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Let $$(P_t)$$ ( P t ) be the transition semigroup of the Markov family $$(X^x(t))$$ ( X x ( t ) ) defined by SDE $$\begin{aligned} \mathrm{d}X= b(X)\mathrm{d}t + \mathrm{d}Z, \qquad X(0)=x, \end{aligned}$$ d X = b ( X ) d t + d Z , X ( 0 ) = x , where $$Z=\left( Z_1, \ldots , Z_d\right) ^*$$ Z = Z 1 , … , Z d ∗ is a system of independent real-valued Lévy processes. Using the Malliavin calculus we establish the following gradient formula $$\begin{aligned} \nabla P_tf(x)= {\mathbb {E}}\, f\left( X^x(t)\right) Y(t,x), \qquad f\in B_b({\mathbb {R}}^d), \end{aligned}$$ ∇ P t f ( x ) = E f X x ( t ) Y ( t , x ) , f ∈ B b ( R d ) , where the random field Y does not depend on f. Moreover, in the important cylindrical $$\alpha $$ α -stable case $$\alpha \in (0,2)$$ α ∈ ( 0 , 2 ) , where $$Z_1, \ldots , Z_d$$ Z 1 , … , Z d are $$\alpha $$ α -stable processes, we are able to prove sharp $$L^1$$ L 1 -estimates for Y(t, x). Uniform estimates on $$\nabla P_tf(x)$$ ∇ P t f ( x ) are also given.

In this paper, we study a first extension of the theory of mild solutions for Hamilton–Jacobi–Bellman (HJB) equations in Hilbert spaces to the case where the domain is not the whole space. More precisely, we consider a half-space as domain, and a semilinear HJB equation. Our main goal is to establish the existence and the uniqueness of solutions to such HJB equations, which are continuously differentiable in the space variable. We also provide an application of our results to an exit-time optimal control problem, and we show that the corresponding value function is the unique solution to a semilinear HJB equation, possessing sufficient regularity to express the optimal control in feedback form. Finally, we give an illustrative example.

We consider singular SDEs like \begin{equation} \label{ss} dX_t = b(t, X_t) dt + A X_t dt + \sigma(t) d{L}_t , \;\; t \in [0,T], \;\; X_0 =x \in {\mathbb R}^n, \end{equation} where $A$ is a real $n \times n $ matrix, i.e., $A \in {\mathbb R}^n \otimes {\mathbb R}^n$, $b$ is bounded and Holder continuous, $\sigma : [0,\infty) \to {\mathbb R}^n \otimes {\mathbb R}^d $ is a locally bounded function and $L= ({L}_t)$ is an ${\mathbb R}^d$-valued Levy process, $1 \le d \le n$. We show that strong existence and uniqueness together with $L^p$-Lipschitz dependence on the initial condition $x $ imply Davie's uniqueness or path by path uniqueness. This extends a result of [E. Priola, AIHP, 2018] proved when $n=d$, $A=0$ and $\sigma(t) \equiv I $. We apply the result to some singular degenerate SDEs associated to the kinetic transport operator $ \frac{1}{2} \triangle_v f + $ ${v \cdot \partial_{x}f} $ $+F(x,v)\cdot \partial_{v}f $ when $n =2d $ and $L$ is an ${\mathbb R}^d$-valued Wiener process. For such equations strong existence and uniqueness are known under Holder type conditions on $b$. We show that in addition also Davie's uniqueness holds.

We prove strong well-posedness for a class of stochastic evolution equations in Hilbert spaces H when the drift term is Holder continuous. This class includes examples of semilinear stochastic damped wave equations which describe elastic systems with structural damping (for such equations even existence of solutions in the linear case is a delicate issue) and semilinear stochastic 3D heat equations. In the deterministic case, there are examples of non-uniqueness in our framework. Strong (or pathwise) uniqueness is restored by means of a suitable additive Wiener noise. The proof of uniqueness relies on the study of related systems of infinite dimensional forward-backward SDEs (FBSDEs). This is a different approach with respect to the well-known method based on the Ito formula and the associated Kolmogorov equation (the so-called Zvonkin transformation or Ito-Tanaka trick). We deal with approximating FBSDEs in which the linear part generates a group of bounded linear operators in H; such approximations depend on the type of SPDEs we are considering. We also prove Lipschitz dependence of solutions from their initial conditions.

