We consider the problem of absolute continuity for the one-dimensional SDE \[X_t=x+\int_0^ta(X_s) ds+Z_t,\] where $Z$ is a real Lévy process without Brownian part and $a$ a function of class $\mathcal{C}^1$ with bounded derivative. Using an elementary stratification method, we show that if the drift $a$ is monotonous at the initial point $x$, then $X_t$ is absolutely continuous for every $t>0$ if and only if $Z$ jumps infinitely often. This means that the drift term has a regularizing effect, since $Z_t$ itself may not have a density. We also prove that when $Z_t$ is absolutely continuous, then the same holds for $X_t$, in full generality on $a$ and at every fixed time $t$. These results are then extended to a larger class of elliptic jump processes, yielding an optimal criterion on the driving Poisson measure for their absolute continuity.
We consider the Itô SDE with non-degenerate diffusion coefficient and measurable drift coefficient. Under the condition that the gradient of the diffusion coefficient and the divergences of the diffusion and drift coefficients are exponentially integrable with respect to the Gaussian measure, we show that the stochastic flow leaves the reference measure absolutely continuous.
We consider infinitely divisible distributions with symmetric Lévy measure and study the absolute continuity of them with respect to the Lebesgue measure. We prove that if $η(r)=\int_{|x|\le r} x^2 ν(dx)$ where $ν$ is the Lévy measure, then $\int_0^1 \frac{r}{η(r)}dr <\infty$ is a sufficient condition for absolute continuity. As far as we know, our result is not implied by existing results about absolute continuity of infinitely divisible distributions.
In this paper, we prove that the laws of perturbed diffusion processes and perturbed reflected diffusion processes are absolutely continuous with respect to the Lebesgue measure. The main tool we use is the Malliavin calculus.
We derive equivalent conditions for the (local) absolute continuity of two laws of semimartingales on random sets. Our result generalizes previous results for classical semimartingales by replacing a strong uniqueness assumption by a weaker uniqueness assumption. The main tool is a generalized Girsanov's theorem, which relates laws of two possibly explosive semimartingales to a candidate density process. Its proof is based on an extension theorem for consistent families of probability measures. Moreover, we show that in a one-dimensional Itô-diffusion setting our result reproduces the known de…
In this paper, we focus on the existence of the density for the law of the solutions to parabolic stochastic partial differential equations with two reflecting walls. The main tool is Malliavin calculus.
We consider piecewise deterministic Markov processes with degenerate transition kernels of the "house-of-cards"-type. We use a splitting scheme based on jump times to prove the absolute continuity, as well as some regularity, of the invariant measure of the process. Finally, we obtain finer results on the regularity of the one-dimensional marginals of the invariant measure, using integration by parts with respect to the jump times.
In this note, we show that the limiting spectral distribution of symmetric random matrices with stationary entries is absolutely continuous under some sufficient conditions. This result is applied to obtain sufficient conditions on a probability measure for its free multiplicative convolution with the semicircle law to be absolutely continuous.
We investigate the regularity of the law of Wong-Zakai-type approximations for Itô stochastic differential equations. These approximations solve random differential equations where the diffusion coefficient is Wick-multiplied by the smoothed white noise. Using a criteria based on the Malliavin calculus we establish absolute continuity and a Fokker-Planck-type equation solved in the distributional sense by the density. The parabolic smoothing effect typical of the solutions of Itô equations is lacking in this approximated framework; therefore, in order to prove absolute continuity, the initial …
We show that the limiting eigenvalue distribution of random symmetric Toeplitz matrices is absolutely continuous with density bounded by 8, partially answering a question of Bryc, Dembo and Jiang (2006). The main tool used in the proof is a spectral averaging technique from the theory of random Schrödinger operators. The similar question for Hankel matrices remains open.
In this paper we study mutual absolute continuity and singularity of probability measures on the path space which are induced by an isotropic stable Lévy process and the purely discontinuous Girsanov transform of this process. We also look at the problem of finiteness of the relative entropy of these measures. An important tool in the paper is the question under which circumstances the a.s. finiteness of an additive functional at infinity implies the finiteness of its expected value.
