Limit Theorems for Cylindrical Martingale Problems Associated with Lévy Generators

Abstract

We prove limit theorems for cylindrical martingale problems associated with Lévy generators. Furthermore, we give sufficient and necessary conditions for the Feller property of well-posed problems with continuous coefficients. We discuss two applications. First, we derive continuity and linear growth conditions for the existence of weak solutions to infinite-dimensional stochastic differential equations driven by Lévy noise. Second, we derive continuity, local boundedness and linear growth conditions for limit theorems and the Feller property of weak solutions to stochastic partial differential equations driven by Wiener noise.

Appendix A. Some Facts on Cylindrical Martingale Problems

To keep this article self-continuous, we collect some results on cylindrical martingale problems, which are used in the proofs of Theorems 1 and 2. Let $M^f$ be defined as in (2.2), $\tau _z$ be defined as in (3.2) and recall that

$$\begin{aligned} \mathcal {D}&= \big \{ g(\langle \cdot , y^*\rangle ) :g \in C^2_c(\mathbb {R}), y^* \in D(A^*)\big \},\\ \mathcal {B}&= \big \{ g (\langle \cdot , y^*\rangle ):g \in C^2_b(\mathbb {R}), y^* \in D(A^*) \big \}. \end{aligned}$$

Proposition 3

The following are equivalent:

• (i) $P \in \mathcal {M}(A, b, a, K, \eta )$.

• (ii) $P \circ X^{-1}_0 = \eta$, and for all $n \in \mathbb {N}$ and $f \in \mathcal {D}$, the process $M^f_{\cdot \wedge \tau _n}$ is a $P$-martingale.

Proof

See [3, Lemma 4.7]. $\square$

Proposition 4

• (i) For all $f \in \mathcal {B}, t \in \mathbb {R}_+, n \in \mathbb {N}$ and all bounded $G \subset \mathbb {B}$, it holds that

  $$\begin{aligned} \sup _{x \in G} \big | \mathcal {K} f (x) \big | + \sup _{s \in [0, t]}\sup _{\omega \in \Omega } \big | M^f_{s \wedge \tau _n(\omega )}(\omega )\big | < \infty . \end{aligned}$$

• (ii) If $P\in \mathcal {M}(A, b, a, K, \eta )$, then for all $n \in \mathbb {N}$ and $f \in \mathcal {B}$ the process $M^f_{\cdot \wedge \tau _n}$ is a $P$-martingale.

Proof

For (i), see [3, Lemma 4.5] including its proof, and for (ii), see [3, Corollary 4.6]. $\square$

Proposition 5

Suppose that for all $x \in \mathbb {B}$ the MP $(A, b, a, K, \varepsilon _x)$ has a unique solution $P_x$.

• (i) The map $x \mapsto P_x$ is Borel, and for all Borel probability measures $\eta$ on $\mathbb {B}$, the MP $(A, b, a, K, \eta )$ has a unique solution given by $\int P_x \eta (\mathrm{d}x)$.

• (ii) Let $\rho$ be a stopping time and $P$ be a probability measure on $(\Omega , \mathcal {F})$ such that $P \circ X^{-1}_ 0 = \eta$, and for all $f \in \mathcal {D}$, the process $M^f_{\cdot \wedge \rho }$ is a local $P$-martingale. Then, $P = \int P_x \eta (\mathrm{d}x)$ on $\mathcal {F}_\rho$.

Proof

For (i), see [3, Theorem 3.2], and for (ii), see [3, Proposition 4.13]. $\square$