We show uniqueness in law for the critical SPDE $$ dX_t = AX_t dt + (-A)^{1/2}F(X(t))dt + dW_t,\;\; X_0 =x \in H, $$ where $A$ $ : dom(A) \subset H \to H$ is a negative definite self-adjoint operator on a separable Hilbert space $H$ having $A^{-1}$ of trace class and $W$ is a cylindrical Wiener process on $H$. Here $F: H \to H $ can be continuous with at most linear growth (some functions $F$ which grow more than linearly can also be considered). This leads to new uniqueness results for generalized stochastic Burgers' equations and for three-dimensional stochastic Cahn-Hilliard type equations which have interesting applications. To get weak uniqueness we also establish a new optimal regularity result for the Kolmogorov equation $ \lambda u - Lu = f$ on $H$, where $\lambda >0$, $ f: H \to {\mathbb R}$ is Borel and bounded and $L$ is the Ornstein-Uhlenbeck operator related to the SPDE when $F=0$. In particular we show that the first derivative $Du : H \to H$ verifies $Du(x) \in \text{dom}((-A)^{1/2})$, for any $x \in H,$ and moreover $$ \sup_{x \in H} |(-A)^{1/2}Du (x)|_H = \| (-A)^{1/2}Du \|_{0} \le C \, \| f\|_{0}. $$, Comment: to appear in AOP

Let $z = (x,y) \in {\mathbb R}^d \times {\mathbb R}^{N-d}$, with $1 \le d < N$. We prove a priori estimates of the following type : $$ \|\Delta_{x}^{\frac \alpha 2} v \|_{L^p({\mathbb R}^N)} \le c_p \Big \| L_{x } v + \sum_{i,j=1}^{N}a_{ij}z_{i}\partial_{z_{j}} v \Big \|_{L^p({\mathbb R}^N)}, \;\; 1

We establish weak well-posedness for critical symmetric stable driven SDEs in R d with additive noise Z, d $\ge$ 1. Namely, we study the case where the stable index of the driving process Z is $\alpha$ = 1 which exactly corresponds to the order of the drift term having the coefficient b which is continuous and bounded. In particular, we cover the cylindrical case when Zt = (Z 1 t ,. .. , Z d t) and Z 1 ,. .. , Z d are independent one dimensional Cauchy processes. Our approach relies on L p-estimates for stable operators and uses perturbative arguments. 1. Statement of the problem and main results We are interested in proving well-posedness for the martingale problem associated with the following SDE: (1.1) X t = x + t 0 b(X s)ds + Z t , where (Z s) s$\ge$0 stands for a symmetric d-dimensional stable process of order $\alpha$ = 1 defined on some filtered probability space ($\Omega$, F, (F t) t$\ge$0 , P) (cf. [2] and the references therein) under the sole assumptions of continuity and boundedness on the vector valued coefficient b: (C) The drift b : R d $\rightarrow$ R d is continuous and bounded. 1 Above, the generator L of Z writes: L$\Phi$(x) = p.v. R d \{0} [$\Phi$(x + z) -- $\Phi$(x)]$\nu$(dz), x $\in$ R d , $\Phi$ $\in$ C 2 b (R d), $\nu$(dz) = d$\rho$ $\rho$ 2$\mu$ (d$\theta$), z = $\rho$$\theta$, ($\rho$, $\theta$) $\in$ R * + x S d--1. (1.2) (here $\times$, $\times$ (or $\times$) and | $\times$ | denote respectively the inner product and the norm in R d). In the above equation, $\nu$ is the L{\'e}vy intensity measure of Z, S d--1 is the unit sphere of R d and$\mu$ is a spherical measure on S d--1. It is well know, see e.g. [20] that the L{\'e}vy exponent $\Phi$ of Z writes as: (1.3) $\Phi$($\lambda$) = E[exp(i $\lambda$, Z 1)] = exp -- S d--1 | $\lambda$, $\theta$ |$\mu$(d$\theta$) , $\lambda$ $\in$ R d , where $\mu$ = c 1$\mu$ , for a positive constant c 1 , is the so-called spectral measure of Z. We will assume some non-degeneracy conditions on $\mu$. Namely we introduce assumption (ND) There exists $\kappa$ $\ge$ 1 s.t. (1.4) $\forall$$\lambda$ $\in$ R d , $\kappa$ --1 |$\lambda$| $\le$ S d--1 | $\lambda$, $\theta$ |$\mu$(d$\theta$) $\le$ $\kappa$|$\lambda$|. 1 The boundedness of b is here assumed for technical simplicity. Our methodology could apply, up to suitable localization arguments, to a drift b having linear growth.

