Conference Agenda

Dunkl processes are a special typ of multidimensional jump diffusions, whose radial part is a multidimensional Bessel process, well known in the framework of interacting particle systems and closely related to random matrix theory. In addition to Bessel processes Dunkl processes possess jumps, namely reflections, which produce sign changes and changes of coordinate processes and lead to a martingale structure of the process. In one dimension this property is closely related to Gilat's theorem, which states that every positive submartingale is the absolute value of a martingale. Namely a one dimensional Dunkl process is the unique martingale corresponding to the positive submartingale of the classical Bessel process.

In this talk we will analyse the jump mechanism of the Dunkl process and the relation to the martingale property. Furthermore, we will show how this provides inside in constructions of martingales associated to Gilat's theorem.

We consider the Neumann problem for the fractional Laplacian introduced by S. Dipierro, X. Ros-Oton and E. Valdinoci. Following the probabilistic interpretation of the corresponding heat equation given by the authors we construct a stochastic process $X_t$ such that $X_t$ starting in $D$ moves as the isotropic stable process until the first exit time from $D$. At the exit time it jumps out of $D$ according to the L\'evy measure $\nu$ of stable process. It stays at the exit point $y$ for an exponential time with mean $1/ν(y, D)$ then jumps back to $D$ and restarts.

We investigate some fundamental properties of $X_t,$ the corresponding semigroup and bilinear form. In particular we prove that the lifetime of $X$ is infinite for $\alpha\in (0, 1]$ and finite for $\alpha\in (1, 2).$ In the latter case we have $\lim_{t\to\zeta} X_t = 0$, where $\zeta$ denotes the lifetime. We prove that for suffciently regular functions $f$ the function $u = Gf$ is the solution of the Neumann problem, where $G$ is the 0-potential of $X$, i.e., $Gf(x) = E_x \int_0^\infty f(X_t) \, dt.$ This is a joint work with Krzysztof Bogdan and Damian Fafula from Wroclaw University of Science and Technology.

Consider a Volterra Gaussian process of the form $X(t)=\int^t_0K(t,s)dW(s),$ where $W$ is a Wiener process and $K$ is a square integrable Volterra kernel. In this talk, we study weighted multiple self-intersection local times formally defined as $$ T^{\rho}_k=\int_{\Delta_k}\rho(X(t_1))\prod^{k-1}_{i=1}\delta_0(X(t_{i+1})-X(t_i))dt_1\ldots dt_k, $$ where $\Delta_k=\{0\leqslant t_1\leqslant\ldots \leqslant t_k\leqslant 1\}$ and $\rho$ is a weight function. The variable $T^{\rho}_k$ measures how much time the process $X$ spends in small neighbourhoods of its self-intersection points. As geometric characteristics of nonsmooth stochastic processes [1], self-intersection local times are an important and interesting notion related to the stochastic process and have been studied in different directions.

To give a rigorous definition to the variable $T^{\rho}_k$, it is natural to approximate the delta-function and prove the existence of the limit. In this talk, we discuss conditions on the process $X$ and the weight function $\rho$ that guarantee the existence of the limit in mean square. Next, we highlight a class of Volterra Gaussian processes generated by continuously differentiable kernels $K(t-s),\ t,s\in[0,1]$ such that $K(0)\neq 0.$ This class of Volterra Gaussian processes displays similarity to the Wiener process. It is not difficult to check that for a planar Wiener process $W$, even a double self-intersection local time does not exist. Indeed, one can check that $$ E\int_{\Delta_2}\delta_0(W(t_2)-W(t_1))dt_1dt_2=\int_{\Delta_2}\frac{1}{2\pi(t_2-t_1)}dt_1dt_2=\infty. $$ Nevertheless, the Rosen renormalized self-intersection local time $$ \int_{\Delta_k}\prod^{k-1}_{i=1}(\delta_0(W(t_{i+1})-W(t_i))-E\delta_0(W(t_{i+1})-W(t_i)))dt_1\ldots dt_k $$ exists [2]. In this talk, we construct the Rosen renormalized self-intersection local time of multiplicity $k$ for planar Volterra Gaussian processes [3].

References

1. J.-F. Le Gall, Wiener sausage and self-intersection local times, Journal of Functional Analysis 88, 1990, 299-341.

2. J. Rosen, A renormalized local time for multiple intersections of planar Brownian motion, Séminaire de Probabilités, XX, 1984/85, Lecture Notes in Math., vol. 1204, Springer, Berlin, 1986, pp. 515–531.

3. O. Izyumtseva, W. R. KhudaBukhsh, Local time of self-intersection and sample path properties of Volterra Gaussian processes https://www.arxiv.org/pdf/2409.04377).

We study the relationship between different kinds of convergence of finite signed measures and discuss their metrizability. In particular, we study the concept of basic convergence recently introduced by A. A. Khartov and introduce the related concept of almost basic convergence. We discover that a sequence of finite signed measures converges vaguely if and only if it is locally uniformly bounded in variation and the corresponding sequence of distribution functions either converges in Lebesgue measure up to constants, converges basically, or converges almost basically.