Feller And Strong Feller Properties On Diffusions

Feller property and strong Feller one are significant in the study of Markov processes. In this paper, we study these properties for diffusions. For the following stochastic differential equation (SDE) in R¹ :if either Okabe-Shimizu condition or Perkins condition holds, then any k-point motion ((Y_t(x₁),…, Y_t(x_k))_t≥0)_{(x1,…,xk)∈R^k} for the unique strong solution (Y_t(x))_t≥0 to the SDE has a Feller semigroup on (C₀(R^k), ||·||) (see Chapter 3). Removing the drift term in above SDE, we prove that for anyσ∈C_b (R¹) satisfyingthe weak solution (?) to the obtained SDE is Fellerian on (C₀ (R¹), ||·||) under a mild condition, but the (?) diffusion Y obtained by absorbing previous diffusion at Z(σ) is Fellerian on (C₀ (R¹), ||·||) without additional conditions (refer to Chapter 4). Finally, though the generator of any k-point motion of stochastic flows with k≥2 is degenerated and may not be hypoelliptic. its strong Feller property still holds in some cases (see Chapter 5).