SPDEs
Mathematical challenges
Stochastic partial differential equations describe spatially extended systems under random forcing. Our work ranges from conservative and degenerate equations to singular and supercritical regimes in which classical solution concepts no longer apply.
We develop notions of solution, a priori estimates and compactness methods that respect conservation laws and nonlinear geometry. We study well-posedness, regularity, invariant behaviour and scaling limits, as well as situations in which noise regularizes an otherwise unstable or ill-posed evolution. A recurring objective is to connect SPDEs to the microscopic particle systems from which their fluctuations emerge and to devise numerical approximations that preserve the structure of the underlying equation.
Current directions
- Conservative, degenerate and singular SPDEs
- Well-posedness and regularity
- Regularization and stabilization by noise
- Supercritical equations
- Fluctuation limits and invariant behaviour
Related people
Related projects
Selected publications
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Parabolic-hyperbolic splitting in support propagation for stochastic porous media equations
Preprint · arXiv:2609.11468 (2026). Max Sauerbrey and Joshua Utley study support propagation and waiting-time phenomena for stochastic porous media equations with conservative noise.
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The Schrödinger equation with fluctuating nonlinearity in the energy space
We study nonlinear Schrödinger equations with nonlinear Stratonovich noise in their energy space $H^1(\mathbb R^d;\mathbb C)$. By combining the stochastic Strichartz estimates derived in [Potential Anal. 41 (2014), pp.\ 269–315] with the approach from [Ann.\ Inst.\ H.\ Poincaré Phys.\ Théor.\…
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Finite speed of propagation and waiting time phenomena for stochastic porous media equations with nonlinear conservative noise
Starting from localized energy estimates, we prove finite speed of propagation for kinetic solutions to stochastic porous media equations with nonlinear conservative noise, the existence and uniqueness of which has recently been established. In particular, we propose a novel iteration technique which…
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The stochastic Cahn-Hilliard equation in critical spaces
We study stochastic Cahn-Hilliard equations in bounded smooth domains with a double-well potential, transport-type noise, and natural Neumann boundary conditions in dimensions $d\le 4$. By employing stochastic maximal regularity techniques and deriving suitable energy estimates, we prove local and global well-posedness. The…
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The stochastic Keller–Segel system in critical spaces
Accepted / forthcoming · Oberwolfach Report for the seminar Stochastic Partial Differential Equations in Critical Spaces. We study stochastic, parabolic-parabolic Keller–Segel equations on the $d$-dimensional torus in scaling critical Besov spaces, for $d \geq 3$. Using stochastic maximal regularity estimates, we prove local well-posedness of the equation, i.e.,…
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Effective fluctuating continuum models for stochastic gradient descent
Workshop report · Oberwolfach Reports 23(1), 821–822 (2026). Fluctuating continuum descriptions of stochastic gradient descent.
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Introduce thermal noise before discretizing, not afterwards!
Workshop report · Oberwolfach Reports 23(1), 803–806 (2026). Incorporating thermal noise before discretising a continuum model.
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Probabilistically Strong Solutions to Stochastic Euler Equations
In this paper, we establish the existence of probabilistically strong, measure-valued solutions for the stochastic incompressible Navier–Stokes equations and prove their convergence, in the vanishing viscosity limit, to probabilistically strong solutions for the stochastic incompressible Euler equations. In particular, this solves the…