Chernoff-Mehler Approximation for Lévy Processes with Drift
Max Nendel
TL;DR
The paper develops a Chernoff-Mehler approximation scheme for Lévy processes with drift on $\mathbb{R}^d$ by analyzing transition operators $$(P_t f)(x)=\int f(\psi_t(x)+y)\,\mu_t(dy)$$ in the space $\operatorname{C}_{\rm b}(\mathbb{R}^d)$ under the mixed topology. Under three core conditions (M) boundedness, (T) tightness, and (D) uniform Lipschitz control for the deterministic flow $\psi_t$, it proves that null sequences yield subsequences converging to the transition semigroup of a Lévy process with drift, described by the SDE $dX_t^x = F(X_t^x)\,dt + dY_t$ where $Y_t$ has a Lévy triplet $(b,\sigma,\nu)$ and $F$ is Lipschitz. The work provides a detailed Lévy triplet identification via a Courrège-type analysis, shows equivalences and stronger convergence results (including for $M^*,T^*$) in spaces like $L_p$ and $\operatorname{UC}_{\rm b}$, and validates practical instances such as Lipschitz flows, Euler schemes, Runge-Kutta methods, and the Central Limit Theorem within the same framework. This unifies stochastic and deterministic discretizations under a common semigroup approximation theory in bi-continuous settings, with potential applications to numerical schemes for Lévy-driven dynamics and SPDE approximations. Key contributions include explicit criteria linking the measure family $\{\mu_t\}$ and the deterministic map family $\{\psi_t\}$ to the generator of the limiting process, and the demonstration that prominent numerical schemes fit into the proposed scheme as special cases.
Abstract
In this paper, we study an approximation scheme for Lévy processes with drift in terms of a representation that is akin to the celebrated Mehler formula for Lévy-Ornstein-Uhlenbeck processes. The approximation scheme is based on a variant of the Chernoff product formula on the space of bounded continuous functions. In a first step, we provide sufficient and necessary conditions for arbitrary families of probability measures, indexed by positive real numbers, to give rise to a convolution semigroup via a Chernoff approximation on the space of bounded continuous functions, equipped with the mixed topology. In this context, we provide explicit criteria both for the convergence of subsequences and the entire family, and discuss fine properties related to the domain of the associated generator of the Lévy process and the infinitesimal behavior of the approximating family of measures. In a second step, we enrich the family of measures by a deterministic component and derive explicit conditions that ensure both the convergence of subsequences and the entire family to a Lévy process with drift under a Chernoff approximation. In a series of examples, we show that our general conditions on the dynamics are satisfied, for example, by flows of Lipschitz ordinary differential equations, Euler schemes, and arbitrary Runge-Kutta methods, and that the Central Limit Theorem can be subsumed under our framework.