Generalized backward doubly stochastic differential equations and SPDEs with nonlinear Neumann boundary conditions

In this paper a new class of generalized backward doubly stochastic differential equations is investigated. This class involves an integral with respect to an adapted continuous increasing process. A probabilistic representation for viscosity solutions of semi-linear stochastic partial differential equations with a Neumann boundary condition is given.Comment: Published at http://dx.doi.org/10.3150/07-BEJ5092 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

On the Besov regularity of the bifractional Brownian motion

Our aim in this paper is to improve H\"{o}lder continuity results for the bifractional Brownian motion (bBm) $(B^{\alpha,\beta}(t))_{t\in[0,1] }$ with $0<\alpha<1$ and $0<\beta\leq 1$. We prove that almost all paths of the bBm belong (resp. do not belong) to the Besov spaces $\mathbf{Bes}(\alpha \beta,p)$ (resp. $\mathbf{bes}(\alpha \beta,p)$) for any $\frac{1}{\alpha \beta}<p<\infty$, where $\mathbf{bes}(\alpha \beta,p)$ is a separable subspace of $\mathbf{Bes}(\alpha \beta,p)$. We also show the It\^{o}-Nisio theorem for the bBm with $\alpha \beta>\frac{1}{2}$ in the H\"{o}lder spaces $\mathcal{C}^{\gamma}$, with $\gamma<\alpha \beta$.Comment: 20 page

Local times for systems of non-linear stochastic heat equations

peer reviewedWe consider u(t, x) = (u1(t, x) , ⋯ , ud(t, x)) the solution to a system of non-linear stochastic heat equations in spatial dimension one driven by a d-dimensional space-time white noise. We prove that, when d≤ 3 , the local time L(ξ, t) of {u(t,x),t∈[0,T]} exists and L(· , t) belongs a.s. to the Sobolev space Hα(Rd) for α<4-d2, and when d≥ 4 , the local time does not exist. We also show joint continuity and establish Hölder conditions for the local time of {u(t,x),t∈[0,T]}. These results are then used to investigate the irregularity of the coordinate functions of {u(t,x),t∈[0,T]}. Comparing to similar results obtained for the linear stochastic heat equation (i.e., the solution is Gaussian), we believe that our results are sharp. Finally, we get a sharp estimate for the partial derivatives of the joint density of (u(t1, x) - u(t, x) , ⋯ , u(tn, x) - u(tn-1, x)) , which is a new result and of independent interest

On the uniform Besov regularity of local times of general processes

Our main purpose is to use a new condition, $\alpha$-local nondeterminism, which is an alternative to the classical local nondeterminism usually utilized in the Gaussian framework, in order to investigate Besov regularity, in the time variable $t$ uniformly in the space variable $x$, for local times $L(x, t)$ of a class of continuous processes. We also extend the classical Adler's theorem [1, Theorem 8.7.1] to the Besov spaces case. These results are then exploited to study the Besov irregularity of the sample paths of the underlying processes. Based on similar known results in the case of the bifractional Brownian motion, we believe that our results are sharp. As applications, we get sharp Besov regularity results for some classical Gaussian processes and the solutions of systems of non-linear stochastic heat equations. The Besov regularity of their corresponding local times is also obtained

On Besov regularity and local time of the solution to the stochastic heat equation

peer reviewedSharp Besov regularities in time and space variables are investigated for (Formula presented.), the mild solution to the stochastic heat equation driven by space–time white noise. Existence, Hölder continuity, and Besov regularity of local times are established for (Formula presented.) viewed either as a process in the space variable or time variable. Hausdorff dimensions of their corresponding level sets are also obtained