Article

PDF
Access to the PDF text
Advertising


Free Article !

Comptes Rendus Mathématique
Volume 355, n° 2
pages 170-175 (février 2017)
Doi : 10.1016/j.crma.2016.12.011
Received : 15 February 2016 ;  accepted : 5 January 2017
Large deviations of a velocity jump process with a Hamilton–Jacobi approach
Grandes déviations pour un processus à sauts de vitesse avec une approche de Hamilton–Jacobi
 

Nils Caillerie
 Université de Lyon, Université Claude-Bernard Lyon-1, CNRS UMR 5208, Institut Camille-Jordan, 43 bd du 11-Novembre-1918, 69622 Villeurbanne cedex, France 

Abstract

We study a random process on   moving in straight lines and changing randomly its velocity at random exponential times. We focus more precisely on the Kolmogorov equation in the hyperbolic scale  , with  , before proceeding to a Hopf–Cole transform, which gives a kinetic equation on a potential. We show convergence as   of the potential towards the viscosity solution to a Hamilton–Jacobi equation   where the Hamiltonian may lack   regularity, which is quite unseen in this type of studies.

The full text of this article is available in PDF format.
Résumé

Nous nous intéressons à un processus aléatoire sur   qui alterne des phases de mouvements rectilignes uniformes et change de vitesse à des temps exponentiels. Nous étudions plus précisément l'équation de Kolmogorov après rééchelonnement hyperbolique  ,  , puis nous effectuons une transformée de Hopf–Cole qui nous donne une équation cinétique suivie par un potentiel. Nous montrons la convergence pour   de ce potentiel vers la solution de viscosités d'une équation de Hamilton–Jacobi   où le hamiltonien peut présenter une singularité  , ce qui est assez inédit dans ce type d'études.

The full text of this article is available in PDF format.
Version française abrégée

Nous nous donnons une densité de probabilité   et nous notons V son support. Nous supposons que V est compact et que 0 appartient à l'intérieur de l'enveloppe convexe de V , que l'on note  . Pour  , nous notons  . Nous étudions le mouvement de particules dans   suivant le processus de Markov déterministe par morceaux défini comme suit : une particule donnée se déplace de manière rectiligne uniforme avec une vitesse   tirée aléatoirement en suivant la loi de probabilité  . À des temps exponentiels de paramètre 1, la particule change de direction en tirant une nouvelle vitesse tirée selon la loi  . Afin d'étudier des résultats de larges déviations du processus similairement aux techniques développées dans [[3], [4], [5], [6], [7], [8]], nous nous intéressons à l'équation de Chapman–Kolmogorov forward suivie par la densité de particules après un rééchelonement hyperbolique  ,   :
∂tfε+v⋅∇xfε=1ε(M(v)ρε−fε),(t,x,v)∈R+×Rn×V. Nous étudions plus particulièrement l'équation vérifiée par un potentiel   obtenu après passage par une transformée de Hopf–Cole :  . Nous cherchons alors une éventuelle limite pour  . Nous procédons à un développement WKB :  , ce qui amène, en posant   et  , à la résolution d'un problème spectral dans l'espace des mesures positives : chercher   un couple valeur/vecteur propres associé à l'opérateur  . On obtient une équation de Hamilton–Jacobi  . Pour   et   sur son support, le vecteur propre Q a une densité et conduit à un hamiltonien H défini par l'équation implicite∫VM(v)1+H(p)−v⋅pdv=1. La positivité de Q garantit que  . En dimension supérieure toutefois, et même si  , cette équation peut ne pas avoir de solution   lorsque p devient grand. Cela se manifeste pour le vecteur propre par une concentration de la mesure Q autour des valeurs v qui annulent  , ce qui force  . Cette transition entraîne une singularité   du hamiltonien.

Nous démontrons la convergence de   vers φ , où φ est solution de viscosité [[5]] de l'équation de Hamilton–Jacobi en utilisant la méthode de la fonction test perturbée [[7]].

Introduction

We continue the work initiated in [[1], [2]]. Let   be a probability density function. We suppose that the support of M , which we denote V , is compact and that 0 belongs to the interior of  , the convex hull of V . We denote by   the Euclidian norm in   and by ⋅ the canonical scalar product. For  , we define
(1)μ(p):=max{v⋅p|v∈Conv(V)},   and  .

