A Skew Stochastic Heat Equation

We consider a stochastic heat equation driven by a space-time white noise and with a singular drift, where a local-time in space appears. The process we study has an explicit invariant measure of Gibbs type, with a non-convex potential. We obtain existence of a Markov solution, which is associated with an explicit Dirichlet form. Moreover, we study approximations of the stationary solution by means of a regularization of the singular drift or by a finite-dimensional projection.     

Sharp Large Deviation Estimates for the Stochastic Heat Equation

We consider the family {X , 0} of solution to the heat equation on [0,T]×[0,1] perturbed by a small space-time white noise, that is _t X = X +b({X })+({X }) . Then, for a large class of Borelian subsets of the continuous functions on [0,T]×[0,1], we get an asymptotic expansion of P({X }A) as 0. This kind of expansion has been handled for several stochastic systems, ranging from Wiener integrals to diffusion processes.     

On the Positivity of the Stochastic Heat Equation

We study the positivity preserving properties of the heat equation with a white noise potential and random initial condition. Moreover, we find a generalized Feynman--Kac formula for the solution of the problem using methods from the white noise analysis. The initial condition can anticipate the driving white noise. We show that the solution is positive, when the random initial condition is positive. For the case of a time-dependent white noise potential, we give a special representation of the solution together with regularity results.     

Stochastic Heat Equation with Multiplicative Fractional-Colored Noise

We consider the stochastic heat equation with multiplicative noise \(u_{t}=\frac{1}{2}\Delta u+u\dot{W}\) in ?₊×? ^d , whose solution is interpreted in the mild sense. The noise \(\dot{W}\) is fractional in time (with Hurst index H≥1/2), and colored in space (with spatial covariance kernel f). When H>1/2, the equation generalizes the Itô-sense equation for H=1/2. We prove that if f is the Riesz kernel of order α, or the Bessel kernel of order α<d, then the sufficient condition for the existence of the solution is d≤2+α (if H>1/2), respectively d<2+α (if H=1/2), whereas if f is the heat kernel or the Poisson kernel, then the equation has a solution for any d. We give a representation of the kth order moment of the solution in terms of an exponential moment of the “convoluted weighted” intersection local time of k independent d-dimensional Brownian motions.     

Large Time Asymptotic Properties of the Stochastic Heat Equation

Journal of Theoretical Probability - In this article, we study the large time asymptotic behavior of the stochastic heat equation.     

On Transformations in the Feynman-Kac Formula and Quantummechanical N-Body Systems

Coordinate transformations in the Feynman-Kac formula are considered. The contractive semigroup with transformed coordinates is expressed by an integral with a correspondingly transformed diffusion measure. This representation is used for N-body systems transformed to clustered Jacobi coordinates implying a decomposition of the Feynman-Kac formula.     

Perturbation of Transition Functions and a Feynman-Kac Formula for the Incorporation of Mortality

Markov transition kernels are perturbed by output kernels with a special emphasis on building mortality into structured population models. A Feynman-Kac formula is derived which illustrates the interplay of mortality with a Markov process associated with the unperturbed kernel. partially supported by NSF grants DMS-0314529 and SES-0345945 partially supported by NSF grants DMS-9706787 and DMS-0314529     

Sticky-Reflected Stochastic Heat Equation Driven by Colored Noise

Ukrainian Mathematical Journal - We prove the existence of a sticky-reflected solution to the heat equation on the space interval [0, 1] driven by colored noise. The process can be interpreted as...     

Feynman-Kac Formula for Left Continuous Additive Functionals and Extended Kato Class Measures

The perturbation of the generator of a Borel right process by a signed measure is investigated, using probabilistic and analytic potential theoretical methods. We establish a Feynman-Kac formula associated with measures charging no polar set and belonging to an extended Kato class. A main tool of this approach is the validity of a Khas’minskii Lemma for Stieltjes exponentials of positive left continuous additive functionals.      

Additive Functionals of the Solution to Fractional Stochastic Heat Equation

We give sharp regularity results for the solution to the linear stochastic heat equation with fractional noise in time. We apply these result to prove the local nondeterminism of this process and to study the existence and the joint continuity of its local times.     

Moderate Deviations for Stochastic Heat Equation with Rough Dependence in Space

In this paper, we establish a moderate deviation principle for the stochastic heat equation driven by a Gaussian noise which is white in time and which has     

Weak Convergence of a Numerical Method for a Stochastic Heat Equation

Weak convergence with respect to a space of twice continuously differentiable test functions is established for a discretisation of a heat equation with homogeneous Dirichlet boundary conditions in one dimension, forced by a space-time Brownian motion. The discretisation is based on finite differences in space and time, incorporating a spectral approximation in space to the Brownian motion.     

