On nonlocal Hamilton-Jacobi equations related to jump processes, some recent results Daria Ghilli Institute of mathematics and scientic computing, Graz, Austria 25/11/2016 Workshop "Numerical methods for Hamilton-Jacobi equations in optimal control and related elds", RICAM Linz
The Neumann boundary value problem { u(x) I[u](x) + H(x, Du) = 0 in Ω, u n = 0 on Ω the diusion I is a nonlocal operator, in our context a singular integral term, H is a nonlinear Hamiltonian, Ω R n is a general domain (enough smooth).
The Neumann boundary value problem { u(x) I[u](x) + H(x, Du) = 0 in Ω, u n = 0 on Ω the diusion I is a singular integral term related to discontinuous jumping processes, H is a nonlinear Hamiltonian, Ω R n is a general domain (enough smooth).
Probabilistic version: Lévy processes Right continuous, limits on the left. Examples: Brownian motion (the only continuous Lévy process), composed Poisson. Innitesimal generator: nonlocal operator of Lévy type
Infinitesimal generator Def: if X t a stochastic process starting at x and u is smooth if X t is continuous: The general form is 1 Lu(x) = lim t 0 t [E(u(X t) u(x))]. L[u](x) = 1/2Tr(σσ T D 2 u) + b Du(x), L[u](x) = 1/2Tr(σσ T D 2 u) + b Du(x) + T [u](x), Tr(σσ T D 2 u), σσ T is simmetric and positive: classical diusion (type laplacian); b Du(x): transport term (drift); T [u](x) : jump term
Nonlocal operators with Lévy type terms T [u](x) = [u(x + z) u(x)]dµ(z), R n where µ is a Lévy measure: R n (1 z 2 )µ(dz) < (2.1) Example: µ(dz) = dz z σ (0, 2); n+σ If u is C 1, T [u] is well-dened if z is µ-integrable in 0. µ(dz) = dz z σ (0, 1); n+σ T well-dened if u C 2 and µ is simmetric satisfying (2.1).
Fully nonlinear framework: basic references Fully nonlinear equations in all the space (without boundary condition): Very rst beginning, rst order equations: Soner 1986, Sayah 1991 Second order: Alvarez Tourin 1996, Jacobsen Karlsen 2006, Barles, Imbert 2008, Barles Chasseigne Imbert Ciomaga, Caerelli Sivestre 2006, 2009, Garroni Menaldi 2002 Numerical results: La Cioma Jakobsen Karlsen, Biswas Jakobsen Karlsen, Camilli Jakobsen, Huang Obermann.. Dirichlet problem for general second-order fully nonlinear HJB, Barles Chasseigne Imbert 2007 2008.
Fully nonlinear framework: viscosity solutions The solutions in general are not regular, in this way we give a sense to the dierential terms. Useful to deal with the integral terms when the measure is singular, by replacing u with a regular function: R n [u(x + z) u(x)]dµ(z) In case of boundary conditions, useful denition of "generalized" boundary conditions (see later in the talk)
Neumann condition: probabilistic approach In the classical probabilistic approach, Neumann problems are associated to stochastic processes being reected on the boundary: KEY RESULT: for a PDE with Neumann or oblique boundary conditions, there is a unique underlying reection process. Any consistent approximation converges to it (Lions-Snitzman, Barles-Lions). REMARK: This relies on the underlying stochastic processes being continuous (at least in the case of normal reections).
Neumann boundary condition (Ω ="halfspace") New nonlocal fenomena: Several ways to keep the process inside the domain The choice of the reection inuences the PDE inside the domain
Linear PIDE-BCGJ G. Barles, E.Chasseigne, C. Georgeline, E. R. Jakobsen, 2014, "On neumann type problems for non-local equations set in a half space" linear equations: u(x) I[u](x) + f(x) = 0 + Neumann BC domains with at boundary, i.e. the halfspace. (At least) four dierent and coherent (w.r.t the classical case) models of reection.
Mirror reection:
Normal projection, Lions-Sznitman: Close to the Lions-Sznitman approach. Problem investigated by Barles, Georgeline, Jakobsen 2013 in the framework of fully non-linear equations in general domains.
Stick on the wall, eas on the window: Outwards jumps are stopped where the jump path hits the boundary, and then the process is restarted there.
Contraction in the normal direction. Natural ways to dene "reections" (in particular mirror reection). To the best of our knowledge, processes with generators of the form "mirror" and "eas" have not been considered yet. It could be problematic to work with in general domains due to the possibility of multiple reections
Censored process: don't jump outside, censored: Any outwards jump is cancelled (censored) and the process is restarted (resurrected) at the origin of that jump.
