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Multidimensional stochastic differential equations with distributional drift. (English) Zbl 1355.60074

This paper deals with a multidimensional stochastic differential equation with distributional drift \(dX_{t}=b(t,X_{t})dt+dW_{t}\), \(X_0=x_0\), where \(b\) is a vector field on \([0,T]\times R^{d}\), \(d\geq1\), which is a distribution in space and weakly bounded in time, that is, \(b\in L^{\infty}([0,T];S'(R^{d};R^{d}))\), where \(S'(R^{d})\) is the space of tempered distributions.
The authors prove existence and uniqueness of a mild solution to the parabolic partial differential equation \(\partial_{t}u+L^{b}u-(\lambda+1)u=-b\) on \([0,T]\times R^{d}\), \(u(T)=0\) on \(R^{d}\), where \(L^{b}u=(1/2)\Delta u+b\cdot\nabla u\), \(\lambda>0\). Using the mild solution of this parabolic differential equation, the notion of virtual solution to the considered stochastic differential equation with distributional drift is introduced. Theorems on the existence and uniqueness of the virtual solution to this equation are proved. The authors show that the virtual solution is the limit of classical solutions of regularized stochastic differential equations.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H30 Applications of stochastic analysis (to PDEs, etc.)
35K10 Second-order parabolic equations
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