Brownian Motion with Singular Time-Dependent Drift

Peng Jin ()
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Peng Jin: Bergische Universität Wuppertal

Journal of Theoretical Probability, 2017, vol. 30, issue 4, 1499-1538

Abstract: Abstract In this paper, we study weak solutions for the following type of stochastic differential equation where $$b: [0,\infty ) \times \mathbb {R}^{d}\rightarrow \mathbb {R}^{d}$$ b : [ 0 , ∞ ) × R d → R d is a measurable drift, $$W=(W_{t})_{t \ge 0}$$ W = ( W t ) t ≥ 0 is a d-dimensional Brownian motion and $$(s,x)\in [0,\infty ) \times \mathbb {R}^{d}$$ ( s , x ) ∈ [ 0 , ∞ ) × R d is the starting point. A solution $$X=(X_t)_{t \ge s}$$ X = ( X t ) t ≥ s for the above SDE is called a Brownian motion with time-dependent drift b starting from (s, x). Under the assumption that |b| belongs to the forward-Kato class $$\mathcal {F}\mathcal {K}_{d-1}^{\alpha }$$ F K d - 1 α for some $$\alpha \in (0,1/2)$$ α ∈ ( 0 , 1 / 2 ) , we prove that the above SDE has a unique weak solution for every starting point $$(s,x)\in [0,\infty ) \times \mathbb {R}^{d}$$ ( s , x ) ∈ [ 0 , ∞ ) × R d .

Keywords: Stochastic differential equations; Singular drift; Kato class; Weak solutions; Martingale problem; Resolvent; 60H10; 60J60 (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/s10959-016-0687-3

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