We prove, with a purely analytic technique, a one-side Liouville theorem for a class of Ornstein--Uhlenbeck operators ${\mathcal L_0}$ in $\mathbb{R}^N$, as a consequence of a Liouville theorem at “$t=- \infty$” for the corresponding Kolmogorov operators ${\mathcal L_0} - \partial_t$ in $\mathbb{R}^{N+1}$. In turn, this last result is proved as a corollary of a global Harnack inequality for non-negative solutions to $({\mathcal L_0} - \partial_t) u = 0$ which seems to have an independent interest in its own right. We stress that our Liouville theorem for ${\mathcal L_0}$ cannot be obtained by a probabilistic approach based on recurrence if $N>2$. We provide a self-contained proof of a Liouville theorem involving recurrent Ornstein--Uhlenbeck stochastic processes in the Appendix.

We consider a non-local operator L α which is the sum of a fractional Laplacian △ α / 2 , α ∈ ( 0 , 1 ) , plus a first order term which is measurable in the time variable and locally β-Holder continuous in the space variables. Importantly, the fractional Laplacian Δ α / 2 does not dominate the first order term. We show that global parabolic Schauder estimates hold even in this case under the natural condition α + β > 1 . Thus, the constant appearing in the Schauder estimates is in fact independent of the L ∞ -norm of the first order term. In our approach we do not use the so-called extension property and we can replace △ α / 2 with other operators of α-stable type which are somehow close, including the relativistic α-stable operator. Moreover, when α ∈ ( 1 / 2 , 1 ) , we can prove Schauder estimates for more general α-stable type operators like the singular cylindrical one, i.e., when △ α / 2 is replaced by a sum of one dimensional fractional Laplacians ∑ k = 1 d ( ∂ x k x k 2 ) α / 2 .

A result of A.M. Davie [Int. Math. Res. Not. 2007] states that a multidimensional stochastic equation $dX_t = b(t, X_t)\,dt + dW_t$, $X_0=x$, driven by a Wiener process $W= (W_t)$ with a coefficient $b$ which is only bounded and measurable has a unique solution for almost all choices of the driving Brownian path. We consider a similar problem when $W$ is replaced by a L\'evy process $L= (L_t)$ and $b$ is $\beta$-H\"older continuous in the space variable, $ \beta \in (0,1)$. We assume that $L_1$ has a finite moment of order $\theta$, for some ${\theta}>0$. Using also a new c\`adl\`ag regularity result for strong solutions, we prove that strong existence and uniqueness for the SDE together with $L^p$-Lipschitz continuity of the strong solution with respect to $x $ imply a Davie's type uniqueness result for almost all choices of the L\'evy paths. We apply this result to a class of SDEs driven by non-degenerate $\alpha$-stable L\'evy processes, $\alpha \in (0,2)$ and $\beta > 1 - \alpha/2$., Comment: To appear in AIHP

We prove gradient estimates for transition Markov semigroups ( P t ) associated to SDEs driven by multiplicative Brownian noise having possibly unbounded C 1 -coefficients, without requiring any monotonicity type condition. In particular, first derivatives of coefficients can grow polynomially and even exponentially. We establish pointwise estimates with weights for D x P t φ of the form t | D x P t φ ( x ) | ≤ c ( 1 + | x | k ) ‖ φ ‖ ∞ , t ∈ ( 0 , 1 ] , φ ∈ C b ( R d ) , x ∈ R d . We use two main tools. First, we consider a Feynman–Kac semigroup with potential V related to the growth of the coefficients and of their derivatives for which we can use a Bismut–Elworthy–Li type formula. Second, we introduce a new regular approximation for the coefficients of the SDE. At the end of the paper we provide an example of SDE with additive noise and drift b having sublinear growth together with its derivative such that uniform estimates for D x P t φ without weights do not hold.

