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We consider the stochastic differential equation d X t = A ( X t − ) d Z t , X 0 = x , driven by cylindrical α -stable process Z t in , where α ∈ (0,1) and d ≥ 2. We assume that the determinant of A ( x ) = ( a i j ( x )) is bounded away from zero, and a i j ( x ) are bounded and Lipschitz continuous. We show that for any fixed γ ∈ (0, α ) the semigroup P t of the process X t satisfies | P t f ( x ) − P t f ( y ) | ≤ c t − γ / α | x − y | γ | | f | | ∞ $|P_{t} f(x) - P_{t} f(y)| \le c t^{-\gamma /\alpha } |x - y|^{\gamma } ||f||_{\infty }$ for arbitrary bounded Borel function f . Our approach is based on Levi’s method.

Strong Feller Property for SDEs Driven by Multiplicative Cylindrical Stable Noise