CS25: Volterra Gaussian Processes

Organizer: Olga Iziumtseva (University of Nottingham)

Self-intersection local times of Volterra Gaussian processes in stochastic flows

Olga Iziumtseva

Self-intersection local times of a random process are random variables that measure how much time the process spends in small neighborhoods of its self-intersection points. Being the geometric characteristics of nonsmooth random processes \([1]\), it found applications in studying of polymer models. Weighted self-intersection local times for a random process \(x(t),\ t\in[0,1]\) are formally defined as \[T_k(\rho)=\int_{\Delta_k}\rho(x(t_1))\prod^{k-1}_{i=1}\delta_0(x(t_{i+1})-x(t_i))dt_1\ldots dt_k,\] where \(\Delta_k=\{0\leqslant t_1\leqslant\ldots \leqslant t_k\leqslant 1\}\), \(\rho\) is a weight function and \(\delta_0\) is the Dirac delta function at zero. To give a rigorous definition to the variable \(T_k(\rho)\), it is natural to approximate the delta function and prove the existence of the limit. In this talk, we study multiple self-intersection local times for Volterra Gaussian processes in the stochastic flow generated by the equation with interaction introduced by A. Dorogovtsev in \([2]\) \[\begin{cases} du(t,x)=a(u(t,x),\mu_t)dt+\int_{\mathbb{R}^d}b(u(t,x),\mu_t,z)W(dt,dz)\\ u(0,x)=x,\ x\in \mathbb{R}^d\\ \mu_t=\mu_0\circ u(t,\cdot)^{-1}, \end{cases}\] where \(W\) is a Brownian sheet on \(\mathbb{R}^d\times [0,\infty)\) and \(\mu_0\) is a probability measure on \(\mathbb{R}^d.\) Firstly, for a class of weights we describe conditions on a general Gaussian process in \(\mathbb{R}^d\) which guarantee the existence of its weighted self-intersection local times. Then, we define a class of Volterra Gaussian processes (i.e. Gaussian processes that admit the representation \[x(t)=\int^t_0k(t,s)dw(s),\ t\in[0,1]\] with some Wiener process \(w\) and some square integrable Volterra kernel \(k\)), which suits required conditions. It allows us to check the existence of weighted self-intersection local times for this class of Volterra Gaussian processes. Finally, we apply the developed technique for a Volterra Gaussian process \(x\) evolving in the stochastic flow of interacting particles generated by the equation with interaction and prove the existence of multiple self-intrersection local times for the process \(u(t,x(s)),\ s\in[0,1].\)

Bibliography

\([1]\) J.-F. Le Gall. “Wiener sausage and self-intersection local times.”Journal of Functional Analysis, vol. 88, 1990, pp. 299–341.

\([2]\) Dorogovtsev, Andrey A. (2023). Measure-valued Processes and Stochastic Flows. De Gruyter.

\([3]\) 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.

Volterra Gaussian Processes as the fluctuations of the total quasi-steady-state-approximation of Michaelis–Menten enzyme kinetics

Wasiur KhudaBukhsh

Volterra Gaussian processes appear in the asymptotic analysis of many biological systems. In this talk, I will present one such example from the field of systems biology. We consider a stochastic model of the Michaelis-Menten (MM) enzyme kinetic reactions in terms of Stochastic Differential Equations (SDEs) driven by Poisson Random Measures (PRMs). It has been argued that among various Quasi-Steady State Approximations (QSSAs) for the deterministic model of such chemical reactions, the total QSSA (tQSSA) is the most accurate approximation, and it is valid for a wider range of parameter values than the standard QSSA (sQSSA). While the sQSSA for this model has been rigorously derived from a probabilistic perspective at least as early as 2006 in Ball et al. (2006), a rigorous study of the tQSSA for the stochastic model appears missing. We fill in this gap by deriving it as a Functional Law of Large Numbers (FLLN), and also studying the fluctuations around this approximation as a Functional Central Limit Theorem (FCLT).

Bibliography

\([1]\) Ganguly, A. and KhudaBukhsh, W. R. (2025). Asymptotic Analysis of the Total Quasi-Steady State Approximation for the Michaelis–Menten Enzyme Kinetic Reactions. arXiv: https://arxiv.org/abs/2503.20145

Strong solutions for singular SDEs driven by long-range dependent fractional Brownian motion and other Volterra processes

Ercan Sönmez

We investigate the well-posedness of stochastic differential equations driven by fractional Brownian motion, focusing on the long-range dependent case \(H \in (\frac12, 1)\). While existing results on regularization by such noise typically require Hölder continuity of the drift, we establish new strong existence and uniqueness results for certain classes of singular drifts, including discontinuous and highly irregular functions. More generally, we treat stochastic differential equations with additive noise given by a broader class of Volterra processes satisfying suitable kernel conditions, which, in addition to fractional Brownian motion, also includes the Riemann–Liouville process as a special case. Our approach relies on probabilistic arguments. This is a joint work with Maximilian Buthenhoff.