New Mathematical Proof Helps To Solve Equations With Random Components
Arthur T Knackerbracket has processed the following story:
Whether it's physical phenomena, share prices or climate models-many dynamic processes in our world can be described mathematically with the aid of partial differential equations. Thanks to stochastics-an area of mathematics which deals with probabilities-this is even possible when randomness plays a role in these processes.
Something researchers have been working on for some decades now are so-called stochastic partial differential equations. Working together with other researchers, Dr. Markus Tempelmayr at the Cluster of Excellence Mathematics Munster at the University of Munster has found a method which helps to solve a certain class of such equations.
The basis for their work is a theory by Prof. Martin Hairer, recipient of the Fields Medal, developed in 2014 with international colleagues. It is seen as a great breakthrough in the research field of singular stochastic partial differential equations. "Up to then," Tempelmayr explains, "it was something of a mystery how to solve these equations. The new theory has provided a complete 'toolbox,' so to speak, on how such equations can be tackled."
The problem, Tempelmayr continues, is that the theory is relatively complex, with the result that applying the 'toolbox' and adapting it to other situations is sometimes difficult.
"So, in our work, we looked at aspects of the 'toolbox' from a different perspective and found and proved a method which can be used more easily and flexibly."
[...] Stochastic partial differential equations can be used to model a wide range of dynamic processes, for example, the surface growth of bacteria, the evolution of thin liquid films, or interacting particle models in magnetism. However, these concrete areas of application play no role in basic research in mathematics as, irrespective of them, it is always the same class of equations which is involved.
The mathematicians are concentrating on solving the equations in spite of the stochastic terms and the resulting challenges such as overlapping frequencies which lead to resonances.
[...] The approach they took was not to tackle the solution of complicated stochastic partial differential equations directly, but, instead, to solve many different simpler equations and prove certain statements about them.
"The solutions of the simple equations can then be combined-simply added up, so to speak-to arrive at a solution for the complicated equation which we're actually interested in." This knowledge is something which is used by other research groups who themselves work with other methods.
More information: Pablo Linares et al, A diagram-free approach to the stochastic estimates in regularity structures, Inventiones mathematicae (2024). DOI: 10.1007/s00222-024-01275-z
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