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Register a new userOn strong solutions of It^os equations with a$,in W^{1}_{d}$ and b$,in L_{d}$
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N.V. Krylov
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We consider It^o uniformly nondegenerate equations with time independent coefficients, the diffusion coefficient in $W^{1}_{d,loc}$, and the drift in $L_{d}$. We prove the unique strong solvability for any starting point and prove that as a function of the starting point the solutions are Holder continuous with any exponent $<1$. We also prove that if we are given a sequence of coefficients converging in an appropriate sense to the original ones, then the solutions of approximating equations converge to the solution of the original one.
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This paper is a natural continuation of cite{Kr_20_2}, where strong Markov processes are constructed in time inhomogeneous setting with Borel measurable uniformly bounded and uniformly nondegenerate diffusion and drift in $L_{d+1}(mathbb{R}^{d+1})$. Here we study some properties of these processes such as the probability to pass through narrow tubes, higher summability of Greens functions, and so on. The results seem to be new even if the diffusion is constant.
This paper is a natural continuation of [8], where strong Markov processes are constructed in time inhomogeneous setting with Borel measurable uniformly bounded and uniformly nondegenerate diffusion and drift in $L_{d+1}(mathbb{R}^{d+1})$. Here we study some properties of these processes such as higher summability of Greens functions, boundedness of resolvent operators in Lebesgue spaces, establish It^os formula, and so on.
We prove the solvability of It^o stochastic equations with uniformly nondegenerate, bounded, measurable diffusion and drift in $L_{d+1}(mathbb{R}^{d+1})$. Actually, the powers of summability of the drift in $x$ and $t$ could be different. Our results seem to be new even if the diffusion is constant. The method of proving the solvability belongs to A.V. Skorokhod. Weak uniqueness of solutions is an open problem even if the diffusion is constant.
We consider elliptic equations with operators $L=a^{ij}D_{ij}+b^{i}D_{i}-c$ with $a$ being almost in VMO, $bin L_{d}$ and $cin L_{q}$, $cgeq0$, $d>qgeq d/2$. We prove the solvability of $Lu=fin L_{p}$ in bounded $C^{1,1}$-domains, $1<pleq q$, and of $lambda u-Lu=f$ in the whole space for any $lambda>0$. Weak uniqueness of the martingale problem associated with such operators is also obtained.
This paper is a natural continuation of cite{Kr_20_2} and cite{Kr_21_1} where strong Markov processes are constructed in time inhomogeneous setting with Borel measurable uniformly bounded and uniformly nondegenerate diffusion and drift in $L_{d+1}(mathbb{R}^{d+1})$ and some properties of their Greens functions and probability of passing through narrow tubes are investigated. On the basis of this here we study some further properties of these processes such as Harnack inequality, Holder continuity of potentials, Fanghua Lin estimates and so on.
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