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A (rough) pathwise approach to a class of non-linear stochastic partial differential equations. (English) Zbl 1219.60061
The authors of the present paper study a stochastic parabolic evolution equation of the form \(\partial_tu=F(t,x,Du,D^2u)dt+Du(t,x) V(x)\circ dB_t,\, t\geq 0,\, u(0,x)=u_0(x),\, x\in R^d,\) where \(B\) is a multi-dimensional Brownian motion and the stochastic integral is understood in Stratonovich’s sense. Motivated by an essentially pathwise approach by P.-L. Lions and P. E. Souganidis [C. R. Acad. Sci., Paris, Sér. I, Math. 326, No. 9, 1085–1092 (1998; Zbl 1002.60552)], the authors apply T. J. Lyons’ rough path analysis [Rev. Mat. Iberoam. 14, No. 2, 215–310 (1998; Zbl 0923.34056)].
The authors’ core arguments are purely deterministic and allow even to consider instead of the driving Brownian motion \(B\) a function \(z\in C^{0,p-\text{var}}([0,T];G^{[p]}(R^d)).\) A continuous dependence property of the solution \(u\) of the above equation driven by \(z\) on the initial condition \(u_0\) gives the possibility for various probabilistic applications.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
34F05 Ordinary differential equations and systems with randomness
35K55 Nonlinear parabolic equations
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