Math Problems Vault

Problem 36.1

Suppose $P_t$ is a family of sub-Markov transition probabilities and we define $\bar{P}_t$ by (36.12). Show that $P_t$ is a family of Markov transition probabilities. Show that $\mathbb {P}^{\Delta }\left(X_t \neq \Delta \right.$ for some $\left.t>0\right)=0$ , i.e., starting at $\Delta$ , the process stays there forever.

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Problem 36.2

Show that if $P_t(x, A)$ is defined by (19.17), and $P_t f(x)=\int f(y) P_t(x, d y)$ , then $P_t$ maps $C_0$ into $C_0$ -

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Problem 36.3

Show that (36.14) equals (36.15).

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Problem 36.4

Show that $P_t$ defined by (36.13) satisfies all the parts of Assumption 36.1.

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Problem 36.5

(1)(2)

Suppose $\left\{ \mu_t, t \geq 0\right\}$ is a tight family of probability measures on the real line. Suppose there exists a function $\psi : \mathbb {R} \rightarrow \mathbb {C}$ such that the Fourier transforms of the $\mu_t$ have the following form:

\int e^{i u x} \mu _t(d x)=e^{t \psi (u)}, \quad t \geq 0, u \in \mathbb {R} .

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(1)

Prove that $\mu_t$ converges weakly to $\mu_0$ as $t \rightarrow 0$ . Note that $\mu_0$ is the same as point mass at 0 .

(2)

Define the operators $P_t$ by

P_t f(x)=\int f(x-y) \mu _t(d y) \text{. }

Prove that the $P_t$ form a strongly continuous semigroup of contraction operators mapping $C_0$ into $C_0$ . Conclude that there exists a strong Markov process whose semigroup is given by the $P_t$ .

This semigroup is called a convolution semigroup because $\mu_{t+s}=\mu_t * \mu_s$ , in the sense of convolution of measures. We will see later that these are associated with Lévy processes.