Distribution dependent SDEs driven by additive continuous noise

We study distribution dependent stochastic differential equation driven by a continuous process, without any specification on its law, following the approach initiated in [17]. We provide several criteria for existence and uniqueness of solutions which go beyond the classical globally Lipschitz setting. In particular we show well-posedness of the equation, as well as almost sure convergence of the associated particle system, for drifts satisfying either Osgood-continuity, monotonicity, local Lipschitz or Sobolev differentiability type assumptions

Distribution dependent SDEs driven by additive fractional Brownian motion

We study distribution dependent stochastic differential equations with irregular, possibly distributional drift, driven by an additive fractional Brownian motion of Hurst parameter $$H\in (0,1)$$. We establish strong well-posedness under a variety of assumptions on the drift; these include the choice $$\begin{aligned} B(\cdot ,\mu )=(f*\mu )(\cdot ) + g(\cdot ), \quad f,\,g\in B^\alpha _{\infty ,\infty },\quad \alpha >1-\frac{1}{2H}, \end{aligned}$$B(·,μ)=(f∗μ)(·)+g(·),f,g∈B∞,∞α,α>1-12H,thus extending the results by Catellier and Gubinelli (Stochast Process Appl 126(8):2323–2366, 2016) to the distribution dependent case. The proofs rely on some novel stability estimates for singular SDEs driven by fractional Brownian motion and the use of Wasserstein distances

Stochastic PDEs and weakly interacting particle systems

We study two problems relating to weakly interacting particle systems in the presence of exogenous noise. Part I presents a pathwise regularisation by noise result for McKean–Vlasov equations with singular interaction kernels. Using ideas from the theory of averaged fields and non-linear Young integrals we obtain well-posedness of the McKean–Vlasov systems, particle approximations and mean field convergence. In Part II we study a family of semi-linear, convection-diffusion SPDEs that are closely related to PDEs coming from the theory of collision-less kinetics. We study these equations in the presence of additive spacetime white noise. In one dimension we show global well-posedness and exponential ergodicity for an equation with a cubic non-linearity and repulsive sign choice. In two dimensions we show local well-posedness for a renormalised equation.</p

Distribution dependent SDEs driven by additive fractional Brownian motion

AbstractWe study distribution dependent stochastic differential equations with irregular, possibly distributional drift, driven by an additive fractional Brownian motion of Hurst parameter $$H\in (0,1)$$ H ∈ ( 0 , 1 ) . We establish strong well-posedness under a variety of assumptions on the drift; these include the choice $$\begin{aligned} B(\cdot ,\mu )=(f*\mu )(\cdot ) + g(\cdot ), \quad f,\,g\in B^\alpha _{\infty ,\infty },\quad \alpha &gt;1-\frac{1}{2H}, \end{aligned}$$ B ( · , μ ) = ( f ∗ μ ) ( · ) + g ( · ) , f , g ∈ B ∞ , ∞ α , α &gt; 1 - 1 2 H , thus extending the results by Catellier and Gubinelli (Stochast Process Appl 126(8):2323–2366, 2016) to the distribution dependent case. The proofs rely on some novel stability estimates for singular SDEs driven by fractional Brownian motion and the use of Wasserstein distances. </jats:p

A stochastic model of chemorepulsion with additive noise and nonlinear sensitivity

We consider a stochastic partial differential equation (SPDE) model for chemorepulsion, with non-linear sensitivity on the one-dimensional torus. By establishing an a priori estimate independent of the initial data, we show that there exists a pathwise unique, global solution to the SPDE. Furthermore, we show that the associated semi-group is Markov and possesses a unique invariant measure, supported on a Hölder–Besov space of positive regularity, which the solution law converges to exponentially fast. The a priori bound also allows us to establish tail estimates on the Lp norm of the invariant measure which are heavier than Gaussian

Distribution dependent SDEs driven by additive continuous noise

AMC

Distribution dependent SDEs driven by additive fractional Brownian motion

AbstractWe study distribution dependent stochastic differential equations with irregular, possibly distributional drift, driven by an additive fractional Brownian motion of Hurst parameter $$H\in (0,1)$$ H ∈ ( 0 , 1 ) . We establish strong well-posedness under a variety of assumptions on the drift; these include the choice $$\begin{aligned} B(\cdot ,\mu )=(f*\mu )(\cdot ) + g(\cdot ), \quad f,\,g\in B^\alpha _{\infty ,\infty },\quad \alpha &gt;1-\frac{1}{2H}, \end{aligned}$$ B ( · , μ ) = ( f ∗ μ ) ( · ) + g ( · ) , f , g ∈ B ∞ , ∞ α , α &gt; 1 - 1 2 H , thus extending the results by Catellier and Gubinelli (Stochast Process Appl 126(8):2323–2366, 2016) to the distribution dependent case. The proofs rely on some novel stability estimates for singular SDEs driven by fractional Brownian motion and the use of Wasserstein distances. </jats:p

Distribution dependent SDEs driven by additive fractional Brownian motion

AMC