We focus on the motion dynamics in   of particles given by the following piecewise deterministic Markov process: a particle moves successively in straight lines with velocity v , chosen randomly with probability distribution  . At random exponential times (with parameter 1), the particle changes its velocity, choosing randomly a new velocity with distribution  . The Chapman–Kolmogorov forward equation associated with the probability density function   of this process is given by:
(2)∂tf+v⋅∇xf=Mρ−f,(t,x,v)∈R+×Rn×V, where  . In order to investigate large deviation principles for the process, one can study the large scale hyperbolic limit   with  . In this scale, the kinetic equation ((2)) reads:(3)∂tfε+v⋅∇xfε=1ε(Mρε−fε),(t,x,v)∈R+×Rn×V. Then, we perform the following Hopf–Cole transformation:  , where we expect the potential   to become independent of v as  . Such techniques have already been studied for a more general case of Markov process with a finite discrete set of states in [[3]] and, from a probabilistic point of view, in [[8]].

Here, assume that the initial condition is well-prepared, i.e. it does not depend on v :  . We believe that the conclusion of this paper is not dramatically modified if  . Indeed, the only expected change concerns the initial condition of the Hamilton–Jacobi equation, which should be independent of v . This is left for future work. The equation satisfied by   reads
(4)∂tφε+v⋅∇xφε=∫VM(v′)(1−eφε−φ′εε)dv′,(t,x,v)∈R+×Rn×V. As in [[9]], the limit potential satisfies a Hamilton–Jacobi equation. Surprisingly enough, our Hamiltonian may lack   regularity as we will show in Proposition 2.

Theorem 1

Under the previous assumptions,   converges locally uniformly on   toward φ, where φ does not depend on v. Moreover, φ is the viscosity solution to the following Hamilton–Jacobi equation:
(5)∂tφ(t,x)+H(∇xφ(t,x))=0,(t,x)∈R+×Rn, with initial condition   and a Hamiltonian H given as follows: if  , then  . Else,   is uniquely determined by the following formula: (6)∫VM(v)1+H(p)−v⋅pdv=1.

A corollary to Theorem 1 is that Theorem 1.1 from [[2]] is only correct when  , since there is no solution to ((6)) when  . The present result establishes the appropriate statement in the case  . Interestingly enough, the proof appears quite different due to the apparition of Dirac masses in the velocity variable in the expression of the corrector.

Identification of the Hamiltonian

In order to identify the limit  , we perform the formal expansion   where Q is to be determined. Plugging this ansatz into the kinetic formulation ((3)) and writing   and  , we get (at the formal limit  ) the following spectral problem:
(7)(1+H−v⋅p)Q=∫VM(v′)Q(v′)dv′. A similar spectral problem has been studied in [[4]] in a more general case. The positivity of Q yields   for all   hence  . Suppose  . Then,   for all   and  . Integrating against M with respect to v , we obtain the following problem: find H such that  . If  , by monotonicity, such H exists and is unique. Equation ((6)), however, does not have an   solution for  . Similarly to [[4]], we look for solutions in a larger set, namely the set of positive measures. Then, a solution to the spectral problem is the eigenvalue   associated with the positive measure  , where   and   is the Dirac measure centered in  . Here is an example where  :

Example 1

Let   and   where   is the Lebesgue measure of the n-dimensional unit ball. Then,  . Indeed, for  , we have   and   hence
∫VM(v)μ(p)−v⋅pdv=1|p|ωn∫B(0,1)11−v1dv=ωn−1|p|ωn∫−11(1−v12)n−121−v1dv1=1|p|×nn−1. By rotational invariance, we conclude that  . The Fig. 1 gives illustrations of the Hamiltonian and μ as functions of the radius of p, in the cases   and  . In the cases   we can see the   singularity where  .



Fig. 1


Fig. 1. 

Blue plain lines: Hamiltonian for n =1,3 and  . Black dotted lines:  . Lignes pleines bleues : Hamiltonien pour n =1,3 et  . Lignes noires en pointillés :  .

Zoom



Fig. 1


Fig. 1. 

Blue plain lines: Hamiltonian for n =1,3 and  . Black dotted lines:  . Lignes pleines bleues : Hamiltonien pour n =1,3 et  . Lignes noires en pointillés :  .

Zoom

Proposition 2

The following properties hold:

(i)
The set   is convex.
(ii)
The function H is continuous and convex.
(iii)
If  , then H is not  . More precisely,H has a jump discontinuity at  .

Proof

Let us first notice that μ is positively 1-homogeneous. Moreover, it is convex since it is a supremum of linear functions.