We show among other things how knowing Schauder or Sobolev-space estimates for the one-dimensional heat equation allows one to derive their multidimensional analogs for equations with coefficients depending only on time variable with the same constants as in the case of the one-dimensional heat equation. The method is quite general and is based on using the Poisson stochastic process. It also applies to equations involving non-local operators. It looks like no other method is available at this time and it is a very challenging problem to find a purely analytic approach to proving such results. We only give examples of applications of our results. Their proofs will appear elsewhere.

Starting from a microscopic model for a system of neurons evolving in time which individually follow a stochastic integrate-and-fire type model, we study a mean-field limit of the system. Our model is described by a system of SDEs with discontinuous coefficients for the action potential of each neuron and takes into account the (random) spatial configuration of neurons allowing the interaction to depend on it. In the limit as the number of particles tends to infinity, we obtain a nonlinear Fokker-Planck type PDE in two variables, with derivatives only with respect to one variable and discontinuous coefficients. We also study strong well-posedness of the system of SDEs and prove the existence and uniqueness of a weak measure-valued solution to the PDE, obtained as the limit of the laws of the empirical measures for the system of particles.

We prove uniqueness in law for possibly degenerate SDEs having a linear part in the drift term. Diffusion coefficients corresponding to non-degenerate directions of the noise are assumed to be continuous. When the diffusion part is constant we recover the classical degenerate Ornstein-Uhlenbeck process which only has to satisfy the Hormander hypoellipticity condition. In the proof we also use global L p -estimates for hypoelliptic Ornstein-Uhlenbeck operators recently proved in Bramanti et al. (Math. Z. 266, 789–816 2010) and adapt the localization procedure introduced by Stroock and Varadhan. Appendix contains a quite general localization principle for martingale problems.

International audience; We consider regularity properties of stochastic kinetic equations with multiplicative noise and drift term which belongs to a space of mixed regularity (Lp-regularity in the velocity-variable and Sobolev regularity in the space-variable). We prove that, in contrast with the deterministic case, the SPDE admits a unique weakly differentiable solution which preserves a certain degree of Sobolev regularity of the initial condition without developing discontinuities. To prove the result we also study the related degenerate Kolmogorov equation in Bessel-Sobolev spaces and construct a suitable stochastic flow.

We show among other things how knowing Schauder or Sobolev-space estimates for the one-dimensional heat equation allows one to derive their multidimensional analogs for equations with coefficients depending only on time variable with the {\em same\/} constants as in the case of the one-dimensional heat equation. The method is based on using the Poisson stochastic process. It looks like no other method is available at this time and it is a very challenging problem to find a purely analytic approach to proving such results., 37 pages, more discussion and references added, some errors corrected

We prove that semilinear stochastic abstract wave equations, including wave and plate equations, are well-posed in the strong sense with an α-Holder continuous drift coefficient, if α ∈ ( 2 / 3 , 1 ) . The uniqueness may fail for the corresponding deterministic PDE and well-posedness is restored by adding an external random forcing of white noise type. This shows a kind of regularization by noise for the semilinear wave equation. To prove the result we introduce an approach based on backward stochastic differential equations. We also establish regularizing properties of the transition semigroup associated to the stochastic wave equation by using control theoretic results.

