\def\R{\mathbb R}
We give a characterization of G-regularity for super-Brownian motion and the Brownian snake. More precisely, we define a capacity on $E=(0,\infty)\times \R^d$, which is not invariant by translation. We then prove that the hitting probability of a Borel set $A\subset E$ for the graph of the Brownian snake starting at $(0,0)$ is comparable, up to multiplicative constants, to its capacity. This implies that super-Brownian motion started at time 0 at the Dirac mass $δ_0$ hits immediately $A$ (that is $(0,0)$ is G-regular for $A^c$) if and only if its capacity is infinite. As a direct consequence, if $Q\subset E$ is a domain such that $(0,0)\in \partial Q$, we give a necessary and sufficient condition for the existence on $Q$ of a positive solution of $\partial_t u+{1/2}Δu =2u^2$ which blows up at $(0,0)$. We also give an estimation of the hitting probabilities for the support of super-Brownian motion at fixed time. We prove that if $d\geq 2$, the support of super-Brownian motion is intersection-equivalent to the range of Brownian motion.
We derive integral formulas, involving the Airy function, for moments of the time a two-sided Brownian motion with parabolic drift attains its maximum.
We study the first exit time of a multi-dimensional fractional Brownian motion from unbounded domains. In particular, we are interested in the upper tail of the corresponding distribution when the domain is parabola-shaped.
We study the maximum of a Brownian motion with a parabolic drift; this is a random variable that often occurs as a limit of the maximum of discrete processes whose expectations have a maximum at an interior point. We give series expansions and integral formulas for the distribution and the first two moments, together with numerical values to high precision.
We study the pointwise regularity of the Multifractional Brownian Motion and in particular, we get the existence of slow points. It shows that a non self-similar process can still enjoy this property. We also consider various extensions of our results in the aim of requesting a weaker regularity assumption for the Hurst function without altering the regularity of the process.
In the Reed-Frost model, an example of an SIR epidemic model, one can examine a statistic that counts the number of concurrently infected individuals. This statistic can be reformulated as a statistic on the \ER random graph $G(n,p)$. Within the critical window of Aldous and Martin-Löf, i.e. when $p = p(n) = n^{-1}+λn^{-4/3}$, the cumulative sum of this statistic converges weakly to the integral of a Brownian motion with parabolic drift. This same statistic exhibits a deterministic scaling limit when $p = (1+λ\varepsilon_n)/n$ whenever $\varepsilon_n\to 0$ and $n^{1/3}\varepsilon_n\to\infty$.
In this paper, we obtain the existence and uniqueness theorem for backward stochastic differential equation driven by G-Brownian motion (G-BSDE) under degenerate case. Moreover, we propose a new probabilistic method based on the representation theorem of G-expectation and weak convergence to obtain the regularity of fully nonlinear PDE associated to G-BSDE.
Let $B_s$ be a $d$-dimensional Brownian motion and $ω(dx)$ be an independent Poisson field on $\mathbb{R}^d$. The almost sure asymptotics for the logarithmic moment generating function [\log\math bb{E}_0\exp\biggl{\pmθ\int_0^t\bar{V}(B_s) ds\biggr}\qquad (t\to\infty)] are investigated in connection with the renormalized Poisson potential of the form [\bar{V}(x)=\int_{\mathbb{R}^d}{\frac{1}{|y-x|^p}}[ω(dy)-dy],\qquad x\in\mathbb{R}^d.] The investigation is motivated by some practical problems arising from the models of Brownian motion in random media and from the parabolic Anderson models.
In this work, we will show the existence and uniqueness of the solution to the semi linear stochastic differential equations driven by weighted fractional Brownian motion with delay. We also prove smoothness of the density of the solution with respect to Lebesgue's measure on R^d for $d \geq 1$.
We apply improved elliptic regularity results to a concrete symmetric Dirichlet form and various non-symmetric Dirichlet forms with possibly degenerate symmetric diffusion matrix. Given the (non)-symmetric Dirichlet form, using elliptic regularity results and stochastic calculus we show weak existence of the corresponding singular stochastic differential equation for any starting point in some subset E of R^d. As an application of our approach we can show the existence of n-regularized Liouville Brownian motion only via Dirichlet form theory starting from all points in R^2.
In this paper we prove, for small Hurst parameters, the higher order differentiability of a stochastic flow associated with a stochastic differential equation driven by an additive multi-dimensional fractional Brownian noise, where the bounded variation part is given by the local time of the unknown solution process. The proof of this result relies on Fourier analysis based variational calculus techniques and on intrinsic properties of the fractional Brownian motion.
By using Malliavin calculus and multiple Wiener-Itô integrals, we study the existence and the regularity of stochastic currents defined as Skorohod (divergence) integrals with respect to the Brownian motion and to the fractional Brownian motion. We consider also the multidimensional multiparameter case and we compare the regularity of the current as a distribution in negative Sobolev spaces with its regularity in Watanabe space.
We study the Hardy-Hénon parabolic equations on $\mathbb{R}^{N}$ ($N=2, 3$) under the effect of an additive fractional Brownian noise with Hurst parameter $H>\max\left(1/2, N/4\right).$ We show local existence and uniqueness of a mid $L^{q}$-solution under suitable assumptions on $q$.
