arXiv subjects
Publications and source records attributed to Luciano Tubaro.
In this paper, we provide an alternative proof of a result due to Gross (1960s), extending it to the Fréchet space setting. Our argument adapts the proof technique developed by Bogachev in the Banach case, as presented in the monograph Gaussian Measures, to the more general framework of separable Fréchet spaces, simplifying several steps along the way.
The "Riemann-type integrals" of the title are not a single new integral, but the family of classical constructions -- Riemann, Riemann-Stieltjes, Young-Kondurar, complex-analytic, and Itô -- unified by one elementary method. We give a simple argument computing the Riemann integral of a polynomial directly, establishing the fundamental theorem of calculus for polynomials and then continuous functions. The same idea extends to the real line, the complex plane, the multidimensional case, the Riemann-Stieltjes integral, and the Itô integral of a polynomial of a Wiener process (more generally, a continuous semimartingale), recovering the classical Itô formula. The contribution is methodological: a single algebraic telescoping identity, paired with an estimate on a quadratic remainder term, drives every case. What changes is only the remainder's fate: it vanishes identically for polynomials on $\mathbb{R}$; it vanishes in the limit under a quadratic-variation condition $Q(π)\to0$ on $\mathbb{C}$ and for Hölder paths; and for a continuous semimartingale it survives as the quadratic variation producing the Itô correction -- except for a planar Brownian motion, or more generally a conformal martingale, where an algebraic cancellation inside the complex square removes it again. We regard this last point -- that the Itô correction and its cancellation in the conformal case are two instances of the same quadratic remainder, not unrelated facts -- as the paper's most distinctive observation. The manuscript has three parts: Part I develops the core method on $\mathbb{R}$ and $\mathbb{C}$; Part II extends it deterministically to the multidimensional, Riemann-Stieltjes, and Hölder (Young-Kondurar) cases; Part III extends it to the stochastic setting.
We consider a well posed SPDE$\colon dZ=(AZ+b(Z)) dt+dW(t),\,Z_0=x, $ on a separable Hilbert space $H$, where $A\colon H\to H$ is self-adjoint, negative and such that $A^{-1+β}$ is of trace class for some $β>0$, $b\colon H\to H$ is Lipschitz continuous and $W$ is a cylindrical Wiener process on $H$. We denote by $W_A(t)=\int_0^te^{(t-s)A}\,dW(s),\,t\in[0,T],$ the stochastic convolution. We prove, with the help of a formula for nonlinear transformations of Gaussian integrals due to R. Ramer, the following identity $$(P\circ Z_x^{-1})(Φ) =\int_XΦ(h+e^{\cdot A}x)\, \exp\left\{ -\tfrac12|γ_x(h)|^2_{ H_{Q_T}} + I(γ_x)(h)\right\} N_{Q_T}(dh), $$ where $ N_{Q_T}$ is the law of $W_A$ in $C([0,T],H)$, $ H_{Q_T}$ its Cameron--Martin space, $$ [γ_x(k)](t)=\int_0^t e^{(t-s)A}b(k(s)+e^{sA}x) ds,\quad t\in[0,T], \; k \in C([0,T],H) $$ and $I(γ_x) $ is the Itô integral of $γ_x$. Some applications are discussed; in particular, when $b$ is dissipative we provide an explicit formula for the law of the stationary process and the invariant measure $ν$ of the Markov semigroup $(P_t)$. Some concluding remarks are devoted to a similar problem with colored noise.
These Lecture Notes are a brief introduction to the Malliavin calculus. In particular, different notions of Malliavin derivative found in the literature are considered and compared.
We consider a sequence of fractional Ornstein-Uhlenbeck processes, that are defined as solutions of a family of stochastic Volterra equations with kernel given by the Riesz derivative kernel, and leading coefficients given by a sequence of independent Gamma random variables. We construct a new process by taking the empirical mean of this sequence. In our framework, the processes involved are not Markovian, hence the analysis of their asymptotic behaviour requires some ad hoc construction. In our main result, we prove the almost sure convergence in the space of trajectories of the empirical means to a given Gaussian process, which we characterize completely.
We prove the existence and a new representation formula for the gradient of the semigroup associated to an Ornstein-Uhlenbeck in a bounded convex domain in d dimensions.
