Isotropic-like Markov generators on ultrametric spaces

Seminarium: 
Analiza harmoniczna
Osoba referująca: 
Alexander Bendikov (Uniwersytet Wrocławski)
Data spotkania seminaryjnego: 
czwartek, 25. Maj 2017 - 14:15
Sala: 
603
Opis: 

Let (X,d) be a locally compact separable ultrametric space.
Given a measure m and a symmetric measurable function J(x,y) we consider
the linear operator L^{J}f(x)=∫(f(x)-f(y))J(x,y)dm(y) defined on the set D
of all locally constant functions f having compact support. According to
, when J(x,y) is an isotropic function
satisfying certain conditions, the operator (-L^{J},D) is essentially
self-adjoint and extends in L²(X,m) as a self-adjoint Markov generator,
its Markov semigroup exp(-tL^{J}) admits a continuous transition density
(heat kernel) p^{J}(t,x,y) w.r.t. m. When J(x,y) is not isotropic but
uniformly in x,y comparable to the isotropic function J(x,y) as above the
operator (-L^{J},D) extends in L²(X,m) as a self-adjoint Markov generator,
the Markov semigroup exp(-tL^{J}) admits a continuous heat kernel
p^{J}(t,x,y) w.r.t. m, and the function p^{J}(t,x,y) is uniformly
comparable in t,x,y to the function p^{J}(t,x,y), the heat kernel of the
Markov semigroup exp(-tL^{J}). We illustrate our exposition by a number of
examples.