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SEMINARS
Path-by-Path Uniqueness for SDEs in finite and infinite dimensional spaces
时间  Datetime
2018-07-02 14:30 — 15:30 
地点  Venue
Middle Lecture Room
报告人  Speaker
Lukas Wresch
单位  Affiliation
Bielefeld University
邀请人  Host
张登
报告摘要  Abstract

Consider the following SDE in a separable Hilbert space 

$$ \mathrm dX_t = - A X_t \mathrm dt  +  f(t,X_t)\mathrm dt + \mathrm dW_t , $$

where $A$ is a positive, linear operator, $f$ is a bounded Borel measurable function and $W$ a cylindrical Wiener process. If the components of $f$ decay to 0 in a faster than exponential way we establish path-by-path uniqueness for mild solutions of this SDE. This extends A. M. Davie’s result from $\mathbb R^d$ to Hilbert space-valued stochastic differential equations. 

In this talk we consider the so-called path-by-path approach where the above SDE is considered as a random integral equation with parameter $\omega\in\Omega$. We show that there is a set $\Omega'$ of measure 1 such that for every $\omega\in\Omega'$ the corresponding integral equation for this $\omega$ has at most one solution. This notion of uniqueness (called path-by-path uniqueness) is much stronger than the usual pathwise uniqueness considered in the theory of SDEs.