We consider linear systems on a separable Hilbert space $H$, which are null controllable at some time $T_0>0$ under the action of a point or boundary control. Parabolic and hyperbolic control systems usually studied in applications are special cases. To every initial state $ y_0 \in H$ we associate the minimal "energy" needed to transfer $ y_0 $ to $ 0 $ in a time $ T \ge T_0$ ("energy" of a control being the square of its $ L^2 $ norm). We give both necessary and sufficient conditions under which the minimal energy converges to $ 0 $ for $ T\to+\infty $. This extends to boundary control systems the concept of null controllability with vanishing energy introduced by Priola and Zabczyk (Siam J. Control Optim. 42 (2003)) for distributed systems. The proofs in Priola-Zabczyk paper depend on properties of the associated Riccati equation, which are not available in the present, general setting. Here we base our results on new properties of the quadratic regulator problem with stability and the Linear Operator Inequality., In this version we have also added a section on examples and applications of our main results. This version is similar to the one which will be published on "SIAM Journal on Control and Optimization" (SIAM)

We study the Cauchy problem involving non-local Ornstein–Uhlenbeck operators in finite and infinite dimensions. We prove classical solvability without requiring that the Levy measure corresponding to the large jumps part has a first finite moment. Moreover, we determine a core of regular functions which is invariant for the associated transition Markov semigroup. Such a core allows to characterize the marginal laws of the Ornstein–Uhlenbeck stochastic process as unique solutions to Fokker–Planck–Kolmogorov equations for measures.

We prove exponential convergence to the invariant measure, in the total variation norm, for solutions of SDEs driven by α -stable noises in finite and in infinite dimensions. Two approaches are used. The first one is based on Liapunov’s function approach by Harris, and the second on Doeblin’s coupling argument in [8] . Irreducibility and uniform strong Feller property play an essential role in both approaches. We concentrate on two classes of Markov processes: solutions of finite dimensional equations, introduced in [27] , with Holder continuous drift and a general, non-degenerate, symmetric α -stable noise, and infinite dimensional parabolic systems, introduced in [29] , with Lipschitz drift and cylindrical α -stable noise. We show that if the nonlinearity is bounded, then the processes are exponential mixing. This improves, in particular, an earlier result established in [28] , with a different method.

We consider a SDE with a smooth multiplicative non-degenerate noise and a possibly unbounded Holder continuous drift term. We prove the existence of a global flow of diffeomorphisms by means of a special transformation of the drift of Ito–Tanaka type. The proof requires non-standard elliptic estimates in Holder spaces. As an application of the stochastic flow, we obtain a Bismut–Elworthy–Li type formula for the first derivatives of the associated diffusion semigroup.

This Note is concerned with the properties of solutions to a linear evolution equation perturbed by a cylindrical Levy process. It turns out that solutions, under rather weak requirements, do not have a cadlag modification. Some natural open questions are also stated.

We consider non-degenerate SDEs with a $\beta$-Holder continuous and bounded drift term and driven by a Levy noise $L$ which is of $\alpha$-stable type. If $\alpha \in [1,2)$ and $\beta \in (1 - \frac{\alpha}{2},1) $ we show pathwise uniqueness and existence of a stochastic flow. We follow the approach of [Priola, Osaka J. Math. 2012] improving the assumptions on the noise $L$. In our previous paper $L$ was assumed to be non-degenerate, $\alpha$-stable and symmetric. Here we can also recover relativistic and truncated stable processes and some classes of temperated stable processes.

We prove L p -parabolic a-priori estimates for \(\partial _{t}u +\sum _{ i,j=1}^{d}c_{ij}(t)\partial _{x_{i}x_{j}}^{2}u = f\) on \(\mathbb{R}^{d+1}\) when the coefficients c ij are locally bounded functions on \(\mathbb{R}\). We slightly generalize the usual parabolicity assumption and show that still L p -estimates hold for the second spatial derivatives of u. We also investigate the dependence of the constant appearing in such estimates from the parabolicity constant. The proof requires the use of the stochastic integral when p is different from 2. Finally we extend our estimates to parabolic equations involving non-degenerate Ornstein-Uhlenbeck type operators.

We consider an Ornstein-Uhlenbeck process with values in R^n driven by a L\'evy process (Z_t) taking values in R^d with d possibly smaller than n. The L\'evy noise can have a degenerate or even vanishing Gaussian component. Under a controllability condition and an assumption on the L\'evy measure of (Z_t), we prove that the law of the Ornstein-Uhlenbeck process at any time t>0 has a density on R^n. Moreover, when the L\'evy process is of $\alpha$-stable type, $\alpha \in (0,2)$, we show that such density is a $C^{\infty}$-function.

