lower semicon-tinuous function and let τ > be such that (1 + τ λ ) − ≤ . For all δ > and all x ∈ D ( ∂f ) , x δ ∈ D ( ∂f ) , with Γ δ as in (9) , one has Γ δ ( x , x δ ) ≤ δ min i ∈{ ,δ } |∇ f | ( x i ) + (cid:18) δ + 12 δτ (cid:19) | x δ − x | + (cid:18) δ + 40 τ δ (cid:19) |∇ f ( x δ ) − ∇ f ( x ) | . Proof.
We use Theorem 2.1 to interpolate between x δ and x as follows: set ˜ γ ( t ) := (cid:18) − tδ (cid:19) ( x + τ ∇ f ( x )) + tδ ( x δ + τ ∇ f ( x δ )) , ξ ( t ) := ∇ f τ (˜ γ ( t )) , and γ ( t ) := J τ (˜ γ ( t )) = ˜ γ ( t ) − τ ξ ( t ) , where the second equality follows from (6).Since ξ (0) = ∇ f τ ( x + τ ∇ f ( x )) = ∇ f ( x ) and a similar property holds at time δ , the path γ is admissible. Let us now estimate the action of the path γ .Kinetic term (we use our Lipschitz bound for ∇ f τ to deduce that | ˙ ξ ( t ) | ≤ τ | ˙˜ γ ( t ) | ): Z δ | ˙ γ | d t ≤ Z δ | ˙˜ γ | d t + 2 τ Z δ | ˙ ξ | d t ≤ Z δ | ˙˜ γ | d t = 20 δ | ( x δ + τ ∇ f ( x δ )) − ( x + τ ∇ f ( x )) | ≤ δ | x δ − x | + 40 τ δ |∇ f ( x δ ) − ∇ f ( x ) | . Gradient term (we use the first inequality in (8), our Lipschitz bound for ∇ f τ , and finally (7)): Z δ |∇ f | ( γ )d t ≤ Z δ |∇ f τ | (˜ γ )d t ≤ Z δ (cid:18) |∇ f τ | (˜ γ (0)) + 3 τ | ˜ γ ( t ) − ˜ γ (0) | ) (cid:19) d t ≤ Z δ (cid:26) |∇ f | ( x ) + 18 τ t δ | ( x δ + τ ∇ f ( x δ )) − ( x + τ ∇ f ( x )) | (cid:27) d t ≤ δ |∇ f | ( x ) + 6 δτ | ( x δ + τ ∇ f ( x δ )) − ( x + τ ∇ f ( x )) | ≤ δ |∇ f | ( x ) + 12 δτ | x δ − x | + 12 δ |∇ f ( x δ ) − ∇ f ( x ) | . We get the result by gathering these two estimates, and by remarking that in the second line,we could have controlled |∇ f | τ (˜ γ ( t )) by its value at time δ instead of its value at time .6hoosing δ = τ , bounding |∇ f | ( x i ) , i = 0 , by the max of these two values, and using |∇ f ( x δ ) − ∇ f ( x ) | ≤ i ∈{ , } |∇ f | ( x i ) , we will apply the interpolation lemma in the form Γ δ ( x , x δ ) ≤ τ | x δ − x | + 210 τ max i ∈{ ,δ } |∇ f | ( x i ) . (10) In this section, f h , f denote generic proper, λ -convex and lower semicontinuous functions from H to ( −∞ , ∞ ] .Mosco convergence is a particular case of Γ -convergence, where the topologies used for the lim sup and the lim inf inequalities differ. Definition 4.1 (Mosco convergence) . We say that f h Mosco converge to f whenever:(a) for all x ∈ H there exist x h → x strongly with lim sup h f h ( x h ) ≤ f ( x ); (b) for all sequences ( x h ) ⊂ H weakly converging to x , one has lim inf h f h ( x h ) ≥ f ( x ) . It is easy to check that for sequences of λ -convex functions, Mosco convergence implies thepointwise convergence of J f h ,τ to J f,τ , contrarily to usual Γ -convergence. Indeed, for τ > admissible, (a) grants lim sup h →∞ min y ∈ H f h ( y ) + | y − x | τ ≤ min y ∈ H f ( y ) + | y − x | τ , while (b) grants lim inf h →∞ min y ∈ H f h ( y ) + | y − x | τ ≥ min y ∈ H f ( y ) + | y − x | τ , and the weak convergence of minimizers y h to the minimizer y . Eventually, the convergence ofthe energies together with lim inf h →∞ f h ( y h ) ≥ f ( y ) and lim inf h →∞ | y h − x | ≥ | y − x | grants that both liminf are limits, and that the convergence of y h is strong.Recall that given x h, , x h, ∈ H , the functionals Θ f h ,x h, ,x h, defined in (2), are obtainedfrom I f h by adding endpoints constraints. Θ f,x ,x is defined analogously.We say that Θ f h ,x h, ,x h, Γ -converge to Θ f,x ,x in the C ([0 , H ) topology if(a) for all γ ∈ C ([0 , H ) there exist γ h ∈ C ([0 , H ) converging to γ with lim sup h →∞ Θ f h ,x h, ,x h, ( γ h ) ≤ Θ f,x ,x ( γ ); (b) for all sequences ( γ h ) ⊂ C ([0 , H ) converging to γ one has lim inf h →∞ Θ f h ,x h, ,x h, ( γ h ) ≥ Θ f,x ,x ( γ ) .
