Exit times of diffusions with incompressible drifts

Transcription

1 Exit times of diffusions with incompressible drifts Andrej Zlatoš University of Chicago Joint work with: Gautam Iyer (Carnegie Mellon University) Alexei Novikov (Pennylvania State University) Lenya Ryzhik (Stanford University) Paris VI & VII, June 4, 2010

2 Drift-enhanced diffusions We study the effects of incompressible drifts u on diffusion. Without incompressibility assumption u = 0, effects may be arbitrary Typically, an incompressible drift enhances diffusion due to extra mixing We will show that the last statement is not always true! (At least in some sense.)

3 Drifts and principal eigenvalues Long time behavior of solutions of φ t + u φ = φ on a bounded domain Ω R n with Dirichlet boundary conditions is governed by the principal eigenvalue λ u of L u = + u. Namely, φ(t, ) e λut. If ψ u is the (normalized) principal eigenfunction, then λ u = Ω ψ u L u ψ u dx = Ω ψ u 2 dx w 2 inf w H0 1(Ω) w 2 = λ 0 by incompressibility of u. So addition of any incompressible u 0 increases the rate of decay of φ (diffusion is always enhanced in this sense).

4 Drifts and explosions So we have that ψ + u ψ = λψ has a positive solution if and only if λ = λ u, with λ u λ 0. Now consider positive solutions of the explosion problem ψ + u ψ = λe ψ on Ω, with Dirichlet boundary conditions (cold boundary). There is and explosion threshold λ u such that positive solutions exist when λ < λ u and do not exists when λ > λu For u = 0: Joseph-Lundgren, Keener-Keller, Crandall-Rabinowitz (1970s) For general u: Berestycki-Novikov-Kiselev-Ryzhik (2009)

5 Drifts and explosions Is it true that λ u λ0 for all u (and all Ω)? Intuition says yes because mixing should increase interaction between the hot spots inside Ω and the cold boundary Ω. Berestycki-Kagan-Joulin-Sivashinsky (1997) say no, based on a numerical treatment of the case of Ω being a long rectangle. What is going on with the intuitive picture of drift-enhanced diffusion?

6 Drifts and exit times Consider a diffusing particle, starting at x Ω and subject to the drift u. That is, consider the stochastic process dx x t = u(x x t )dt + 2dW t and X x 0 = x where W t is the Brownian motion (factor 2 can be scaled out). If τ u (x) 0 is the expected exit time of X x t from Ω, then τ u + u τ u = 1 on Ω, τ u = 0 on Ω. Main problem: Which incompressible drift (if any) maximizes τ u in some sense? When is it u = 0? We will look at τ u as the measure of size of τ u. We can answer the second question in R 2 in this sense.

7 Results Theorem Let Ω R 2 be a bounded simply connected domain with a C 1 boundary. Then u = 0 maximizes τ u within the class of incompressible drifts if and only if Ω is a disc. A stronger version of one direction of this result extends to R n : Theorem Let Ω R n be a bounded domain with a C 1 boundary and u an incompressible drift on Ω. Then for any p [1, ], τ u p τ 0,Ω p where Ω R n is a ball with Ω = Ω and τ 0,Ω is the expected exit time for zero drift on Ω.

8 Proof of Theorem 2 Let τ = τ u and consider its symmetric rearrangement τ on Ω. If Ω h = {x Ω τ(x) > h} and Ω h = {x Ω τ (x) > h}, then Ω h is the ball (same center as Ω ) with Ω h = Ω h. Isoperimetric inequality gives Ω h τ dσ Ω h dσ τ = Ω h 2 Ω h 2 τ dσ Ω h Co-area formula gives 1 τ dσ = d dh Ω h = d dh Ω 1 h = Ω h τ dσ So Ω h Ω h τ dσ Ω h τ dσ = Ω h = Ω h and hence Ω h dσ τ dτ ( ) (τ ) 1 (h) = 1 dr Ω h τ dσ Ω h ( ) Ω Ω h h = dτ0,ω (τ ) 1 (h) dr Therefore τ τ 0,Ω, and the result follows.

