2025 CMS Winter Meeting

Toronto, Dec 5 - 8, 2025

Abstracts        
Probability and PDEs
Org: Vincent Martinez (CUNY Hunter College), Geordie Richards (University of Guelph) and Philippe Sosoe (Cornell University)
[PDF]
MUSTAFA AVCI, Athabasca University
A viscosity solution approach to the Feynman-Kac formula for a one-dimensional parabolic PDE with variable exponent coefficient  [PDF]

This work establishes the existence and uniqueness of a solution for a one-dimensional parabolic Cauchy problem set on the positive half-line involving coefficients with variable exponent whose generator is associated with a stochastic differential equation involving state-dependent variable exponent. The problem is analyzed within the framework of viscosity solutions, addressing cases where classical solutions may not exist due to insufficient coefficient regularity. We demonstrate that the unique viscosity solution is given by the Feynman-Kac formula, thereby establishing a rigorous link between the probabilistic representation and the analytical solution. A key element of the proof relies on the property that the associated stochastic process remains strictly positive on its state space $(0,\infty)$, which allows for the application of local ellipticity arguments despite potential degeneracy at the boundary. The analysis is completed by applying the standard parabolic regularity theory to show that the viscosity solution possesses local Sobolev regularity in $W_{m,loc}^{2,1}$.

{\bf Keywords.} stochastic process; viscosity solutions; Feynman-Kac formula; degenerate parabolic PDE; the comparison principle; the dynamic programming principle; local ellipticity.

YURI BAKHTIN, Courant Institute, NYU
Differentiability of the effective Lagrangian for HJB equations in dynamic random environments  [PDF]

We prove differentiability of the effective Lagrangian for continuous space-time multidimensional directed variational problems in random dynamic environments with positive dependence range in space and time. Thus the limiting fundamental solutions in the associated homogenization problems for HJB equations are classical. For several continuous models of FPP and LPP type, our method provides differentiability of limit shapes and shape functions. This is joint work with Douglas Dow.

RALUCA BALAN, University of Ottawa
Recent advances for SPDEs with L\'evy noise  [PDF]

In this talk, we introduce a new class of processes that can be used as noise for stochastic partial differential equations (SPDEs). This noise is called the ``L\'evy colored noise'', and is constructed from a L\'evy white noise using the convolution with a suitable spatial kernel. We assume that the L\'evy measure of the noise has finite variance. The stochastic integral with respect to this noise is constructed similarly to the integral with respect to the spatially-homogeneous Gaussian case considered in Dalang (1999). Using Rosenthal's inequality, we provide an upper bound for the $p$-th moment of the stochastic integral with respect to this noise. Then, we analyze the existence of moments for linear and non-linear SPDE with this noise, considering as examples the stochastic heat and wave equations. This talk is based on joint work with Juan Jim\'enez.

BJOERN BRINGMANN, Princeton University
Global well-posedness of the stochastic Abelian-Higgs equations in two dimensions  [PDF]

There has been much recent progress on the local solution theory for geometric singular SPDEs. However, the global theory is still largely open. In this talk, we discuss the global well-posedness of the stochastic Abelian-Higgs model in two dimension, which is a geometric singular SPDE arising from gauge theory. This is joint work with S. Cao.

FRANCESCO CELLAROSI, Queen's University
Stochastic Calculus for the Theta Process  [PDF]

A key step in stochastic analysis is the ``art" of giving meaning to integrals against a random function. This is not a trivial task, since interesting random functions have poor regularity and Riemann-type approaches typically do not work. Sometimes, we may exploit the martingale-like properties of our random functions to give meaning to integrals against them. Alternatively, we may employ Rough Path Theory to define stochastic integration, trading some of the nice probabilistic properties for analytical and algebraic ones. I will outline how this is done in the case of the Theta Process. This is a stochastic process of number-theoretical origin that shares several (but not all!) properties with the Brownian Motion, and classical probabilistic tools to define a stochastic calculus for this process cannot be used. Joint work with Zachary Selk (https://arxiv.org/abs/2406.05523)

YU-TING CHEN, University of Victoria

DUNCAN DAUVERGNE, University of Toronto

NATHAN GLATT-HOLTZ, Indiana University

FAUZIA JABEEN, Toronto Metropolitan University

CHRISTOPHER KENNEDY, Queen's University

ARJUN KRISHNAN, University of Rochester

ZAIB UN NISA MEMON, Toronto Metropolitan University
A hybrid method for stochastic simulations of reaction-diffusion epidemic models  [PDF]

Reactive Multiparticle Collision (RMPC) Dynamics, a particle-based method, is able to keep track of every single individual in a population. However, tracking of infectious individuals becomes infeasible as the cases increase, in which case a compartment-based method, such as Inhomogeneous Stochastic Simulation Algorithm (ISSA), is typically used. This motivated the development of a temporally coupled RMPC-ISSA framework. The hybrid method results in significant acceleration of the simulations of reaction-diffusion epidemic models compared to RMPC-only simulations. This is joint work with K. Rohlf.

MIHAI NICA, University of Guelph

JEREMY QUASTEL, University of Toronto

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