General sufficient conditions are given for absolute continuity and convergence in variation of the distributions of the unctionals on a probability space, generated by a Poisson point measure. The phase space of the Poisson point measure is supposed to be of the form (0,\infty)\times U, and its intensity measure to be equal dtΠ(du). We introduce the family of time stretching transformations of the configurations of the point measure. The sufficient conditions for absolute continuity and convergence in variation are given in the terms of the time stretching transformations and the relative dif…
We study the absolute continuity with respect to the Lebesgue measure of the distribution of the nodal volume associated with a smooth, non-degenerated and stationary Gaussian field $(f(x), {x \in \mathbb R^d})$. Under mild conditions, we prove that in dimension $d\geq 3$, the distribution of the nodal volume has an absolutely continuous component plus a possible singular part. This singular part is actually unavoidable baring in mind that some Gaussian processes have a positive probability to keep a constant sign on some compact domain. Our strategy mainly consists in proving closed Kac--Rice…
Let $X$ be a $\mathbb{C}$-valued random variable with the property that $$X \ \text{ has the same law as }\ \sum_{j\ge1} T_j X_j$$ where $X_j$ are i.i.d.\ copies of $X$, which are independent of the (given) $\mathbb{C}$-valued random variables $ (T_j)_{j\ge1}$. We provide a simple criterion for the absolute continuity of the law of $X$ that requires, besides the known conditions for the existence of $X$, only finiteness of the first and second moment of $N$ - the number of nonzero weights $T_j$. Our criterion applies in particular to Biggins' martingale with complex parameter.
The paper is concerned with one-dimensional two-sided Ornstein-Uhlenbeck type processes with delay or anticipation. We prove existence and uniqueness requiring almost sure boundedness on the left half-axis in case of delay and almost sure boundedness on the right half-axis in case of anticipation. For those stochastic processes $(X,P_μ)$ we calculate the Radon-Nikodym density under time shift of trajectories, $P_μ(dX_{\cdot -t})/P_μ(dX)$, $t\in {\Bbb R}$.
We prove results for random walks in dynamic random environments which do not require the strong uniform mixing assumptions present in the literature. We focus on the "environment seen from the walker"-process and in particular its invariant law. Under general conditions it exists and is mutually absolutely continuous to the environment law. With stronger assumptions we obtain for example uniform control on the density or a quenched CLT. The general conditions are made more explicit by looking at hidden Markov models or Markov chains as environment and by providing simple examples.
Consider two laws \(P\) and \(Q\) of multidimensional possibly explosive diffusions with common diffusion coefficient \(\mathfrak{a}\) and drift coefficients \(\mathfrak{b}\) and \(\mathfrak{b} + \mathfrak{a} \mathfrak{c}\), respectively, and the law \(P^\circ\) of an auxiliary diffusion with diffusion coefficient \(\langle \mathfrak{c},\mathfrak{a}\mathfrak{c}\rangle^{-1}\mathfrak{a}\) and drift coefficient \(\langle \mathfrak{c}, \mathfrak{a}\mathfrak{c}\rangle^{-1}\mathfrak{b}\). We show that \(P \ll Q\) if and only if the auxiliary diffusion \(P^\circ\) explodes almost surely and that \(P\…
We consider the two dimensional Navier-Stokes equations in vorticity form with a stochastic forcing term given by a gaussian noise, white in time and coloured in space. First, we prove existence and uniqueness of a weak (in the Walsh sense) solution process $ξ$ and we show that, if the initial vorticity $ξ_0$ is continuous in space, then there exists a space-time continuous version of the solution. In addition we show that the solution $ξ(t,x)$ (evaluated at fixed points in time and space) is locally differentiable in the Malliavin calculus sense and that its image law is absolutely continuous…
A non linear Ito equation in a Hilbert space is studied by means of Girsanov theorem. We consider a non linearity of polynomial growth in suitable norms, including that of quadratic type which appears in the Kuramoto-Sivashinsky equation and in the Navier-Stokes equation. We prove that Girsanov theorem holds for the 1-dimensional stochastic Kuramoto-Sivashinsky equation and for a modification of the 2- and 3-dimensional stochastic Navier-Stokes equation. In this way, we prove existence and uniqueness of solutions for these stochastic equations. Moreover, the asymptotic behaviour for large time…
Let $μ$ and $ν$ be fixed probability measures on a filtered space $(Ω, {\cal F}, ({\cal F}_t)_{t\in {\bf R}^{+}})$. Denote by $μ_T $ and $ν_T $ (respectively, $μ_{T-} $ and $ν_{T-} $) the restrictions of the measures $μ$ and $ν$ on ${\cal F}_T $ (respectively, on ${\cal F}_{T-} $) for a stopping time $T$. We find the Hahn decomposition of $μ_T $ and $ν_T $ using the Hahn decomposition of the measures $μ$, $ν$, and the Hellinger process $h_t$ in the strict sense of order 1/2. The norm of the absolutely continuous component of $μ_{T-} $ with respect to $ν_{T-} $ is computed in terms of density p…
We study existence and regularity of the density for the solution $u(t,x)$ (with fixed $t > 0$ and $x \in D$) of the heat equation in a bounded domain $D \subset \mathbb R^d$ driven by a stochastic inhomogeneous Neumann boundary condition with stochastic term. The stochastic perturbation is given by a fractional Brownian motion process. Under suitable regularity assumptions on the coefficients, by means of tools from the Malliavin calculus, we prove that the law of the solution has a smooth density with respect to the Lebesgue measure in $\mathbb R$.
