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Theme for TIFR Centre For Applicable Mathematics, Bangalore

Abstract: It is well known that ordinary differential equations (ODEs) regularize when an additional forcing by Brownian motion is added. Indeed, an ODE driven by a function, $$b$$, might have multiple solutions or no solutions when $$b$$ is a bounded measurable function. However, once the random forcing by Brownian motion $$(B_t)_{t\ge 0}$$ is added, the corresponding stochastic differential equation will have a unique strong solution even for bounded measurable $$b$$ without any additional assumptions on continuity. This phenomenon is called in the literature regularization by noise''. For more general forcing and for equations driven by distributions one needs precise regularity estimates on the associated resolvents to make the above program precise.