9 Proof of Theorem 1 Assume that Ω R 2 is not a disc. Opposite direction follows from Theorem 2. We will show τ u > τ 0 for some incompressible u. We will only consider drifts with u ν = 0 on Ω (tangential to the boundary). Main idea: Look at infinite amplitude drift limit τ u = lim A τ Au : τ Au + Au τ Au = 1 on Ω, τ Au = 0 on Ω. Show that there is u such that τ u > τ 0 = τ 0. Then for A 1. τ Au > τ 0

10 Infinite amplitude drifts Stream function of u tangential to Ω is ψ such that u = ψ = ( ψ x2,ψ x1 ) and ψ = 0 on Ω. So level sets of ψ are streamlines of u. It follows from results of Freidlin-Wentzell (1993) that the limit τ u = lim A τ Au exists and is uniform in Ω if the stream function ψ of u has a single critical point in Ω (i.e., ψ is a hill or a valley ). Then τ u = 0 on Ω and τ u satisfies the Freidlin problem τ u (y) = ψ(y) 0 Ω ψ,h Ω ψ,h ψ dx dh with Ω ψ,h = {x Ω ψ(x) > h}. So τ u is constant on Ω ψ,h. Let x 0 be the critical point of ψ and define I(ψ) = τ ψ = τ ψ (x 0 ) = ψ 0 Ω ψ,h Ω ψ,h ψ dx dh If ψ maximizes τ u, then ψ is a critical point of I.

11 Variation of the drift For any smooth vector field w supported inside Ω, let d dt Y t x = w(yt x ) and Y0 x = x, and define ψ w ε (x) = ψ(y x ε ). If ψ is a critical point of I, then d dε I(ψw ε ) ε=0 = 0 for each w. This can be showed to be equivalent to the PDE 2 φ(x) = 1+ φ(x) 2 ( Ω φ,φ(x) with φ = τ ψ a reparametrization of ψ. )( ) 1 dσ φ dσ φ Ω φ,φ(x)

12 Domains with τ 0 having a single critical point Now assume τ 0 maximizes I and has a single critical point. Then τ 0 = 1 and τ 0 τ 0 = 0 give τ 0 + A τ 0 τ 0 = 1. So φ = τ τ 0 = τ A τ 0 = τ 0 and 2 φ(x) = 1+ φ(x) 2 ( Ω φ,φ(x) becomes 2 = 1 + τ 0 2 F(τ 0 ). )( ) 1 dσ φ dσ φ Ω φ,φ(x) Then θ(x) = G(τ 0 (x)) with G(s) = s 0 F(r) 1/2 dr solves 1 = θ 2 (and θ = 0 on Ω) If Ω is a not a disc, solutions have more than one interior singularity. Since τ 0 is analytic, θ can only be singular at x 0.

13 General domains Lemma For any Ω, the set of maxima (minima) of τ 0 is discrete. This uses analyticity of τ 0 and simple connectedness of Ω. Pick one maximum x 0, then h close to τ 0 (x 0 ), and let Ω be the component of Ω τ 0,h containing x 0 (and no other critical points). Lemma If u = 0 maximizes τ u on Ω, then it maximizes τ u on Ω. If not, take v supported in Ω with τ v Ω L (Ω ) > τ 0 Ω L (Ω ). Let u 0 = τ 0, and u = u 0 in Ω \ Ω and u = v in Ω. Then τ Au τ 0 = τ Au τ Au 0 0 on Ω \ Ω as A. So τ Au h on Ω and hence τ Au h + τ v Ω on Ω. Then τ Au L (Ω) h+ τ v Ω L (Ω ) > h+ τ 0 Ω L (Ω ) = τ 0 Ω L (Ω). So if u = 0 maximizes τ u, then Ω must be a disc and τ 0 radial on it. Analyticity of τ 0 shows that Ω must be a disc.

14 An example τ 0 and the maximizer τ u for Ω an ellipse (maximizer is obtained by solving the level set PDE numerically). Level sets of the maximizer near its maximum are circular. This is the case for general solutions of the level set PDE.

15 Open problems Existence/uniqueness of the solutions of the PDE 2 φ(x) = 1+ φ(x) 2 ( Ω φ,φ(x) )( ) 1 dσ φ dσ φ Ω φ,φ(x) τ u = lim A τ Au is well defined for general u satisfying some non-degeneracy assumptions (not only those whose stream function has a single critical point). If general u maximizes τ u, does φ = τ u satisfy the PDE (with integration over connected components of level sets of φ)? Are there maximizers of τ u and τ u for general Ω? How about other norms, e.g., τ u p? One direction true. How about more dimensions (no stream functions there)? One direction true.