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Rearranged Stochastic Heat Equation
François Delarue1, William R P Hammersley1
1Laboratoire J.A. Dieudonné, CNRS, Université Côte d'Azur, Nice, France.
Probability Theory and Related Fields
|February 27, 2025
Summary
This study constructs a strong Feller semigroup for probability measures, mapping functions to Lipschitz continuous ones with integrable blow-up. The method uses a rearranged stochastic heat equation and an Euler scheme for robust mathematical analysis.
Area of Science:
- Stochastic Analysis
- Probability Theory
- Partial Differential Equations
Background:
- Feller semigroups are fundamental in the study of Markov processes.
- Constructing semigroups with specific functional analytic properties, like Lipschitz continuity, is challenging.
- Stochastic heat equations provide a framework for modeling random evolution.
Purpose of the Study:
- To explicitly construct a strong Feller semigroup on the space of probability measures.
- To ensure the semigroup maps bounded measurable functions to Lipschitz continuous functions.
- To analyze the behavior of the Lipschitz constant in small time.
Main Methods:
- Utilizing a rearranged stochastic heat equation driven by colored noise.
- Employing an Euler scheme alternating between flat dynamics and a rearrangement operation.
- Adapting techniques from the Bismut-Elworthy-Li formula for Lipschitz property analysis.
Main Results:
- A novel construction of a strong Feller semigroup is presented.
- The semigroup exhibits the desired mapping property into Lipschitz continuous functions.
- The Lipschitz constant demonstrates integrable blow-up behavior in small time.
Conclusions:
- The proposed Euler scheme is proven to be tight.
- A consistent theory for the limiting reflected equation, including the reflection term, is established.
- The construction successfully yields a strong Feller semigroup with specific functional analytic properties.
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