Math 2.0

Probability theory and stochastic processes Definition

Poisson boundary

Builds on Random walk on a groupProbability measureLp spaceLiouville property of graphs

Definition

Let Γ\Gamma be a countable group and let μ\mu be a probability measure on Γ\Gamma. It defines the random walk Xn=g1g2⋯gnX_n = g_1 g_2 \cdots g_n with independent steps gkg_k of law μ\mu. A function f ⁣:Γ→ℝf \colon \Gamma \to \R is μ\mu-harmonic if f(g)=∑h∈Γμ(h)f(gh)f(g) = \sum_{h \in \Gamma} \mu(h) f(gh) for every g∈Γg \in \Gamma. Let H∞(Γ,μ)H^\infty(\Gamma, \mu) be the space of bounded μ\mu-harmonic functions. A Poisson boundary of (Γ,μ)(\Gamma, \mu) is a standard probability space (B,ν)(B, \nu) with a measurable action of Γ\Gamma such that:

  1. ν\nu is μ\mu-stationary: ∑h∈Γμ(h) h∗ν=ν\sum_{h \in \Gamma} \mu(h)\, h_* \nu = \nu.
  2. The Poisson transform φ↦fφ\varphi \mapsto f_\varphi, with fφ(g)=∫Bφ(gξ) dν(ξ)f_\varphi(g) = \int_B \varphi(g \xi) \, d\nu(\xi), is a bijection from L∞(B,ν)L^\infty(B, \nu) onto H∞(Γ,μ)H^\infty(\Gamma, \mu).

Remarks

Clause 1 makes each fφf_\varphi harmonic: ∑hμ(h)∫Bφ(ghξ) dν(ξ)=∫Bφ(gξ) d(∑hμ(h)h∗ν)(ξ)=fφ(g)\sum_h \mu(h) \int_B \varphi(gh\xi) \, d\nu(\xi) = \int_B \varphi(g\xi) \, d\Big(\sum_h \mu(h) h_* \nu\Big)(\xi) = f_\varphi(g). The Poisson boundary exists and is unique up to a Γ\Gamma-equivariant measure isomorphism. One construction takes the space of ergodic components of the shift on the space of sample paths of the walk, with the image of the path measure. Example: let Γ=ℤd\Gamma = \Z^d and let the support of μ\mu generate ℤd\Z^d as a group. By the Choquet–Deny theorem the bounded harmonic functions are constant, so a one-point space is a Poisson boundary. Non-example: for the free group F2F_2 on a,ba, b and μ\mu uniform on {a,a−1,b,b−1}\set{a, a^{-1}, b, b^{-1}}, a one-point space fails clause 2, since there are nonconstant bounded harmonic functions. In that case a Poisson boundary is the space of infinite reduced words in a±1,b±1a^{\pm 1}, b^{\pm 1} with the law of the limit of XnX_n. For the simple random walk on a Cayley graph, a trivial Poisson boundary is the same as the Liouville property of the graph.

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To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Liouville property of graphs, concept, no evidence levelLiouville property of…conceptLp space, concept, no evidence levelLp spaceconceptProbability measure, concept, no evidence levelProbability measureconceptRandom walk on a group, concept, no evidence levelRandom walk on a groupconceptPoisson boundary, definition, no evidence levelPoisson boundarydefinition
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  1. Liouville property of graphs concept
  2. Lp space concept
  3. Probability measure concept
  4. Random walk on a group concept
  5. Poisson boundary definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Poisson_boundary
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UsesRandom walk on a group, Probability measure, Lp space, Liouville property of graphs

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