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Register a new userIt^os formula for the $L_{p}$-norm of stochastic $W^{1}_{p}$-valued processes
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Authors
N.V. Krylov
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We prove It^os formula for the $L_{p}$-norm of a stochastic $W^{1}_{p}$-valued processes appearing in the theory of SPDEs in divergence form.
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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.
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.
This paper establishes It^os formula along a flow of probability measures associated with gene-ral semimartingales. This generalizes existing results for flow of measures on It^o processes. Our approach is to first prove It^os formula for cylindrical polynomials and then use function approximation and localization techniques for the general case. This general form of It^os formula enables derivation of dynamic programming equations and verification theorems for McKean-Vlasov controls with jump diffusions and for McKean-Vlasov mixed regular-singular control problems. It also allows for generalizing the classical relation between the maximum principle and the dynamic programming principle to the McKean-Vlasov singular control setting, where the adjoint process is expressed in term of the derivative of the value function with respect to probability measures.
We propose to study a new type of Backward stochastic differential equations driven by a family of It^os processes. We prove existence and uniqueness of the solution, and investigate stability and comparison theorem.
Let $mathcal{M}(Omega, mu)$ denote the algebra of all scalar-valued measurable functions on a measure space $(Omega, mu)$. Let $B subset mathcal{M}(Omega, mu)$ be a set of finitely supported measurable functions such that the essential range of each $f in B$ is a subset of ${ 0,1 }$. The main result of this paper shows that for any $p in (0, infty)$, $B$ has strict $p$-negative type when viewed as a metric subspace of $L_{p}(Omega, mu)$ if and only if $B$ is an affinely independent subset of $mathcal{M}(Omega, mu)$ (when $mathcal{M}(Omega, mu)$ is considered as a real vector space). It follows that every two-valued (Schauder) basis of $L_{p}(Omega, mu)$ has strict $p$-negative type. For instance, for each $p in (0, infty)$, the system of Walsh functions in $L_{p}[0,1]$ is seen to have strict $p$-negative type. The techniques developed in this paper also provide a systematic way to construct, for any $p in (2, infty)$, subsets of $L_{p}(Omega, mu)$ that have $p$-negative type but not $q$-negative type for any $q > p$. Such sets preclude the existence of certain types of isometry into $L_{p}$-spaces.
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