7n connection with the proof of property (a) it is useful to introduce the functional Γ − lim sup h →∞ Θ f h ,x h, ,x h, ( γ ) := inf (cid:26) lim sup h →∞ Θ f h ,x h, ,x h, ( γ h ) : γ h → γ (cid:27) so that (a) is equivalent to Γ − lim sup h Θ f h ,x h, ,x h, ≤ Θ f,x ,x . Recall also that the Γ − lim sup is lower semicontinuous, a property that can be achieved, for instance, by a diagonal argument. Proof of Theorem 1.1.
It is clear that the endpoint condition passes to the limit with respect tothe C ([0 , H ) topology, since x h,i converge to x i . Also, it is well known that the action func-tional is lower semicontinuous in C ([0 , H ) . Hence, the Γ -liminf inequality, namely property(b), follows immediately from Fatou’s lemma and the variational characterization (4) of |∇ f | .Indeed, for all y = x and all sequences x h → x lim inf h →∞ |∇ f h | ( x h ) ≥ lim inf h →∞ [ f h ( x h ) − f h ( y h ) + λ | x h − y h | (cid:3) + | x h − y h | ≥ [ f ( x ) − f ( y ) + λ | x − y | (cid:3) + | x − y | . where y h is chosen as in (a) of Definition 4.1. Passing to the supremum, we get the inequality lim inf h |∇ f h | ( x h ) ≥ |∇ f | ( x ) , and this grants the lower semicontinuity of the gradient term in thefunctionals. Notice that this part of the proof works also if we assume only that Γ - lim inf h f h ≥ f ,for the strong topology of H , but the stronger property (namely (b) in Definition 4.1) is necessarybecause we will need in the next step convergence of the resolvents.So, let us focus on the Γ -limsup one, property (a). Fix a path γ with Θ f,x ,x ( γ ) < ∞ , τ > (with (1 + τ λ − ) ≤ if λ < ) and consider the perturbed paths γ τh ( t ) = J f h ,τ ( γ ( t )) , γ τ ( t ) = J f,τ ( γ ( t )) ; using the (1 + τ λ ) − -Lipschitz property of the maps J f,τ , the first inequalityin (8), the convergence of γ τh to γ τ and eventually the second inequality in (8) one gets lim sup h →∞ Z (cid:8) | ˙ γ τh | + |∇ f h | ( γ τh ) (cid:9) d t ≤ lim sup h →∞ Z (cid:26) (1 + τ λ ) − | ˙ γ | + | γ − γ τh | τ (cid:27) d t ≤ Z (cid:26) (1 + τ λ ) − | ˙ γ | + | γ − γ τ | τ (cid:27) d t ≤ (1 + τ λ ) − Z (cid:8) | ˙ γ | + |∇ f | ( γ ) (cid:9) d t. Also, the convergence of resolvents gives lim h →∞ J f h ,τ ( x i ) = J f,τ ( x i ) . Finally, using again the inequalities (8) and once more the convergence of resolvents, we get lim sup h →∞ |∇ f h | ( J f h ,τ ( x i )) ≤ | J f,τ ( x i ) − x i | τ ≤ (1 + τ λ ) − |∇ f | ( x i ) ≤ |∇ f | ( x i ) . Since the endpoints have been slightly modified by the composition with J f h ,τ , we argue asfollows. Denoting by S an upper bound for |∇ f h | ( x h,i ) and |∇ f | ( x i ) , we apply twice theconstruction of Lemma 3.1, with δ = τ , to f h with endpoints x h,i , J f h ,τ ( x i ) , to extend thecurves γ τh , still denoted γ τh , to the interval [ − τ, τ ] , in such a way that (we use (10) in thefirst inequality, and the second