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A particle representation for the heat equation solution

A particle representation for the heat equation solution . Krzysztof Burdzy University of Washington. Fleming-Viot model. N = population size (constant in time ) = ecological niche carrying capacity. Free space Population distribution converges to a fractal structure.

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A particle representation for the heat equation solution

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  1. A particle representation for the heat equation solution Krzysztof Burdzy University of Washington

  2. Fleming-Viot model N = population size (constant in time ) = ecological niche carrying capacity Free space Population distribution converges to a fractal structure Let N go to infinity. Bounded domain Population distribution converges to a density = heat equation solution

  3. Competing species

  4. Selected references • B, Holyst, Ingerman, March (1996) • B, Holyst, March (2000) • B, Quastel (2007) • Grigorescu, Kang (2004, 2006, 2006)

  5. Multiple populations “Minimization of the Renyi entropy production in the space-partitioning process” Cybulski, Babin, and Holyst, Phys. Rev. E 71, 046130 (2005)

  6. The stationary distribution - first Dirichlet eigenvalue in k-th region Conjecture 1: The stationary distribution minimizes Bucur, Buttazzo and Henrot “Existence results for some optimal partition problems” Adv. Math. Sci. Appl. 8 (1998) 571—579 Conti, Terracini and Verzini “On a class of optimal partition problems related to the Fucık spectrum and to the monotonicity formulae” Calc. Var. 22, 45–72 (2005) Conjecture 2: The honeycomb pattern minimizes Special thanks to Luis Caffarelli!

  7. The stationary distribution (2) (*) B, Holyst, Ingerman and March “Configurational transition in a Fleming-Viot-type model and probabilistic interpretation of Laplacian eigenfunctions” J. Phys. A 29, 1996, 2633-2642 Conjecture 3: The critical ratio r for (*) and m populations satisfies

  8. Rigorous results – one population Theorem (B, Holyst, March, 2000) Suppose that the individual trajectories are independent Brownian motions. Then

  9. Idea of the proof A parabolic function (harmonic in space-time): A martingale plus a process with positive jumps:

  10. One population – convergence to the heat equation solution • population size • individual particle mass • empirical density at time • individual trajectories follow Brownian motions Theorem (B, Holyst, March, 2000) If then where is the normalized heat equation solution with

  11. One population – convergence of stationary distributions • population size • individual particle mass • empirical density at time • individual trajectories follow Brownian motions Theorem (B, Holyst, March, 2000) The process has a stationary distribution . Moreover, where is the first Dirichlet eigenfunction.

  12. One population – convergence of stationary distributions – assumptions Assumption: The uniform internal ball condition

  13. Two populations – convergence to the heat equation solution • population size (same for population I and II) • individual particle mass (population I) • individual particle mass (population II) • empirical density at time • individual trajectories follow random walks Theorem (B, Quastel, 2007) If then where is the normalized heat equation solution with

  14. Two populations – convergence to the heat equation solution – assumptions • Trajectories – simple random walks • Trajectories reflect at the domain boundary • (iii) The two populations have equal sizes • (iv) The domain has an analytic boundary

  15. Idea of the proof - n-th Neumann eigenfunction Main technical challenge: bound the clock rate

  16. Spectral representation and L1 Lemma. Suppose that D is a domain with smooth boundary, is the n-th eigenfunction for the Laplacian with Neumann boundary conditions and is a signed measure with a finite total variation.

  17. Diffusion in eigenfunction space Open problem: What is the speed of diffusion?

  18. Invariance principle for reflected random walks • Theorem. Reflected random walk converges • to reflected Brownian motion. • -domains, Stroock and Varadhan (1971) • Uniform domains, B and Chen (2007) Example: Von Koch snowflake is a uniform domain. Counterexample (B and Chen, 2007): Reflected random walk does not converge to reflected Brownian motion in a planar fractal domain.

  19. Myopic conditioning - open, connected, bounded set - Markov process - Brownian motion conditioned by Theorem (B and Chen, 2007). When , converge to reflected Brownian motion in D.

  20. reflected Brownian motion in - hitting time of Problem:

  21. Definition: We will call a bounded set a trap domain if

  22. Hyperbolic blocks

  23. Theorem (B, Chen and Marshall, 2006): A simply connected planar domain is a trap domain if and only if

  24. Horn domain: Corollary: is a trap domain iff Example: Trap domain

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