We study Girsanov's theorem in the context of symmetric Markov processes, extending earlier work of Fukushima-Takeda and Fitzsimmons on Girsanov transformations of ``gradient type.'' We investigate the most general Girsanov transformation leading to another symmetric Markov process. This investigation requires an extension of the forward-backward martingale method of Lyons-Zheng, to cover the case of processes with jumps.
The aim of this paper is to establish some new results on the absolute continuity and the convergence in total variation for a sequence of d-dimensional vectors whose components belong to a finite sum of Wiener chaoses. First we show that the probability that the determinant of the Malliavin matrix of such vectors vanishes is zero or one, and this probability equals to one is equivalent to say that the vector takes values in the set of zeros of a polynomial. We provide a bound for the degree of this annihilating polynomial improving a result by Kusuoka. On the other hand, we show that the conv…
For the class of stochastic partial differential equations studied in [Conus-Dalang,2008], we prove the existence of density of the probability law of the solution at a given point $(t,x)$, and that the density belongs to some Besov space. The proof relies on the method developed in [Debussche-Romito, 2014]. The result can be applied to the solution of the stochastic wave equation with multiplicative noise, Lipschitz coefficients and any spatial dimension $d\ge 1$, and also to the heat equation. This provides an extension of the results proved in [Sanz-Solé and Süß, 2013].
Let $X$ be a $n$-dimensional Ornstein-Uhlenbeck process, solution of the S.D.E. $$\d X_t = AX_t \d t + \d B_t$$ where $A$ is a real $n\times n$ matrix and $B$ a Lévy process without Gaussian part. We show that when $A$ is non-singular, the law of $X_1$ is absolutely continuous in $\r^n$ if and only if the jumping measure of $B$ fulfils a certain geometric condition with respect to $A,$ which we call the exhaustion property. This optimal criterion is much weaker than for the background driving Lévy process $B$, which might be very singular and sometimes even have a one-dimensional discrete jump…
We prove that the weak solution of a uniformly elliptic stochastic differential equation with locally smooth diffusion coefficient and Hölder continuous drift has a Hölder continuous density function. This result complements recent results of Fournier-Printems \cite{F1}, where the density is shown to exist if both coefficients are Hölder continuous and exemplifies the role of the drift coefficient in the regularity of the density of a diffusion.
We consider a supercritical branching process $Z_n$ in a stationary and ergodic random environment $ξ=(ξ_n)_{n\ge0}$. Due to the martingale convergence theorem, it is known that the normalized population size $W_n=Z_n/ (\mathbb E (Z_n|ξ))$ converges almost surely to a random variable $W$. We prove that if $W$ is not concentrated at $0$ or $1$ then for almost every environment $ξ$ the law of $W$ conditioned on the environment $ξ$ is absolutely continuous with a possible atom at $0$. The result generalizes considerably the main result of \cite{kaplan:1974}, and of course it covers the well-known…
We prove existence of solutions to continuity equations in a separable Hilbert space. We look for solutions which are absolutely continuous with respect to a reference measure γwhich is Fomin-differentiable with exponentially integrable partial logarithmic derivatives. We describe a class of examples to which our result applies and for which we can prove also uniqueness. Finally, we consider the case where γis the invariant measure of a reaction-diffusion equation and prove uniqueness of solutions in this case. We exploit that the gradient operator D_x is closable with respect to L^p(H,γ) and …
In the recent article [A. Jentzen, B. Kuckuck, T. Müller-Gronbach, and L. Yaroslavtseva, arXiv:1904.05963 (2019)] it has been proved that the solutions to every additive noise driven stochastic differential equation (SDE) which has a drift coefficient function with at most polynomially growing first order partial derivatives and which admits a Lyapunov-type condition (ensuring the the existence of a unique solution to the SDE) depend in a logarithmically Hölder continuous way on their initial values. One might then wonder whether this result can be sharpened and whether in fact, SDEs from this…
We prove the existence and the Besov regularity of the density of the solution to a general parabolic SPDE which includes the stochastic Burgers equation on an unbounded domain. We use an elementary approach based on the fractional integration by parts.
