Fractional Laplacian Grzegorz Karch 5ème Ecole de printemps EDP Non-linéaire Mathématiques et Interactions: modèles non locaux et applications Ecole Supérieure de Technologie d Essaouira, du 27 au 30 Avril 2015
Laplacian
Discrete random walk State: Let x i = i x with i Z be the location of the particle at time t n = n t. Dynamics: If the particle is in state x i at time step t n, it will jump either to x i 1 or to x i+1 with equal probabilities.
Discrete random walk Define p(m, n) = probability that the particle is in state x m at time step t n. REMARK p(m, n) = ( 1 2 ) n ( ) n = a ( ) n 1 n! 2 a!(n a)! where a = n + m. 2
Discrete random walk and the heat equation Master equation p(m; n) = 1 ) (m 2 p 1, n 1 + 1 ( ) 2 p m + 1; n 1.
Discrete random walk and the heat equation Master equation p(m; n) = 1 ) (m 2 p 1, n 1 + 1 ( ) 2 p m + 1; n 1. We now scale the master equation, using x 0, t 0, x 2 2 t = D. We assume that the scaled probabilities p(m, n) approach a continuous (and even twice differentiable!) function u(x; t) ( ) x u(x; t) = p x, t t
Discrete random walk and the heat equation Then u(x; t) = 1 2 u(x x, t t) + 1 u(x + x, t t), 2
Discrete random walk and the heat equation Then or equivalently, u(x; t) = 1 2 u(x x, t t) + 1 u(x + x, t t), 2 u(x; t) u(x, t t) t = x 2 2 t u(x x, t t) 2u(x, t t) + u(x + x, t t) x 2. In the limit x 0, t 0, and x 2 2 t = D, we obtain the heat equation u t = Du xx.
Discrete random walk and the heat equation In the case of the random walk on the d-dimensional lattice ( x)z d, we obtain ( ) d 2 u u t = D u = D xk 2. k=1
Discrete random walk and the heat equation In the case of the random walk on the d-dimensional lattice ( x)z d, we obtain ( ) d 2 u u t = D u = D xk 2. Fundamental solution N (x, t) = k=1 ( ) 1 x 2 exp (2πDt) n/2 4Dt
Laplace operator & Wiener process Brownian motion one trajectory of a Wiener process
Laplace operator & Wiener process Definition The stochastic process {W (t)} t 0 is called the Wiener process, if it fulfils the following conditions W (0) = 0 with probability equal to one, W (t) has independent increments, trajectories of W are continuous with probability equal to one 0 s t W t W s N (0, t s). For every function u 0 C b (R n ) we define u(x, t) = E x (u 0 (W (t))) = u 0 (x y) N (0, t)(dy), R n where N (0, t)(dy) = (2πt) n/2 e y 2 /(2t) dy. Hence u t = 1 2 u oraz u(x, 0) = u 0(x).
Fractional Laplacian
Random walk and fractional Laplacian Let Π : R d [0, + ) satisfies Π(y) = Π( y) for any y R d, and New notation: h = x, k Z d Π(k) = 1. τ = t Give a small h > 0, we consider a random walk on the lattice hz d. Dynamics at any unit of time τ, a particle jumps from any point of hz d to any other point; the probability for which a particle jumps from the point hk hz d to the point h k is taken to be Π(k k) = Π( k k).
Random walk and fractional Laplacian We call u(x, t) the probability that our particle lies at x hz d at time t Z.
Random walk and fractional Laplacian We call u(x, t) the probability that our particle lies at x hz d at time t Z. u(x, t + τ) = k Z d Π(k)u(x + hk, t). Hence, u(x, t + τ) u(x, t) = k Z d Π(k) ( u(x + hk, t) u(x, t) ).
Random walk and fractional Laplacian Particularly nice asymptotics are obtained in the case and Π(0) = 0. τ = h α and Π(y) = C for y 0 y d+α We observe that Π(k) τ = h d Π(hk).
Random walk and fractional Laplacian Hence u(x, t + τ) u(x, t) τ = Π(k) ( ) u(x + hk, t) u(x, t) τ k Z d = h d Π(hk) ( u(x + hk, t) u(x, t) ) k Z d = h d k Z d ψ(hk, x, t). where ψ(y, x, t) = Π(y) ( u(x + y, t) u(x, t) ).