inequality in (8) in the second one) lim sup h →∞ Z δ − δ (cid:8) | ˙ γ τh | + |∇ f h | ( γ τh ) (cid:9) d t ≤ (1 + τ λ ) − Z (cid:8) | ˙ γ | + |∇ f | ( γ ) (cid:9) d t + 420 τ S + 52 τ n | x − J f,τ ( x ) | + | x − J f,τ ( x ) | o ≤ (1 + τ λ ) − (cid:18)Z (cid:8) | ˙ γ | + |∇ f | ( γ ) (cid:9) d t + 472 τ S (cid:19) t = − τ and t = 1 + τ . The limit of the curves γ τh in [ − τ, τ ] , still denoted γ τ , is the one obtained applying the construction of Lemma 3.1 with x i and J f,τ ( x i ) in the intervals [ − τ, and [1 , τ ] , and which coincides with J f,τ ( γ ( t )) on [0 , .By a linear rescaling of the curves γ τh and γ τ to [0 , we obtain curves ˜ γ τh converging to ˜ γ τ in C ([0 , H ) , with ˜ γ τ convergent to γ as τ → and Γ − lim sup h →∞ Θ f h ,x h, ,x h, (˜ γ τ ) ≤ lim sup h →∞ Θ f h ,x h, ,x h, (˜ γ hτ ) ≤ (1 + O ( τ )) Z (cid:8) | ˙ γ | + |∇ f | ( γ ) (cid:9) d t + O ( τ ) . Eventually, the lower semicontinuity of the Γ -upper limit and the convergence of ˜ γ τ to γ provide: Γ − lim sup h →∞ Θ f h ,x h, ,x h, ( γ ) ≤ Z (cid:8) | ˙ γ | + |∇ f | ( γ ) (cid:9) d t. References [1]
G. Alberti, L. Ambrosio:
A geometric approach to monotone functions in R n . Matem-atische Zeitschrift, (1999), 259–316.[2]
L. Ambrosio, G. Buttazzo, O. Ascenzi:
Lipschitz regularity for minimizers of integralfunctionals with highly discontinuous coefficients.
J. Math. Anal. Appl., (1989), 301–316.[3]
L. Ambrosio, A. Baradat, Y. Brenier:
Monge-Ampère gravitation as a Γ -limit of goodrate functions. Preprint, 2020.[4]
L. Ambrosio, N. Gigli, G. Savaré : Gradient flows in metric spaces and in the space ofprobability measures.
Lectures in Mathematics, ETH Zürich, Birkhäuser (2008).[5]
L. Ambrosio, N. Gigli, G. Savaré : Calculus and heat flow in metric measure spacesand applications to spaces with Ricci bounds from below.
Inventiones Mathematicae, (2014), 289–391.[6]
Y. Brenier:
A double large deviation principle for Monge-Ampère gravitation.
Bull. Inst.Math. Acad. Sin. (N.S.), (2016), 23–41.[7] H. Brezis:
Opérateurs maximaux monotones et semi-groupes de contractions dans lesespaces de Hilbert.
North-Holland Publishing Co., Amsterdam, 1973.[8]
G. Clerc, I. Gentil, G. Conforti:
On the variational interpretation of local logarithmicSobolev inequalities.
Preprint, 2021.[9]
A. Dembo, O. Zeitouni:
Large deviation techniques and applications.
Applications ofMathemmatics , Springer, 1998.[10] P. Dondl, T. Frenzel, A. Mielke:
A gradient system with a wiggly energy and relaxedEDP-convergence.
ESAIM Control Optim. Calc. Var., (2019), paper no. 68, 45pp.[11] A. Mielke, M.A. Peletier, D.R.M. Renger:
On the relation between gradient flowsand the large-deviation principle, with applications to Markov chains and diffusion.
PotentialAnal., (2014), 1293–1327.[12] E. Sandier, S. Serfaty: Γ -Convergence of Gradient Flows with Applications to Ginzburg-Landau. Comm. Pure Appl. Math.,57