This paper develops a new divergence that generalizes relative entropy and can be used to compare probability measures without a requirement of absolute continuity. We establish properties of the divergence, and in particular derive and exploit a representation as an infimum convolution of optimal transport cost and relative entropy. Also included are examples of computation and approximation of the divergence, and the demonstration of properties that are useful when one quantifies model uncertainty.
For a class of partially hyperbolic $C^k$, $k>1$ diffeomorphisms with circle center leaves we prove existence and finiteness of physical (or Sinai-Ruelle-Bowen) measures, whose basins cover a full Lebesgue measure subset of the ambient manifold. Our conditions contain an open and dense subset of all $C^k$ partially hyperbolic skew-products on compact circle bundles. Our arguments blend ideas from the theory of Gibbs states for diffeomorphisms with mostly contracting center direction together with recent progress in the theory of cocycles over hyperbolic systems that call into play geometric pr…
In this article we provide a proof of the so called absolute continuity theorem for random dynamical systems on $R^d$ which have an invariant probability measure. First we present the construction of local stable manifolds in this case. Then the absolute continuity theorem basically states that for any two transversal manifolds to the family of local stable manifolds the induced Lebesgue measures on these transversal manifolds are absolutely continuous under the map that transports every point on the first manifold along the local stable manifold to the second manifold, the so-called Poincaré …
We consider a class of measures called autophage which was introduced and studied by Szekely for measures on the real line. We show that the autophage measures on finite-dimensional vector spaces over real or p-adic field are infinitely divisible without idempotent factors and are absolutely continuous with bounded continuous density. We also show that certain semistable measures on such vector spaces are absolutely continuous.
This thesis develops a new divergence that generalizes relative entropy and can be used to compare probability measures without a requirement of absolute continuity. We establish properties of the divergence, and in particular derive and exploit a representation as an infimum convolution of optimal transport cost and relative entropy. We include examples of computation and approximation of the divergence, and its applications in uncertainty quantification in discrete models and Gauss-Markov models.
We consider the class of non-linear stochastic partial differential equations studied in \cite{conusdalang}. Equivalent formulations using integration with respect to a cylindrical Brownian motion and also the Skorohod integral are established. It is proved that the random field solution to these equations at any fixed point $(t,x)\in[0,T]\times \Rd$ is differentiable in the Malliavin sense. For this, an extension of the integration theory in \cite{conusdalang} to Hilbert space valued integrands is developed, and commutation formulae of the Malliavin derivative and stochastic and pathwise inte…
We prove absolute continuity of the law of the solution, evaluated at fixed points in time and space, to a parabolic dissipative stochastic PDE on $L^2(G)$, where $G$ is an open bounded domain in $\mathbb{R}^d$ with smooth boundary. The equation is driven by a multiplicative Wiener noise and the nonlinear drift term is the superposition operator associated to a real function which is assumed to be monotone, locally Lipschitz continuous, and growing not faster than a polynomial. The proof, which uses arguments of the Malliavin calculus, crucially relies on the well-posedness theory in the mild …
We prove absolute continuity of Gaussian measures associated to complex Brownian bridges under certain gauge transformations. As an application we prove that the invariant measure for the periodic derivative nonlinear Schrödinger equation obtained by Nahmod, Oh, Rey-Bellet and Staffilani in [20], and with respect to which they proved almost surely global well-posedness, coincides with the weighted Wiener measure constructed by Thomann and Tzvetkov [24]. Thus, in particular we prove the invariance of the measure constructed in [24].
