A probabilistic approach to fractional reaction-diffusion equations

Thesis

This thesis consists of two projects. In the first project, we prove a new result relating the solution interface of the scaled fractional Allen–Cahn equation to motion by mean curvature flow. More precisely, we show that, for α ∈ (1, 2) and d ≥ 2, if u is a solution to the scaled fractional Allen–Cahn equation

∂tu=− I(ε)^(α-2)(−∆)^(α/2) u+ε^(-2)u(1−u)(2u-1) t>0, x∈R^d

with a suitably chosen initial condition and scaling then as ε → 0, u converges to the indicator function of a set whose boundary evolves according to motion by mean curvature flow. Here, I(ε) can be chosen from a family of possible ‘scaling functions’, distinguishing our result from known convergence results such as [IS09]. Our proof is purely probabilistic (taking inspiration from [EFP17]) and describes solutions of the fractional Allen–Cahn equation in terms of ternary branching α-stable motions. Finally, we prove that this mean curvature flow behaviour is preserved for a stochastic analogue of the fractional Allen–Cahn equation. We do so via the Spatial Λ-Fleming-Viot process. In our second project, we prove a new result on the spreading speed of solutions to one-dimensional fractional Fisher– KPP type equations with appropriately chosen initial condition, such as the equation

∂tu=−(−∆)^(alpha/2) u+u(1−u) t>0, x∈R.

Taking inspiration from [Pen18], we use the Feynman–Kac formula to prove probabilistically that the front position of solutions to these equations grow exponentially in time, and state the precise order of this spread- ing speed, improving known results such as those from [CR13].

Authors: Becker, K

• DOI: 10.5287/ora-vjjamjgor
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: long range process; stable process; Allen-Cahn equation; mean curvature flow; fractional reaction-diffusion equation
• Subjects: probability