On the infinite dimensional space $E$ of continuous paths from $[0,1]$ to $\mathbb R^n$, $n \ge 3$, endowed with the Wiener measure $μ$, we construct a surface measure defined on level sets of the $L^2$-norm of $n$-dimensional processes that are solutions to a class of stochastic gradient system-type equations, and provide an integration by parts formula involving this surface measure. We follow the approach to surface measures in Gaussian spaces proposed via techniques of Malliavin calculus by Airault and Malliavin in 1988.
We prove that the law of the minimum $m:=\min_{t\in[0,1]} ξ(t)$ of the solution $ξ$ to a one-dimensional ODE with good nonlinearity has continuous density with respect to the Lebesgue measure. As a byproduct of the procedure, we show that the sets $ \{ x\in C([0,1]):\; \min x > r\}$ have finite perimeter with respect to the law $ν$ of the solution $ξ(\cdot)$ in $L^2(0,1)$.
One proves that the $n$-D stochastic controlled equation $dX+AXdt=σ(X)dW+Bu\,dt$, where $σ\in\mbox{Lip}((\R^n,Ł(\R^d,\R^n))$ and the pair $A\inŁ(\R^n)$, $B\inŁ(\R^m,\R^n)$ satisfies the Kalman rank condition, is exactly controllable in each $y\in\R^n$, $σ(y)=0$ on each finite interval $(0,T)$. An application to approximate controllability to stochastic heat equation is given.
We construct surface measures in a Hilbert space endowed with a probability measure $ν$. The theory fits for invariant measures of some stochastic partial differential equations such as Burgers and reaction--diffusion equations. Other examples are weighted Gaussian measures and special product measures $ν$ of non Gaussian measures; in this case we exhibit a Markov process having $ν$ as invariant measure. In any case we prove integration by parts formulae on sublevel sets of good functions (including spheres and hyperplanes) that involve surface integrals.
In this paper, we consider convex sets $K_r = \{g \ge r\}$ in an infinite dimensional Hilbert space, where $g$ is suitably related to a reference Gaussian measure $μ$ in $H$. We first show how to define a surface measure on the level sets $\{g = r\}$ that is related to $μ$. This allows to introduce an integration-by-parts formula in $H$. This formula can be applied in several important constructions, as for instance the case where $μ$ is the law of a (Gaussian) stochastic process and $H$ is the space of its trajectories
This work is concerned with existence of weak solutions to discon- tinuous stochastic differential equations driven by multiplicative Gaus- sian noise and sliding mode control dynamics generated by stochastic differential equations with variable structure, that is with jump nonlin- earity. The treatment covers the finite dimensional stochastic systems and the stochastic diffusion equation with multiplicative noise.
We construct surface measures associated to Gaussian measures in separable Banach spaces, and we prove several properties including an integration by parts formula.
The solution $X_n$ to a nonlinear stochastic differential equation of the form $dX_n(t)+A_n(t)X_n(t)\,dt-\tfrac12\sum_{j=1}^N(B_j^n(t))^2X_n(t)\,dt=\sum_{j=1}^N B_j^n(t)X_n(t)dβ_j^n(t)+f_n(t)\,dt$, $X_n(0)=x$, where $β_j^n$ is a regular approximation of a Brownian motion $β_j$, $B_j^n(t)$ is a family of linear continuous operators from $V$ to $H$ strongly convergent to $B_j(t)$, $A_n(t)\to A(t)$, $\{A_n(t)\}$ is a family of maximal monotone nonlinear operators of subgradient type from $V$ to $V'$, is convergent to the solution to the stochastic differential equation $dX(t)+A(t)X(t)\,dt-\frac12\sum_{j=1}^NB_j^2(t)X(t)\,dt=\sum_{j=1}^NB_j(t)X(t)\,dβ_j(t)+f(t) \,dt$, $X(0)=x$. Here $V\subset H\cong H'\subset V'$ where $V$ is a reflexive Banach space with dual $V'$ and $H$ is a Hilbert space. These results can be reformulated in terms of Stratonovich stochastic equation $dY(t)+A(t)Y(t)\,dt=\sum_{j=1}^NB_j(t)Y(t)\circ dβ_j(t)+f(t)\,dt$.
We consider the stochastic reflection problem associated with a self-adjoint operator $A$ and a cylindrical Wiener process on a convex set $K$ with nonempty interior and regular boundary $Σ$ in a Hilbert space $H$. We prove the existence and uniqueness of a smooth solution for the corresponding elliptic infinite-dimensional Kolmogorov equation with Neumann boundary condition on $Σ$.
We prove, using coupling arguments, exponential convergence to equilibrium for reaction--diffusion and Burgers equations driven by space-time white noise. We use a coupling by reflection.