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Estimates of the Transition Densities for the Reflected Brownian Motion on Simple Nested Fractals

Olszewski Mariusz

Probability and Mathematical Statistics

2019

Vol. 39, Fasc. 2

423--440

We give sharp two-sided estimates for the functions gM (t, x, y) and gM (t, x, y) − g(t, x, y), where gM (t, x, y) are the transition probability densities of the reflected Brownian motion on an M-complex of order M ∈ Z of an unbounded planar simple nested fractal and g(t, x, y) are the transition probability densities of the “free” Brownian motion on this fractal. This is done for a large class of planar simple nested fractals with the good labeling property.

Intrinsic compound kernel estimates for the transition probability density of Lévy-type processes and their applications

Knopova V., Kulik A.

Probability and Mathematical Statistics

2017

Vol. 37, Fasc. 1

53--100

Starting with an integro-differential operator (L, C2∞ (Rn)), we prove that its C∞(Rn)-closure is the generator of a Feller process X, which admits a transition probability density. To construct this transition probability density, we develop a version of the parametrix method and a verification procedure, which proves that the constructed object is the claimed one. As a part of the construction, we prove the intrinsic upper and lower estimates on the density. As an application of the constructed estimates we state the necessary and (separately) sufficient conditions under which a given Borel measure belongs to the Kato and Dynkin classes with respect to the constructed transition probability density.