[Paper Review] Duality for coalescing stochastic flows on the real line

Georgii Riabov|arXiv (Cornell University)|Mar 21, 2019

Mathematical Dynamics and Fractals6 references4 citations

TL;DR

This paper establishes the existence of dual stochastic flows for a class of coalescing stochastic flows on the real line by constructing them as backward perfect cocycles over a metric dynamical system. The key contribution is proving that the dual flow satisfies the duality condition (ψ_{s,t}(x)−y)(x−ψ̃_{t,s}(y))≥0 and applying the framework to Arratia flows with drift, where the dual is shown to be the backward Arratia flow with reversed drift.

ABSTRACT

For a class of coalescing stochastic flows on the real line the existence of dual flows is proved. A stochastic flow and its dual are constructed as a forward and backward perfect cocycles over the same metric dynamical system. The metric dynamical system itself is defined on a new state space for coalescing flows. General results are applied to Arratia flows with drift.

Motivation & Objective

• To establish the existence of dual stochastic flows for a class of coalescing stochastic flows on the real line.
• To construct the dual flow as a backward perfect cocycle over a metric dynamical system.
• To prove that the dual flow satisfies the duality condition (ψ_{s,t}(x)−y)(x−ψ̃_{t,s}(y))≥0 almost surely.
• To apply the general framework to Arratia flows with drift, showing the dual is the backward Arratia flow with drift −a.
• To demonstrate that finite-point motions of the dual flow correspond to weak solutions of SDEs with reversed drift.

Proposed method

• Represent the forward flow ψ as a perfect cocycle φ via ψ_{s,t}(x) = φ(t−s, θ_sω, x), where (θ_h) is a metric dynamical system.
• Construct the dual flow as a backward perfect cocycle ṫφ, with ψ̃_{t,s}(x) = ṫφ(t−s, θ_sω, x).
• Use the duality condition (ψ_{s,t}(x)−y)(x−ψ̃_{t,s}(y))≥0 to define the dual mapping in terms of generalized inverses of the forward flow.
• Prove that the generalized inverses v⁺ and v⁻ are not backward flows individually, but a suitable choice yields a valid backward stochastic flow.
• Apply the framework to Arratia flows with drift a by verifying the conditions of Theorem 2.1 and Lemma 6.1 on the modulus of continuity and transition probabilities.
• Verify that the one-point motion of the dual flow solves the SDE dX(t) = −a(X(t))dt + dW(t), confirming the reversed drift.

Experimental results

Research questions

• RQ1 Does a dual stochastic flow exist for coalescing stochastic flows on R, satisfying the duality condition (ψ_{s,t}(x)−y)(x−ψ̃_{t,s}(y))≥0?
• RQ2 Can the dual flow be constructed as a backward perfect cocycle over a metric dynamical system?
• RQ3 For Arratia flows with drift, is the dual flow the backward Arratia flow with drift −a?
• RQ4 Do the finite-point motions of the dual flow correspond to weak solutions of the SDE with reversed drift?
• RQ5 Is the duality condition preserved under the random dynamical system representation via cocycles?

Key findings

• The dual flow exists and is constructed as a backward perfect cocycle over a metric dynamical system, satisfying the duality condition almost surely.
• For Arratia flows with drift a, the dual flow is the backward Arratia flow with drift −a, and its one-point motion solves dX(t) = −a(X(t))dt + dW(t).
• The transition density of the dual flow satisfies ∂ₜp̃(y,x) = −a(y)∂yp̃(y,x) + ½∂²yp̃(y,x), confirming the reversed drift.
• The generalized inverses v⁺ and v⁻ individually fail the backward evolutionary property, but a proper selection yields a valid backward flow.
• The modulus of continuity condition w_{α,β}(ε,δ) ≥ (ε/(4g⁻¹(ε²δ/32)))² ensures the existence of the random dynamical system representation.
• The duality condition is preserved under the cocycle representation, and the dual flow is Markovian with independent increments before meeting times.