Stochastic equations in infinite dimensions

In this paper we prove the stochastic homeomorphism flow property and the strong Feller property for stochastic differential equations with sigular time dependent drifts and Sobolev diffusion coefficients. Moreover, the local well posedness under local assumptions are also obtained. In particular, we extend Krylov and Rockner's results in [10] to the case of non-constant diffusion coefficients.

Stochastic Homeomorphism Flows of SDEs with Singular Drifts and Sobolev Diffusion Coefficients

In this article we develop a new approach to construct solutions of stochastic equations with merely measurable drift coefficients. We aim at demonstrating the principles of our technique by analyzing strong solutions of stochastic differential equations driven by Brownian motion. An important and rather surprising consequence of our method which is based on Malliavin calculus is that the solutions derived by Veretennikov (Theory Probab Appl 24:354–366, 1979) for Brownian motion with bounded and measurable drift in $\mathbb{R }^{d}$ are Malliavin differentiable. Further, a strength of our approach, which does not rely on a pathwise uniqueness argument, is that it can be transferred and applied to the analysis of various other types of (stochastic) equations: We obtain a Bismut–Elworthy–Li formula (Elworthy and Li, J Funct Anal 125:252–286, 1994) for spatial derivatives of solutions to the Kolmogorov equation with bounded and measurable drift coefficients. To derive the formula, we use that our approach can be applied to obtain Sobolev differentiability in the initial condition in addition to Malliavin differentiability of the associated stochastic differential equations. Another application of our technique is the construction of unique solutions of the stochastic transport equation with irregular vector fields. Moreover, our approach is also applicable to the construction of solutions of stochastic evolution equations on Hilbert spaces.

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A variational approach to the construction and Malliavin differentiability of strong solutions of SDE’s

https://portlandpress.com/biochemsoctrans/article-pdf/4/2/377/550550/bst0040377.pdf

Biology of Cancer (Second Edition)

Scaling techniques were used to analyze the fractal nature of colonies of 15 cell lines growing in vitro as well as of 16 types of tumor developing in vivo. All cell colonies were found to exhibit exactly the same growth dynamics-which correspond to the molecular beam epitaxy (MBE) universality class. MBE dynamics are characterized by 1), a linear growth rate, 2), the constraint of cell proliferation to the colony/tumor border, and 3), surface diffusion of cells at the growing edge. These characteristics were experimentally verified in the studied colonies. That these should show MBE dynamics is in strong contrast with the currently established concept of tumor growth: the kinetics of this type of proliferation rules out exponential or Gompertzian growth. Rather, a clear linear growth regime is followed. The importance of new cell movements-cell diffusion at the tumor border-lies in the fact that tumor growth must be conceived as a competition for space between the tumor and the host, and not for nutrients or other factors. Strong experimental evidence is presented for 16 types of tumor, the growth of which cell surface diffusion may be the main mechanism responsible in vivo. These results explain most of the clinical and biological features of colonies and tumors, offer new theoretical frameworks, and challenge the wisdom of some current clinical strategies.

The universal dynamics of tumor growth

We are concerned with the problem of global well-posedness of the 3D Navier--Stokes equations on the torus with unitary viscosity. While a full answer to this question seems to be out of reach of the current techniques, we establish a regularization by a deterministic vector field. More precisely, we consider the vorticity form of the system perturbed by an additional transport type term. Such a perturbation conserves the enstrophy and therefore a priori it does not imply any smoothing. Our main result is a construction of a deterministic vector field $v=v(t,x)$ which provides the desired regularization of the system and yields global well-posedness for large initial data outside arbitrary small sets. The proof relies on probabilistic arguments developed by Flandoli and Luo, tools from rough path theory by Hofmanova, Leahy and Nilssen and a new Wong--Zakai approximation result, which itself combines probabilistic and rough path techniques.

Global well-posedness of the 3D Navier--Stokes equations perturbed by a deterministic vector field

We consider a class of Hilbert-space valued SDE’s where the drift coefficients are nonLipschitzian in the sense of Holder-continuity. Using a novel technique based on Malliavin calculus we show in this paper the existence and uniqueness of a mild solution to such equations. We emphasize that our approach does not rely on the Yamada-Watanabe principle. Moreover our method gives the important additional insight that the obtained solution is Malliavin differentiable a property which was recently shown to play a crucial role in the study of the geometry of certain optimal causal transference plans, [12].

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Malliavin differentiability and strong solutions for a class of SDE in Hilbert spaces

We prove the existence of an eddy heat diffusion coefficient coming from an idealized model of turbulent fluid. A difficulty lies in the presence of a boundary, with also turbulent mixing and the eddy diffusion coefficient going to zero at the boundary. Nevertheless enhanced diffusion takes place.

Eddy heat exchange at the boundary under white noise turbulence

Realistic stochastic modeling is increasingly requiring the use of bounded noises. In this work, properties and relationships of commonly employed bounded stochastic processes are investigated with...

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Properties of bounded stochastic processes employed in biophysics

Inspired by a PDE–ODE system of aggregation developed in the biomathematical literature, we investigate an interacting particle system representing aggregation at the level of individuals. We prove that the empirical density of the individual converges to the solution of the PDE–ODE system.

Franco Flandoli

Papers

Global well-posedness of the 3D Navier--Stokes equations perturbed by a deterministic vector field

Malliavin differentiability and strong solutions for a class of SDE in Hilbert spaces

Eddy heat exchange at the boundary under white noise turbulence

Properties of bounded stochastic processes employed in biophysics

A particle system approach to aggregation phenomena