Lévy-type Processes and Random Walks

Bjorn Böttcher (Philipps-Universität Marburg)

Let q(x, ξ) : ℝ × ℝ → ℂ be a function such that q(x, ·) is a characteristic exponent of a Lévy process for each fixed x. We define a family of measures {μ_{t,x_{0}}}_{t∈(0,∞],x_{0}∈ℝ} by their characteristic functions

∫ e^{ixξ} μ_{t,x_{0}}(dx) = e^{−tq(x_{0},ξ)}

and a triangular array of dependent random variables (n ∈ ℕ):

X_{t,n,0} := x_{0}, 0 < k ≤ n : X_{t,n,k} ∼ μ_{\frac{t}{n} ,∑_{j=0}^{k−1} X_{t,n,j}}

with row sums

S_{t,x_{0}}^{n} := ∑_{i=0}^{n} X_{t,n,i}.

Question: What conditions on q(x, ξ) ensure that lim_{n→∞} S_{t,x_{0}}^{n} exists?

We present several approaches to this problem, in particular:

1. An analytic approach (by Jacob, Potrykus [3]) expressing the problem by iteration of operators. This technique yields that the corresponding operator limit is a Feller semigroup.

2. A stochastic approach (by Brown, Eagleson [2]), which yields that the distribution of the limit is infinitely divisible.

3. A numerical approach based on ’deterministic processes’.

This problem is motivated by the fact that (imposing some additional conditions) q(x, ξ) is the symbol of a pseudo differential operator which extends to the generator of a Feller semigroup T_{t}. But for most cases it is still unknown how to approximate the transition probabilities of the corresponding Feller process. Nevertheless, in [1] it has been shown that for small t the symbol of T_{t} has essentially the form e^{−tq(x,ξ)}. Thus the above setting seems natural.

References

[1] Böttcher, B. A parametrix construction for the fundamental solution of the evolution equation associated with a pseudo-differential operator generating a markov process. Math. Nachr., 278(11):1235–1241, 2005.

[2] Brown, B. M. and Eagleson, G. K. Martingale convergence to infinitely divisible laws with finite variance. Trans. Amer. Math. Soc., 162:449–453, 1971.

[3] Jacob, N. and Potrykus, A. Roth’s method applied to some pseudodifferential operators with bounded symbols. A case study. In press.