Uniform gradient estimates are derived for diffusion semigroups, possibly with potential, generated by second order elliptic operators having irregular and unbounded coefficients. We first consider the R d -case, by using the coupling method. Due to the singularity of the coefficients, the coupling process we construct is not strongly Markovian, so that additional difficulties arise in the study. Then, more generally, we treat the case of a possibly unbounded smooth domain of R d with Dirichlet boundary conditions. We stress that the resulting estimates are new even in the R d -case and that the coefficients can be Holder continuous. Our results also imply a new Liouville theorem for space–time bounded harmonic functions with respect to the underlying diffusion semigroup.

The paper characterizes some classes of pseudo-differential operators for which there are (or there are not) non-constant bounded harmonic functions. Non-local perturbations of Ornstein–Uhlenbeck operators and operators with dissipative coefficients are considered. The methods used are probabilistic and based on the concept of absorption function and on a new extension of the Bismut–Elworthy–Li formula. The probabilistic interpretation of the Liouville theorem by means of absorption functions for general Markov processes is given as well.

We consider autonomous parabolic Dirichlet problems in a regular unbounded open set Ω⊂RN involving second-order operator A with (possibly) unbounded coefficients. We determine new conditions on the coefficients of A yielding global gradient estimates for the bounded classical solution.

We consider stochastic evolution equations in Hilbert spaces with merely measurable and locally bounded drift term $B$ and cylindrical Wiener noise. We prove pathwise (hence strong) uniqueness in the class of global solutions. This paper extends our previous paper (Da Prato, Flandoli, Priola and M. Rockner, Annals of Prob., published online in 2012) which generalized Veretennikov's fundamental result to infinite dimensions assuming boundedness of the drift term. As in our previous paper pathwise uniqueness holds for a large class, but not for every initial condition. We also include an application of our result to prove existence of strong solutions when the drift $B$ is only measurable, locally bounded and grows more than linearly., The paper will be published in Journal of Theoretical Probability. arXiv admin note: text overlap with arXiv:1109.0363

Linear, null controllable systems, for which an arbitrary initial state can be transferred to the origin with arbitrarily small energy, are characterized. Theorems are stated in terms of an associated algebraic Riccati equation and in terms of the spectrum of the linear part of the system. The results so obtained allow us to determine Ornstein--Uhlenbeck operators for which the Liouville theorem about bounded harmonic functions holds.

We consider an elliptic Dirichlet problem which involves Ornstein–Uhlenbeck operators of special form in a half space of R n . We obtain necessary and sufficient conditions under which global Schauder estimates in spaces of Holder continuous and bounded functions hold. For this purpose we use analytical tools, in particular semigroups and interpolation theory. Moreover we extend a theorem on the analiticity of subordinated semigroups (see Carasso and Kato; Trans. Amer. Math. Soc.327 (1990, 867–877)) to a class of Markov type semigroups. We also provide explicit formulas for the Poisson kernels.

We prove pathwise (hence strong) uniqueness of solutions to stochastic evolution equations in Hilbert spaces with merely measurable bounded drift and cylindrical Wiener noise, thus generalizing Veretennikov's fundamental result on $\mathbb{R}^d$ to infinite dimensions. Because Sobolev regularity results implying continuity or smoothness of functions do not hold on infinite-dimensional spaces, we employ methods and results developed in the study of Malliavin-Sobolev spaces in infinite dimensions. The price we pay is that we can prove uniqueness for a large class, but not for every initial distribution. Such restriction, however, is common in infinite dimensions., Published in at http://dx.doi.org/10.1214/12-AOP763 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We consider a class of degenerate Ornstein-Uhlenbeck operators in $\mathbb{R}^{N}$, of the kind \[\mathcal{A}\equiv\sum_{i,j=1}^{p_{0}}a_{ij}\left( x\right) \partial _{x_{i}x_{j}}^{2}+\sum_{i,j=1}^{N}b_{ij}x_{i}\partial_{x_{j}}\] where $\left( a_{ij}\right) $ is symmetric uniformly positive definite on $\mathbb{R}^{p_{0}}$ ($p_{0}\leq N$), with uniformly continuous and bounded entries, and $\left( b_{ij}\right) $ is a constant matrix such that the frozen operator $\mathcal{A}_{x_{0}}$ corresponding to $a_{ij}\left( x_{0}\right) $ is hypoelliptic. For this class of operators we prove global $L^{p}$ estimates ($1

