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

Abstract

Abstract. We study for a class of symmetric Lévy processes with state space R n the transition density p t (x) in terms of two one-parameter families of metrics, (d t ) t>0 and (δ t ) t>0 . The first family of metrics describes the diagonal term p t (0); it is induced by the characteristic exponent ψ of the Lévy process by d t (x, y) = tψ(x − y). The second and new family of metrics δ t relates to √ tψ through the formulawhere F denotes the Fourier transform. Thus we obtain the following "Gaussian" representation of the transition density:t (x,0) where p t (0) corresponds to a volume term related to √ tψ and where an "exponential" decay is governed by δ 2 t . This gives a complete and new geometric, intrinsic interpretation of p t (x). MSC 2010: Primary: 60J35. Secondary: 60E07; 60E10; 60G51; 60J45; 47D07; 31E05.

Off-Diagonal Heat Kernel Asymptotics of Pseudodifferential Operators on Closed Manifolds and Subordinate Brownian Motion

“…For completeness we mention that there is an alternative geometric viewpoint for stochastic processes on Euclidean space as suggested in [23]. This leads to estimates of the transition density of symmetric Lévy processes given in terms of a natural metric on the real line.…”

Topics on Hamiltonian Dynamics Related to Symbols of Certain Schrodinger Operators Associated with Generators of Levy Processes

“…defines a metric on R n which under natural conditions generates the Euclidean topology, see [48]. One of the key observations in [48] based on [50] was that the diagonal term of p t (x − y), i.e.…”

“…One of the key observations in [48] based on [50] was that the diagonal term of p t (x − y), i.e. p t (0), is then entirely controlled by the volume of the balls B d ψ .…”

“…p t (0), is then entirely controlled by the volume of the balls B d ψ . A further idea in [48] was to try to estimate p t (x) off-diagonal, i.e. for x = 0, by a metric δ t,ψ to obtain estimates of the type p t (x − y) p t (0)e −δ 2 t,ψ (x,y) .…”

“…Examples were provided in [48] and the idea was developed further in [9], see also [8] and the forthcoming paper [20]. In [31], see also [30], [52], and [49], the metric d ψ was investigated in more detail, questions of interest are the convexities of balls, the dependence of isotropy on parameters etc.…”