A parametrix construction for the fundamental solution of the evolution equation associated with a pseudo‐differential operator generating a Markov process

Abstract

Key words Pseudo-differential operator, Markov process, fundamental solution, parametrix for parabolic equation MSC (2000) 35S10, 60J35, 47D07We use the method proposed by H. Kumano-go in the classical case to construct a parametrix of the equation On the one hand we know that e −tq(x,D) is a pseudo-differential operator, but we don't know its symbol.On the other hand it is known that e −tq (x,D) u is a solution to the equationH. Kumanogo constructed an approximation for the symbol of the solution operator to this equation. In his case the operator q(x, D) had a classical symbol, i.e., it was in the class S m ρ,δ . We prove an analogous result for pseudo-differential operators with symbols in W. Hoh's class. This approximation is then also an approximation of the symbol of the Feller semigroup generated by −q(x, D). In particular we will get σ (T t ) (x, ξ) = e −tq(x,ξ) + r 0 (t; x, ξ) ,where the remainder term r 0 satisfies some smallness conditions. *

A geometric interpretation of the transition density of a symmetric Lévy process

“…In [7,6] B. Böttcher has proved that for a large class of operators −q(x, D) the symbol of T t λ t (x, ξ) = e −ixξ E x e iξXt , t > 0, x, ξ ∈ R n , is asymptotically given by λ t (x, ξ)e −tq(x,ξ) + r 0 (t; x, ξ) as t → 0, where r 0 (t; x, ξ) tends for t → 0 weakly to zero in the topology of a certain symbol class.…”

A Parameter-Dependent Symbolic Calculus for Pseudo-Differential Operators With Negative-Definite Symbols

“…However, Corollary 13 states that in a certain (precise) sense for small times t we can approximate σ(T f t )(x, ξ) by σ(S t )(x, ξ) = σ(e −t(f •q)(x,D) )(x, ξ). Of course we have σ(e −t(f •q)(x,D) )(x, ξ) = e −tf (q(x,ξ)) , but according to the results in [2], e −tf (q(x,ξ)) is a certain approximation of σ(e −t(f •q)(x,D) )(x, ξ) for small t. (Note that Böttcher's results in [2] are extensions of Kumano-go's results [20] to Hoh's calculus. ) Thus in the end, our main result gives a first justification for using e −tf (q(x,ξ)) instead of σ(T f t )(x, ξ) (which we do not know!)…”

“…Now it turns out that often such symbols satisfy the conditions we need in Theorem 12 and its corollary. As a concrete example, let us take the Meixner-type symbols in one dimension (compare Böttcher [2] or [4]):…”

“…In [2], Böttcher used Hoh's calculus to construct a parametrix for the associated evolution problem, that is, for the equation…”

Approximation of Feller Processes by Markov Chains With Lévy Increments

“…[11,8]; -solving the associated evolution equation (Kolmogorov's backwards equation), e.g. [3,2,14,15]; -the well-posedness of the martingale problem, e.g. [1,8,20]; -solving a stochastic differential equation, e.g.…”