We show how gradient estimates for transition semigroups can be used to establish exponential mixing for a class of Markov processes in infinite dimensions. We concentrate on semilinear systems driven by cylindrical α-stable noises introduced in [18], α ∈ (0,2). We first prove that if the nonlinearity is bounded, then the system is ergodic and strong mixing. Then we show that the system is exponentially mixing provided that the nonlinearity, or its Lipschitz constant, are sufficiently small.

The motion of a finite number of point vortices on a two-dimensional periodic domain is considered. In the deterministic case it is known to be well posed only for almost every initial configuration. Coalescence of vortices may occur for certain initial conditions. We prove that when a generic stochastic perturbation compatible with the Eulerian description is introduced, the point vortex motion becomes well posed for every initial configuration, in particular coalescence disappears., 23 pages

We introduce a new condition on elliptic operators $L= {1/2}\triangle + b \cdot \nabla $ which ensures the validity of the Liouville property for bounded solutions to $Lu=0$ on $\R^d$. Such condition is sharp when $d=1$. We extend our Liouville theorem to more general second order operators in non-divergence form assuming a Cordes type condition.

We consider a class of degenerate Ornstein–Uhlenbeck operators in $${\mathbb{R}^{N}}$$ , of the kind $$\mathcal{A}\equiv\sum_{i, j=1}^{p_{0}}a_{ij}\partial_{x_{i}x_{j}}^{2} + \sum_{i, j=1}^{N}b_{ij}x_{i}\partial_{x_{j}}$$ where (a ij ), (b ij ) are constant matrices, (a ij ) is symmetric positive definite on $${\mathbb{R} ^{p_{0}}}$$ (p 0 ≤ N), and (b ij ) is such that $${\mathcal{A}}$$ is hypoelliptic. For this class of operators we prove global L p estimates (1

We consider a second-order parabolic equation in $\bR^{d+1}$ with possibly unbounded lower order coefficients. All coefficients are assumed to be only measurable in the time variable and locally H\"older continuous in the space variables. We show that global Schauder estimates hold even in this case. The proof introduces a new localization procedure. Our results show that the constant appearing in the classical Schauder estimates is in fact independent of the $L_{\infty}$-norms of the lower order coefficients. We also give a proof of uniqueness which is of independent interest even in the case of bounded coefficients., Comment: 25 pages

We study existence and uniqueness of solutions for second order ordinary stochastic differential equations with Dirichlet boundary conditions on a given interval. In the first part of the paper we provide sufficient conditions to ensure pathwise uniqueness, extending some known results. In the second part we show sufficient conditions to have the weaker concept of uniqueness in law and provide a significant example. Such conditions involve a linearized equation and are of different type with respect to the ones which are usually imposed to study pathwise uniqueness. This seems to be the first paper which deals with uniqueness in law for (anticipating) stochastic boundary value problems. We mainly use functional analytic tools and some concepts of Malliavin Calculus.

We consider the linear transport equation with a globally Holder continuous and bounded vector field. While this deterministic PDE may not be well-posed, we prove that a multiplicative stochastic perturbation of Brownian type is enough to render the equation well-posed. This seems to be the first explicit example of partial differential equation that become well-posed under the influece of noise. The key tool is a differentiable stochastic flow constructed and analysed by means of a special transformation of the drift of Ito-Tanaka type., Comment: Addition of new parts

We consider a class of semilinear stochastic evolution equations driven by an additive cylindrical stable noise.We investigate structural properties of the solutions like Markov, irreducibility, stochastic continuity, Feller and strong Feller properties, and study integrability of trajectories. The obtained results can be applied to semilinear stochastic heat equations with Dirichlet boundary conditions and bounded and Lipschitz nonlinearities., Comment: This version is almost identical with the one published in Probab. Theory Related Fields. We have also corrected some constants appearing in Theorem 4.16 and Hypothesis 5.6