The paper is concerned with a class of two-sided stochastic processes of the form $X=W+A$. Here $W$ is a two-sided Brownian motion with random initial data at time zero and $A\equiv A(W)$ is a function of $W$. Elements of the related stochastic calculus are introduced. In particular, the calculus is adjusted to the case when $A$ is a jump process. Absolute continuity of $(X,P_{\sbnu})$ under time shift of trajectories is investigated. For example under various conditions on the initial density with respect to the Lebesgue measure, $m=d\bnu/dx$, and on $A$ with $A_0=0$ we verify % {eqnarray*} \…
We show that the law of the KPZ fixed point starting from arbitrary initial condition is absolutely continuous with respect to the law of Brownian motion $B$ on every compact interval. In particular, the Airy$_1$ process is absolutely continuous with respect to $B$ on any compact interval.
We prove a dichotomy between absolute continuity and singularity of the Ginibre point process $\mathsf{G}$ and its reduced Palm measures $\{\mathsf{G}_{\mathbf{x}}, \mathbf{x} \in \mathbb{C}^{\ell}, \ell = 0,1,2\dots\}$, namely, reduced Palm measures $\G_{\mathbf{x}}$ and $\G_{\mathbf{y}}$ for $\mathbf{x} \in \mathbb{C}^{\ell}$ and $\mathbf{y} \in \mathbb{C}^{n}$ are mutually absolutely continuous if and only if $\ell = n$; they are singular each other if and only if $\ell \not= n$. Furthermore, we give an explicit expression of the Radon-Nikodym density $d\G_{\mathbf{x}}/d \G_{\mathbf{y}}$ fo…
We consider the stochastic continuity equation associated to an Itô diffusion with irregular drift and diffusion coefficients. We give regularity conditions under which weak solutions are renormalized in the sense of DiPerna/Lions, and prove well-posedness in $L^p$. As an application, we give a new proof of renormalizability (hence uniqueness) of weak solutions to the stochastic continuity equation when the diffusion matrix is constant and the drift only belongs to $L^q_tL^p$, where $\frac{2}{q} + \frac{n}{p} <1$, without resorting to the regularity of the stochastic flow or a duality method.
We introduce an elementary method for proving the absolute continuity of the time marginals of one-dimensional processes. It is based on a comparison between the Fourier transform of such time marginals with those of the one-step Euler approximation of the underlying process. We obtain some absolute continuity results for stochastic differential equations with Hölder continuous coefficients. Furthermore, we allow such coefficients to be random and to depend on the whole path of the solution. We also show how it can be extended to some stochastic partial differential equations and to some Lévy-…
In this work we prove that for any dimension $d\geq 1$ and any $γ\in (0,1)$ super-Brownian motion corresponding to the log-Laplace equation \begin{equation*} \begin{split} \frac{\partial v(t,x)}{\partial t } & = \frac{1}{2}\bigtriangleup v(t,x) + v^γ(t,x) ,\: (t,x) \in \mathbb{R}_+\times \mathbb{R}^d,\\ v(0,x)&= f(x) \end{split} \end{equation*} is absolutely continuous with respect to the Lebesgue measure at any fixed time $t>0$. Our proof is based on properties of solutions of the \LL\ equation. We also prove that when initial datum $v(0,\cdot)$ is a finite, non-zero measure, then the \LL\ eq…
The existence and uniqueness of mild solutions are proved for a class of degenerate stochastic differential equations on Hilbert spaces where the drift is Dini continuous in the component with noise and Hölder continuous of order larger than $\ff 2 3$ in the other component. In the finite-dimensional case the Dini continuity is further weakened. The main results are applied to solve second order stochastic systems driven by space-time white noises.
Strong convergence results on tamed Euler schemes, which approximate stochastic differential equations with superlinearly growing drift coefficients that are locally one-sided Lipschitz continuous, are presented in this article. The diffusion coefficients are assumed to be locally Lipschitz continuous and have at most linear growth. Furthermore, the classical rate of convergence, i.e. one--half, for such schemes is recovered when the local Lipschitz continuity assumptions are replaced by global and, in addition, it is assumed that the drift coefficients satisfy polynomial Lipschitz continuity.
We study two equivalent characterizations of the strong Feller property for a Markov process and of the associated sub-Markovian semigroup. One is described in terms of locally uniform absolute continuity, whereas the other uses local Orlicz-ultracontractivity. These criteria generalize many existing results on strong Feller continuity and seem to be more natural for Feller processes. By establishing the estimates of the first exit time from balls, we also investigate the continuity of harmonic functions for Feller processes which enjoy the strong Feller property.
