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Construction of a Markov process with prescribed local behavior and state-dependent jump distribution

Let $(E,\mathcal E)$ be a measurable space $\mathcal E_b:=\left\{f:E\to\mathbb R\mid f\text{ is bounded and }\mathcal E\text{-measurable}\right\}$ $(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\...
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