Asymptotic Distributions for Power Variation of the Solution to a Stochastic Heat Equation

Let u = {u(t, x), t ∈ [0, T ], x ∈ R} be a solution to a stochastic heat equation driven by a space-time white noise. We study that the realized power variation of the process u with respect to the time, properly normalized, has Gaussian asymptotic distributions. In particular, we study the realized power variation of the process u with respect to the time converges weakly to Brownian motion.     

Weak Order for the Discretization of the Stochastic Heat Equation Driven by Impulsive Noise

We study the approximation of the distribution of X _T , where (X _t ) _t?∈?[0,?T] is a Hilbert space valued stochastic process that solves a linear parabolic stochastic partial differential equation driven by an impulsive space time noise, $$ dX_t+AX_t\,dt= Q^{1/2}\,dZ_t,\quad X_0=x_0\in H,\quad t\in [0,T]. $$ Here (Z _t ) _t?∈?[0,?T] is an impulsive cylindrical process and the operator Q describes the spatial covariance structure of the noise; we assume that A ^??α has finite trace for some α?>?0 and that A ^β Q is bounded for some β?∈?(α???1, α]. A discretized solution $(X_h^n)_{n\in\{0,1,\ldots,N\}}$ is defined via the finite element method in space (parameter h?>?0) and a θ-method in time (parameter Δt?=?T/N). For $\varphi \in C^2_b(H;{\mathbb R})$ we show an integral representation for the error $|{\mathbb E}\varphi(X^N_h)-{\mathbb E}\varphi(X_T)|$ and prove that $$ \left|{\mathbb E}\varphi\left(X^N_h\right)-{\mathbb E}\varphi(X_T)\right|=O\left(h^{2\gamma}+\left(\Delta t\right)^{\gamma}\right) $$ where γ?<?1???α?+?β. This is the same order of convergence as in the case of a Gaussian space time noise, which has been obtained in a paper by Debussche and Printems (Math Comput 78:845–863, 2009). Our result also holds for a combination of impulsive and Gaussian space time noise.     

The Roughness and Smoothness of Numerical Solutions to the Stochastic Heat Equation

The stochastic heat equation is the heat equation driven by white noise. We consider its numerical solutions using the finite difference method. Its true solutions are H?lder continuous with parameter $(\frac{1}{2}-\epsilon)$ in the space variable, and $(\frac{1}{4}-\epsilon)$ in the time variable. We show that the numerical solutions share this property in the sense that they have non-trivial limiting quadratic variation in x and quartic variation in t. These variations are discontinuous functionals on the space of continuous functions, so it is not automatic that the limiting values exist, and not surprising that they depend on the exact numerical schemes that are used; it requires a very careful choice of scheme to get the correct limiting values. In particular, part of the folklore of the subject says that a numerical scheme with excessively long time-steps makes the solution much smoother. We make this precise by showing exactly how the length of the time-steps affects the quadratic and quartic variations.     

Stochastic Optimal Control for the Stochastic Heat Equation with Exponentially Growing Coefficients and with Control and Noise on a Subdomain

Abstract

We consider stochastic optimal control problems in Banach spaces, related to nonlinear controlled equations with dissipative non linearities: on the nonlinear term we do not impose any growth condition. The problems are treated via the backward stochastic differential equations approach, that allows also to solve in mild sense Hamilton Jacobi Bellman equations in Banach spaces. We apply the results to controlled stochastic heat equation, in space dimension 1, with control and noise acting on a subdomain.     

The Feynman-Kac Formula with a Lebesgue-Stieltjes Measure and Feynman's Operational Calculus

We investigate what happens if in the Feynman-Kac functional, we perform the time integration with respect to a Borel measure η rather than ordinary Lebesgue measure l. Let u(t) be the operator associated with this functional through path integration. We show that u(t), considered as a function of time t, satisfies a certain Volterra-Stieltjes integral equation. This result establishes a “FeynmanKac formula with Lebesgue-Stieltjes measure η.” One recovers the classical Feynman-Kac formula by letting η = l. We deduce from the integral equation that u(t) satisfies a differential equation associated with the continuous part μ of η when η = μ = l, this differential equation reduces to the heat or the Schrödinger equation in the probabilistic or quantum-mechanical case, respectively. Moreover, we observe a new phenomenon, due to the discrete part v of η: the function u(t) undergoes a discontinuity at every point in the support of v, assumed here to be finite. Further, one obtains an explicit expression for u(t) in terms of operators alternatively associated with μ and v. Our results are new even in the probabilistic or “imaginary time” case and allow us to unify various concepts. The derivation of our integral equation has an interesting combinatorial structure and makes essential use of the “generalized Dyson series”— recently introduced by G. W. Johnson and the author—that “disentangle” the operator u(t). We provide natural physical interpretations of our results in both the diffusion and quantum-mechanical cases. We also suggest further connections with Feynman?s operational calculus for noncommuting operators.