We prove that minimal graphs (other than planes) are parabolic in the sense that any bounded harmonic function is determined by its boundary values. The proof relies on using the coupling introduced in the author's earlier paper "A martingale approach to minimal surfaces" to show that Brownian motion on such a minimal graph almost surely strikes the boundary in finite time.
Even though the heat equation with random potential is a well-studied object, the particular case of time-independent Gaussian white noise in one space dimension has yet to receive the attention it deserves. The paper investigates the stochastic heat equation with space-only Gaussian white noise on a bounded interval. The main result is that the space-time regularity of the solution is the same for additive noise and for multiplicative noise in the Wick-Itô-Skorokhod interpretation.
We study the pathwise regularity of the map $$ ϕ\mapsto I(ϕ) = \int_0^T < ϕ(X_t), dX_t>$$ where $ϕ$ is a vector function on $\R^d$ belonging to some Banach space $V$, $X$ is a stochastic process and the integral is some version of a stochastic integral defined via regularization. A \emph{stochastic current} is a continuous version of this map, seen as a random element of the topological dual of $V$. We give sufficient conditions for the current to live in some Sobolev space of distributions and we provide elements to conjecture that those are also necessary. Next we verify the sufficient condi…
By the work of P. Lévy, the sample paths of the Brownian motion are known to satisfy a certain Hölder regularity condition almost surely. This was later improved by Ciesielski, who studied the regularity of these paths in Besov and Besov-Orlicz spaces. We review these results and propose new function spaces of Besov type, strictly smaller than those of Ciesielski and Lévy, where the sample paths of the Brownian motion lie in almost surely. In the same spirit, we review and extend the work of Kamont, who investigated the same question for the multivariate Brownian sheet and function spaces of d…
We consider a possibly strongly degenerate parabolic semilinear problem which can be applied to a differential model for pricing financial derivatives. We prove the asked regularity for applying the Ito's formula which is used for building the differential model.
This paper presents a new estimator of the global regularity index of a multifractional Brownian motion. Our estimation method is based upon a ratio statistic, which compares the realized global quadratic variation of a multifractional Brownian motion at two different frequencies. We show that a logarithmic transformation of this statistic converges in probability to the minimum of the Hurst function, which is, under weak assumptions, identical to the global regularity index of the path.
We study quasilinear degenerate parabolic-hyperbolic stochastic partial differential equations with general multiplicative noise within the framework of kinetic solutions. Our results are twofold: First, we establish new regularity results based on averaging techniques. Second, we prove the existence and uniqueness of solutions in a full $L^1$ setting requiring no growth assumptions on the nonlinearities. In addition, we prove a comparison result and an $L^1$-contraction property for the solutions.
We announce a result on the existence of a unique local solution to a stochastic geometric wave equation on the one dimensional Minkowski space $\mathbb{R}^{1+1}$ with values in an arbitrary compact Riemannian manifold. We consider a rough initial data in the sense that its regularity is lower than the energy critical.
We extend to the vector-valued situation some earlier work of Ciesielski and Roynette on the Besov regularity of the paths of the classical Brownian motion. We also consider a Brownian motion as a Besov space valued random variable. It turns out that a Brownian motion, in this interpretation, is a Gaussian random variable with some pathological properties. We prove estimates for the first moment of the Besov norm of a Brownian motion. To obtain such results we estimate expressions of the form $\E \sup_{n\geq 1}\|ξ_n\|$, where the $ξ_n$ are independent centered Gaussian random variables with va…
We prove Hölder regularity for a general class of parabolic integro-differential equations, which (strictly) includes many previous results. We present a proof which avoids the use of a convex envelop as well as give a new covering argument which is better suited to the fractional order setting. Our main result involves a class of kernels which may contain a singular measure, may vanish at some points, and are not required to be symmetric. This new generality of integro-differential operators opens the door to further applications of the theory, including some regularization estimates for the …
Sobolev regularity theory for stochastic reaction-diffusion-advection equations with spatially homogeneous colored noises and infinitesimal generators of subordinate Brownian motions
A parameter estimation problem is considered for a diagonaliazable stochastic evolution equation using a finite number of the Fourier coefficients of the solution. The equation is driven by additive noise that is white in space and fractional in time with the Hurst parameter $H\geq 1/2$. The objective is to study asymptotic properties of the maximum likelihood estimator as the number of the Fourier coefficients increases. A necessary and sufficient condition for consistency and asymptotic normality is presented in terms of the eigenvalues of the operators in the equation.
We show that solutions of free stochastic differential equations with regular drifts and diffusion coefficients, when considered backwards in time, still satisfy free SDEs for an explicit free Brownian motion and drift. We also study the stochastic integral part with respect to this reversed free Brownian motion of canonical closed martingales. We deduce that conjugate variables computed along a free Brownian motion, an example of such a reversed martingale appearing in the definition of non-microstates free entropy, are in the $L^2$ domain of corresponding free difference quotients for almost…