(i) Let   with  . Since μ is convex, we have for all  
I(τ):=∫VM(v)μ(p)−v⋅p+τ(μ(q)−μ(p)−v⋅(q−p))dv≤∫VM(v)μ((1−τ)p+τq)−v⋅((1−τ)p+τq)dv. Moreover,   and I is differentiable on   with∂τI(τ)=∫VM(v)(μ(p)−v⋅p+τ(μ(q)−μ(p)−v⋅(q−p)))2(μ(p)−μ(q)−v⋅(p−q))dv. It is clear that the sign of   does not change hence  , which proves (i).

(ii) We refer to [[2]], section 1, to prove that H is twice differentiable and strictly convex on   and that
(8)∫VM(v)(1+H(q)−v⋅q)2(∇H(q)−v)dv=0,∀q∈Sing(M)c. In particular,   for all  . It is easy to see that H is continuous in the interior of  . To show continuity of H on  , let   converge to  . If we can extract a subsequence  , then  . If not, then   for m large enough and  . Taking the limit, we get by dominated convergence   hence  .

We now show that H is convex by proving that it is a maximum of convex functions:
(9)H(p)=max(sup{∇H(q)⋅(p−q)+H(q)|q∈Sing(M)c},μ(p)−1),∀p∈Rn. In  , ((9)) holds by convexity of H and  . Let   and  . By convexity of  , there exists a unique   such that  . For all  , we set   and  . By continuity of H ,   hence  . Moreover,   and   are both differentiable and   since  . Hence,  , which ends the proof of (ii).

(iii) Suppose   and H is  . Since   is positive homogeneous of degree 1 on   and since   for all   and  , we know that   for all   hence  , for all  , the inequality being strict on a neighborhood of 0. Then,
(10)p⋅∫VM(v)(1+H(p)−v⋅p)2(∇H(p)−v)dv>0,∀p∈∂Sing(M). By continuity, equations ((8)) and ((10)) are contradictory.  □

Proof of Theorem 1

Let  . We refer to Proposition 2.1 in [[2]] to prove that the Cauchy Problem ((4)) with initial condition   has a unique solution   which is locally (in t ) uniformly (in ε , x and v ) bounded in norm  . In particular, let us mention that
(11)0≤φε(t,⋅,⋅)≤‖φ0‖∞,‖∇vφε(t,⋅,⋅)‖∞≤t‖∇xφ0‖∞. Using the Arzelá–Ascoli theorem, we extract a locally uniformly converging subsequence. We denote by φ the limit. The function φ does not depend on v since   is uniformly bounded on   for all  . We use the perturbed test function method [[7]] to show that φ is a viscosity solution to ((5)). Theorem 1 will follow by uniqueness of the solution [[6]], thanks to the properties of H (see Proposition 2).

Subsolution procedure

Let   be a test function such that   has a local strict maximum at  . We want to show that ψ is a subsolution to ((5)). If  , then we refer to [[2]], section 2, step 2.

Suppose now that  . Let  . Then,  . The uniform convergence of   toward φ ensures that the function   has a local maximum at a point   satisfying   , as  . We then have:
∂tψ(tε,xε)+w⋅∇xψ(tε,xε)=∂tφε(tε,xε)+w⋅∇xφε(tε,xε)=1−∫VM(v′)eφε(tε,xε,w)−φε(tε,xε,v′)εdv′≤1.

Passing to the limit  , we get  . We conclude that φ is a viscosity subsolution to ((5)).

Supersolution procedure

Let   be a test function such that   has a local strict minimum at  . We want to show that ψ is a supersolution to ((5)). If  , then we refer to [[2]], section 2, step 2.

Suppose now that  . Then,   because  . We suppose without loss of generality that the minimum of   is global and that  . Let   with   yet to be determined and
η(v):=ln(μ(∇xψ(t0,x0))−v⋅∇xψ(t0,x0)). Then, η is a continuous function on   and, for all  , we have  . Moreover, η is bounded from below on all compact sets yielding the uniform convergence   on all compact sets of  . Finally,   since  .

The function   has a global strict minimum at  . The first inequality ((11)) ensures that the function   has a local minimum at a point  . As V compact, we can extract a subsequence  , without relabeling, such that  , as  .

If  , then there exists a compact   such that   and the uniform convergence of   towards ψ on A guarantees that  , as  . We then get at point  ,
∂tψ−2C(tε−t0)+vε⋅∇xψ=∂tψε+vε⋅∇xψε=∂tφε+vε⋅∇xφε=1−∫VM′eφε−φ′εεdv′≥1−∫VM(v′)eη(vε)−η(v′)dv′. We take the limit  :∂tψ(t0,x0)+v0⋅∇xψ(t0,x0)≥1−eη(v0)∫VM(v′)e−η(v′)dv′≥1−eη(v0). By construction, for all  , we have   hence, for all  ,we have  . Let  . Since   is a null-set, V is dense in  . Taking the limit  , we get:  .