Good model? dµ α = c α z (N+α) dz α 2 u = f u n = 0 Each model is good!
The censored process-infinitesimal generator We focus on the censored process. Actually we don't jump outside: I[u](x) = P.V. [u(x + z)) u(x)]dµ(z) x+z Ω The Neumann condition inuences the equation inside the domain. No need for conditions on Ω c. Note the bad dependence on x in the domain of integration
Underlying process-probabilistic references Question: is it realized by a concrete Markov process, i.e., does it corresponds to the generator of such a process? In the censored case, the process can be constructed via some probabilistic methods: Refs:K. Bogdan, K. Burdzy and Z. Chen. 2003 Q.Y. Guan. and Z.M. Ma. 2006 M. Fukushima, Y. Oshima and M. Takeda. 1994 N. Jacob 2005 Remark Constructed only via probabilistic methods, no characterization as solution of a stochastic equation.
The Neumann boundary value problem We consider: { u(x) I[u](x) + H(x, Du) = 0 in Ω, u n = 0 on Ω (NBVP) the diusion I is the censored operator: dz I[u](x) = P.V. [u(x + z) u(x)] z n+σ σ (0, 1); x+z Ω H is a nonlinear Hamiltonian, Ω R n is a general domain (enough smooth).
Type of singularity We consider dµ(z) dz 1, σ (0, 1). z n+σ Why σ < 1? Stronger singuarity σ [1, 2): open problem... Remark Even in the case of the halfspace and linear equations (that is, BCGJ), the results could not be optimal. That is, the comparison is established only in some Holder class of functions. Remark Our problem in the case σ (0, 1) is not standard due to: nonlinear term (non trivial to treat even in domains with at boundary); general domain: further diculties due to the general geometry of Ω.
Assumption on the Hamiltonian IMPORTANT: the growth of H in the gradient strictly dominates the nonlocal diusion. either coercive in the gradient term with superfractional growth H(x, p) = b(x) p m + a 1(x) p l + (a 2(x), p) f(x), where m > σ, b(x) b 0 > 0 x Ω, 0 < l < m, a 1, a 2, f, b are continuous and bounded functions, b is Lipschitz continuous. either of Bellman type, not necessarely coercive: H(x, p) = sup{ b(x, α) p l(x, α)}; α A where A is a compact metric space, b : Ω A R n and f : Ω A R are continuous and bounded functions, l is uniforly continuous, b is uniform Lipschitz continuous.
Main results Theorem (COMPARISON-G. 2016) Under the assumptions of the previous slides, let u be a bounded usc subsolution of (NBVP) and v a bounded lsc supersolution of (NBVP). Then u v in Ω. Applications: EXISTENCE, UNIQUENESS By Perron's method for integro-dierential equations, in the class of continuous functions. ASYMPTOTIC BEHAVIOUR FOR THE EVOLUTIVE PROBLEM Convergence as t + of the solutions of the evolutive problem to the associated stationary problem.
Key ideas & ingredients Take Ω the halfspace, that is, Ω = {x 1,, x n R n x n > 0}. Note that {x + z Ω} = {x n + z n 0}. Let x n y n. Preliminary remark I is integrable if z is away from zero. In the proof, we have to deal with dz I[u](x) = [u(x + z) u(x)] x n z n y n z Integrable if x, y are far from the boundary: MAIN DIFFICULTIES NEAR THE BOUNDARY. n+σ dz,
LINEAR (σ < 1, BCGJ): u(x) I[u](x) + f(x) = 0 + NBC Blow-up function on the boundary: The process does not reach the boundary. NONLINEAR (σ < 1, our framework): u(x) I[u](x) + H(x, Du) = 0 + NBC No blow-up: the nonlinear terms push the process to the boundary. Further diculties.
Key ideas & ingredients Flat boundaries: Main idea: localise on equidistant point from the boundary (x n = y n ); key ingredient: superfractional growth of the Hamiltonian (+ σ < 1); once we localize on such points, we can conclude. General domains (enough smooth): (As for at boundaries) localize on equidistant points (now d(x) = d(y)). Further technical diculties due to the general geometry; main idea: prove Lipschitz regularity of the nonlocal terms on equidistant (from the boundary) points. key ingredients: regularity of Ω, σ < 1.
Main references (for the Neumann problem) G. Barles, E. Chasseigne, C. Georgeline, E. R. Jakobsen, On Neumann type problems for non-local equations set in a half space. Trans. Amer. Math. Soc. 366 (2014), no. 9, 4873-4917. D. Ghilli, On Neumann problems for nonlocal Hamilton-Jacobi equations with dominating gradient terms 2016, accepted under minor revision in Calculus of Variations and Partial Dierential Equation.
Thank you for the attention.