# Seminar Lecture Series by Dr. Alexey Rudenko

Dr. Alexey Rudenko from the National Academy of Sciences of Ukraine will give a Series of three Lectures.

**Lecture 1: Monday, 4th November 2019, 9:00-10:00, NHA 1020**

**Title:** Density estimates for Brownian motion on Carnot group

*Abstract: The purpose of the this talk is to discuss Gaussian-like estimates for
the density of diffusion processes, which generators do not satisfy
ellipticity condition. We will consider a special class of Lie groups,
called Carnot groups, and use Ito's definition of Brownian motion on
Lie group to introduce Brownian motion on Carnot group. The estimates
for the density of such processes can be found in terms of a distance
related to the generator of the process. It will be shown that such
estimates can be used to prove the existence of local times for
processes related to Brownian motion on Carnot group, in particular
for Levy area.*

**Lecture 2: Tuesday, 5th November 2019, 9:00-10:00, NHA 1020**

**Title: **Intersections and self-intersections of Markov processes

*Abstract: n this talk we will recall several results about existence and
non-existence of intersections and self-intersections for some classes
of Markov processes, including Brownian motions on Carnot group and
Levy processes. In particular it is well-known that standard Brownian
motion has self-intersections only if underlying space has dimensions
1,2 and 3, but for Brownian motion on Carnot group the situation can
be different: under some conditions there are no self-intersections in
dimension 3. We will discuss some ideas of the proofs, including the
possibility of using intersection and self-intersection local times to
show the existence of intersections and self-intersections.*

**Lecture 3: Wednesday, 6th November 2019, 9:00-10:00, NHA 1020**

**Title: ** Weighted intersection and self-intersection local times for Markov processes

*Abstract: It is well-known that the self-intersection local time for
2-dimensional Brownian motion exists only after renormalization of its
approximations. Instead of using renormalization we introduce another
modification of the local time approximations by adding a weight
function into time integral. We will show that it is possible to
define the corresponding limit in L_2 for some Markov processes under
specific assumptions on their transition density. For Brownian motion
on Carnot group and for some Levy processes this leads to sufficient
conditions in terms of dimensions for the existence of weighted
intersection and self-intersection local times.*