If  , we still have   thanks to the following lemma:

Lemma 3

For  , we have  .

Proof of Lemma 3

We have   by ((11)) and   hence
φε(t,x,v)−ψε(t,x,v)≥−2‖φ0‖∞+C(t−t0)2−εη(v),∀ε>0. Moreover,φε(t0,x0,v)−ψε(t0,x0,v)=φε(t0,x0,v)−φ(t0,x0)−εη(v)≤2‖φ0‖∞−εη(v). Since  , we have   for all   and, thus, the minimum of   cannot be attained for   hence   for all  . At point   we have:∇vφε(tε,xε,vε)=∇vψε(tε,xε,vε)=ε∇vη(vε)=−ε∇xψ(t0,x0)μ(∇xψ(t0,x0))−vε⋅∇xψ(t0,x0). The second estimation ((11)) yields   henceε(t0+1)‖∇xφ0‖∞|∇xψ(t0,x0)|≤μ(∇xψ(t0,x0))−vε⋅∇xψ(t0,x0),⇒εK≥εη(vε)≥εln(ε(t0+1)‖∇xφ0‖∞|∇xψ(t0,x0)|), and   as  .  □

Thanks to Lemma 3, the function   converges uniformly towards   and has a local minimum at   satisfying  , as  . At point  , we have:
∂tψε+vε⋅∇xψε=∂tφε+vε⋅∇xφε=1−∫VM(v′)eφε(tε,xε,vε)−φε(tε,xε,v′)εdv′. The minimal property of   implies at this point:∂tψ(tε,xε)−8‖φ0‖∞(tε−t0)+vε⋅∇xψ(tε,xε)=∂tψε+vε⋅∇xψε≥1−∫VM(v′)eη(vε)−η(v′)dv′≥1−eη(vε). Passing to the limit  , we get  . We conclude that φ is a viscosity supersolution to ((5)).  □


Acknowledgements

This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 639638).

The author also wishes to thank Vincent Calvez and Julien Vovelle for their kind help.

References

Bouin E. A Hamilton–Jacobi approach for front propagation in kinetic equations Kinet. Relat. Models 2015 ;  8 (2) : 255-280 [cross-ref]
Bouin E., Calvez V. A kinetic eikonal equation C. R. Acad. Sci. Paris, Ser. I 2012 ;  350 (5–6) : 243-248 [inter-ref]
P. Bressloff, O. Faugeras, On the Hamiltonian structure of large deviations in stochastic hybrid systems, 2015, hal-01072077v2.
Coville J. Singular measure as principal eigenfunction of some nonlocal operators Appl. Math. Lett. 2013 ;  26 (8) : 831-835 [cross-ref]
Crandall M.G., Lions P.-L. Viscosity solutions of Hamilton–Jacobi equations Trans. Amer. Math. Soc. 1983 ;  277 : 1-42 [cross-ref]
Crandall M.G., Evans L.C., Lions P.L. Some properties of viscosity solutions of Hamilton–Jacobi equations Trans. Amer. Math. Soc. 1984 ;  282 : 487-502 [cross-ref]
Evans L.C. The perturbed test function method for viscosity solutions of nonlinear PDE Proc. R. Soc. Edinb., Sect. A 1989 ;  111 : 359-375 [cross-ref]
Faggionato A., Gabrielli D., Ribezzi Crivellari M. Averaging and large deviations principles for fully piecewise deterministic Markov process and applications to molecular motors Markov Process. Relat. Fields 2010 ;  16 : 497-548
Freidlin M.I., Wentzell A.D. Random Perturbations of Dynamical Systems  New York: Springer-Verlag (1998). 



© 2017  Académie des sciences@@#104156@@
EM-CONSULTE.COM is registrered at the CNIL, déclaration n° 1286925.
As per the Law relating to information storage and personal integrity, you have the right to oppose (art 26 of that law), access (art 34 of that law) and rectify (art 36 of that law) your personal data. You may thus request that your data, should it be inaccurate, incomplete, unclear, outdated, not be used or stored, be corrected, clarified, updated or deleted.
Personal information regarding our website's visitors, including their identity, is confidential.
The owners of this website hereby guarantee to respect the legal confidentiality conditions, applicable in France, and not to disclose this data to third parties.
Close
Article Outline