Related papers: Parameter dependent rough SDEs with applications t…
We consider the perturbation of parabolic operators of the form $\partial_t+P(x,D)$ by large-amplitude highly oscillatory spatially dependent potentials modeled as Gaussian random fields. The amplitude of the potential is chosen so that the…
We present a well-posedness and stability result for a class of nondegenerate linear parabolic equations driven by rough paths. More precisely, we introduce a notion of weak solution that satisfies an intrinsic formulation of the equation…
For parameter identification problems the Fr\'echet-derivative of the parameter-to-state map is of particular interest. In many applications, e.g. in seismic tomography, the unknown quantity is modeled as a coefficient in a linear…
In this paper, we consider the Cauchy problem of semi-linear degenerate backward stochastic partial differential equations (BSPDEs in short) under general settings without technical assumptions on the coefficients. For the solution of…
We prove a center manifold theorem for rough partial differential equations (rough PDEs). The class of rough PDEs we consider contains as a key subclass reaction-diffusion equations driven by nonlinear multiplicative noise, where the…
In this paper, we study well-posedness of McKean-Vlasov stochastic differential equations (SDE) whose drift depends pointwisely on marginal density and satisfies a local integrability condition in time-space variables. The drift and noise…
A new paradigm recently emerged in financial modelling: rough (stochastic) volatility, first observed by Gatheral et al. in high-frequency data, subsequently derived within market microstructure models, also turned out to capture…
The paper investigates existence and uniqueness for a stochastic differential equation (SDE) with distributional drift depending on the law density of the solution. Those equations are known as McKean SDEs. The McKean SDE is interpreted in…
We study the homogenization property of systems of quasi-linear PDEs of parabolic type with periodic coefficients, highly oscillating drift and highly oscillating nonlinear term. To this end, we propose a probabilistic approach based on the…
In this paper, we are interested in path-dependent stochastic differential equations (SDEs) which are controlled by Brownian motion and its delays. Within this non-Markovian context, we give a H \"ormander-type criterion for the regularity…
We consider a $p$-Laplace evolution problem with stochastic forcing on a bounded domain $D\subset\mathbb{R}^d$ with homogeneous Dirichlet boundary conditions for $1<p<\infty$. The additive noise term is given by a stochastic integral in the…
Linearized shallow neural networks that are constructed by fixing the hidden-layer parameters have recently shown strong performance in solving partial differential equations (PDEs). Such models, widely used in the random feature method…
We develop a complete and rigorous mathematical framework for the analysis of stochastic neural field equations under the influence of spatially extended additive noise. By comparing a solution to a fixed deterministic front profile it is…
We give a new take on the error analysis of approximations of stochastic differential equations (SDEs), utilizing and developing the stochastic sewing lemma of L\^e (2020). This approach allows one to exploit regularization by noise effects…
This paper introduces the path derivatives, in the spirit of Dupire's functional It\^o calculus, for the controlled paths in the rough path theory with possibly non-geometric rough paths. The theory allows us to deal with rough integration…
In this paper, we introduce a specific kind of doubly reflected Backward Stochastic Differential Equations (in short DRBSDEs), defined on probability spaces equipped with general filtration that is essentially non quasi-left continuous,…
The purpose of this paper is to establish a well-posedness theory for conservative stochastic partial differential equations on the whole space. This class of stochastic PDEs arises in fluctuating hydrodynamics, and includes the…
Existence, uniqueness, and regularity of a strong solution are obtained for stochastic PDEs with a colored noise $F$ and its super-linear diffusion coefficient: $$ du=(a^{ij}u_{x^ix^j}+b^iu_{x^i}+cu)dt+\xi|u|^{1+\lambda}dF, \quad…
This paper establishes a Freidlin-Wentzell large deviation principle for stochastic differential equations(SDEs) under locally weak monotonicity conditions and Lyapunov conditions. We illustrate the main result of the paper by showing that…
It is known that the transition probabilities of a solution to a classical It\^o stochastic differential equation (SDE) satisfy in the weak sense the associated Kolmogorov equation. The Kolmogorov equation is a partial differential equation…