Probability and partial differential equations in modern applied mathematics

By E. Waymire, Jinqiao Duan. 2005

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Limiting dynamics for stochastic wave equations (2008, background): “The upper semicontinuity of this global random attractor and the global attractor of the heat equation zt − z + f (z) = 0 with Dirichlet boundary condition as ν goes to zero is investigated.”
Rough path analysis via fractional calculus (2006, methodology): “We refer, for instance, to [1], [3], [4] and [7] for recent applications of the rough path analysis to the stochastic calculus with respect to the Wiener process.”
Robust filtering: Correlated noise and multidimensional observation (2012, background): “11This fact is a consequence of the support theorem of Brownian motion in Hölder rough path topology [24]; see also Chapter 13 in [25].”
Semi-martingales and rough paths theory (2005, background): “The sequence (Xn)n∈N of semi-martingales with canonical decompositions Xn = Xn0 + M n + V n is said to satisfy the condition UCV on [0, T ] if (〈Mn〉T )n∈N is tight, (3) (Var1,[0,T ](V n))n∈N is tight.”