Random walk and fractional Laplacian Notice that h d ψ(hk, x, t) ψ(y, x, t) dy k Z d R d when h 0. Consequently, passing to the limit τ = h α 0 we obtain the equation u t (x, t) = ψ(y, x, t) dy, R d that is NOTATION u(x + y, t) u(x, t) u t (x, t) = C R y n+α dy. d u t (x, t) = ( ) α/2 u(x, t)
Fractional Laplacian Now, we compute the Fourier transform of the equation u(x + y, t) u(x, t) u t (x, t) = C R y n+α dy. d to obtain where Fractional Laplacian û t (ξ, t) = C(α, n) ξ α û(ξ, t) e iξ0y 1 C(α, n) = C dy < 0. R y n+α d ( ( ) α/2 v ) (ξ) = ξ α v(ξ).
Laplace operator & Wiener process Brownian motion one trajectory of a Wiener process
Lévy process One trajectory of a Lévy process
Lévy process Two pictures of the same trajectory of a Lévy process
Lévy process Definition The stochastic process {X (t) : t 0} on the probability space (Ω, F, P) is called the Lévy process with values in R n if it fulfils the following conditions: X(0) = 0, P-p.w., for every sequence 0 t 0 < t 1 < < t n random variables X (t 0 ), X (t 1 ) X (t 0 ),..., X (t n ) X (t n 1 ) are independent, the law of X (s + t) X (s) is independent of s, the process X (t) is continuous in probability, namely, lim s t P( X s X t > ε) = 0.
Lévy-Khinchin formula Lévy operator: Lu(x) = b u(x) where d j,k=1 b R d is a given vector, a jk 2 u x j x k (a jk ) d j,k=1 is a given nonnegative definite matrix Π in a Borel measure satisfying Π({0}) = 0 and min(1, η 2 ) Π(dη) < R d R d ( ) u(x η) u(x) Π(dη),
Fractional Laplacian Let in Π(dη) = C(α) η n+α with α (0, 2) ( ) Lu(x) = u(x η) u(x) Π(dη). R d We obtain the α-stable anomalous diffusion equation: u t + ( ) α/2 u = 0
Fundamental solution of the equation u t + ( ) α/2 u = 0 Define the function p α (x, t) by the Fourier transform: p α (ξ, t) = e t ξ α. Note that p 2 (x, t) = (4πt) d/2 e x 2 /(4t). Scaling: p α (x, t) = t d/α P α (xt 1/α ), where (P α )ˇ(ξ) = e ξ α. For every α (0, 2), the function P α is smooth, nonnegative, R d P α (x) dx = 1, and satisfies 0 P α (x) C(1+ x ) (α+d) and P α (x) C(1+ x ) (α+d+1) for a constant C and all x R d.
Maximum principle
Maximum principle Definition The operator A satisfies the positive maximum principle if for any ϕ D(A) the fact 0 ϕ(x 0 ) = sup x R n ϕ(x) for some x 0 R n implies Aϕ(x 0 ) 0. REMARK Aϕ = ϕ or, more generally, Aϕ = ϕ satisfies the positive maximum principle.
Maximum principle THEOREM Denote by L the Lévy diffusion operator. Then A = L satisfies the positive maximum principle. Proof Assume that 0 ϕ(x 0 ) = sup x R n ϕ(x). Then n 2 ϕ(x 0 ) Lϕ(x 0 ) = b ϕ(x 0 ) + a jk x j x k j,k=1 ( ) + ϕ(x 0 η) ϕ(x 0 ) Π(dη) 0. R n
Convexity inequality THEOREM Let u C 2 b (Rn ) and g C 2 (R) be a convex function. Then Lg(u) g (u)lu. Proof. Use the representation Lu(x) = b u(x) and the convexity of g n j,k=1 a jk 2 u x j x k R n ( ) u(x η) u(x) Π(dη). g(u(x η)) g(u(x)) g (u(x))[u(x η) u(x)].
Nonlinear models with fractional Laplacian
Fractal Burgers equation u t + ( ) α/2 u + uu x = 0 where x R.
Self-interacting individuals Differential equations describing the behavior of a collection of self-interacting individuals via pairwise potentials arise in the modeling of animal collective behavior: ocks, schools or swarms formed by insects, fishes and birds. The simplest model: dx j dt = m j K(x i x j ). j i Here, x i is the position of the particle with mass m i. The continuum descriptions u t = (u( K u) ). Here, the unknown function u = u(x, t) 0 is either the population density of a species or the density of particles in a granular media.
Model of chemotaxis u t = ( ) α/2 u (u( K u) ) where α (0, 2].