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This paper is devoted to the continuity of the weak solution for tempered fractional general diffusion equations driven by tempered fractional Brownian motion (TFBM). Based on the Feynman-Kac formula (1.2), by using the Itô isometry for the stochastic integral with respect to TFBM, Parseval's identity and some ingenious calculations, we establish the continuities of the solution with respect to Hurst index H and tempering parameter λ of TFBM. [ABSTRACT FROM AUTHOR]

In a recent paper [7] , a hybrid supercomputing algorithm for elliptic equations has been proposed. The idea is that the interfacial nodal solutions solve a linear system, whose coefficients are expectations of functionals of stochastic differential equations confined within patches of about subdomain size. Compared to standard substructuring techniques, such as the Schur complement method for the skeleton, the hybrid approach produces an explicit and sparse shrunken matrix—hence suitable for substructuring again. The ultimate goal is to push strong scalability beyond the state of the art by leveraging the potential for parallelisation of stochastic calculus. Here, we present a major revamping of that framework, based on the insight of embedding the domain in a cover of overlapping circles (in two dimensions). This allows for efficient Fourier interpolation along the interfaces (now circumferences) and—crucially—for the evaluation of most of the interfacial system entries as the solution of small boundary value problems on a circle. This is both extremely efficient (as they can be solved in parallel and by the pseudospectral method) and free of Monte Carlo error. Stochastic numerics are only needed on the relatively few circles intersecting the domain boundary. In sum, the new formulation is significantly faster, simpler, and more accurate while retaining all of the advantageous properties of PDDSparse. Numerical experiments are included for the purpose of illustration. [ABSTRACT FROM AUTHOR]

For the fractional heat equation \frac {\partial }{\partial t} u(t,x) = -(-\Delta)^{\frac {\alpha }{2}}u(t,x)+ u(t,x)\dot W(t,x) where the covariance function of the Gaussian noise \dot W is defined by the heat kernel, we establish Feynman-Kac formulae for both Stratonovich and Skorohod solutions, along with their respective moments. In particular, we prove that d<2+\alpha is a sufficient and necessary condition for the equation to have a unique square-integrable mild Skorohod solution. One motivation lies in the occurrence of this equation in the study of a random walk in random environment which is generated by a field of independent random walks starting from a Poisson field. [ABSTRACT FROM AUTHOR]

In this paper, we considered the parabolic Anderson model with a class of time-independent generalized Gaussian fields on R d , which included fractional white noise, Bessel field, massive free field, and other nonstationary Gaussian fields. Under the rough initial conditions, we constructed the Feynman-Kac formula as a solution in the Stratonovich integral by Brownian bridge, and then proved the Hölder continuity of the solution with respect to the time variable. As a comparison, we also studied the Hölder continuity under the regular initial conditions that u 0 ≡ C and u 0 ∈ C κ (R d) with κ ∈ (0 , 1 ]. [ABSTRACT FROM AUTHOR]

In this paper, we considered the parabolic Anderson model with a class of time-independent generalized Gaussian fields on $ \mathbb{R}^d $, which included fractional white noise, Bessel field, massive free field, and other nonstationary Gaussian fields. Under the rough initial conditions, we constructed the Feynman-Kac formula as a solution in the Stratonovich integral by Brownian bridge, and then proved the Hölder continuity of the solution with respect to the time variable. As a comparison, we also studied the Hölder continuity under the regular initial conditions that $ u_0\equiv C $ and $ u_0\in C^\kappa(\mathbb{R}^d) $ with $ \kappa\in(0, 1] $.

In this paper, we establish the strong law of large numbers (SLLN) and central limit theorem (CLT) for the time-inhomogeneous stochastic lattice differential equations in different spaces. These equations are driven by a deterministic time periodic force and a white noise. First, under some suitable conditions on the coefficients, we discuss the existence of $ \tau $-periodic solutions in distribution. We further investigate some version of Feynman-Kac formula, which follows a Feynman-Kac evolution operator, as well as a $ \tau $-periodic measure. We demonstrate two main limit theorems for the continuous time-inhomogeneous periodic solution processes, and apply them to two models: a dynamical viral network model and a three-compartment model. [ABSTRACT FROM AUTHOR]

We consider a Riemannian compact manifold $ M $, the associated Laplacian $ \Delta $ and the corresponding Brownian motion $ X_t $, $ t\geq 0. $ Given a Lipschitz function $ V:M\to{\mathbb R} $ we consider the operator $ \frac{1}{2}\Delta+V $, which acts on differentiable functions $ f: M\to{\mathbb R} $ via the expression$ \frac{1}{2} \Delta f(x)+\,V(x)f(x) \,, $for all $ x\in M $.Denote by $ P_t^V $, $ t \geq 0, $ the semigroup acting on functions $ f: M\to{\mathbb R} $ given by$ P_t^V(f)(x): = \mathbb{E}_x\left[e^{\int_0^t V\left(X_r\right) d r} f\left(X_t\right)\right]. $We will derive results that show that this semigroup is a continuous-time version of the discrete-time Ruelle operator.Consider the positive differentiable eigenfunction $ F: M \to \mathbb{R} $ associated with the main eigenvalue $ \lambda $, for the semigroup $ P_t^V $, $ t \geq 0 $. From the function $ F $, in a procedure similar to the one used in discrete-time Thermodynamic Formalism, we can associate by way of a coboundary procedure, a certain stationary Markov semigroup. We show that the probability on the Skorohod space obtained from this new stationary Markov semigroup meets the requirements to be called stationary Gibbs state associated with the potential $ V $. We define entropy, pressure, and the continuous-time Ruelle operator. Also, we present a variational principle of pressure for such a setting. [ABSTRACT FROM AUTHOR]

This paper presents a novel method for analyzing high-dimensional nonlinear stochastic dynamic systems. In particular, we attempt to obtain the solution of the Fokker–Planck–Kolmogorov (FPK) equation governing the response probability density of the system without using the FPK equation directly. The method consists of several important components including the radial basis function neural networks (RBFNN), Feynman–Kac formula and the short-time Gaussian property of the response process. In the area of solving partial differential equations (PDEs) using neural networks, known as physics-informed neural network (PINN), the proposed method presents an effective alternative for obtaining solutions of PDEs without directly dealing with the equation, thus avoids evaluating the derivatives of the equation. This approach has a potential to make the neural network-based solution more efficient and accurate. Several highly challenging examples of nonlinear stochastic systems are presented in the paper to illustrate the effectiveness of the proposed method in comparison to the equation-based RBFNN approach. [ABSTRACT FROM AUTHOR]

In this article, we investigate the following stochastic tempered fractional general diffusion equation driven by tempered fractional Brownian motion (TFBM) ∂ t β , η u (t , x) = L u (t , x) + u (t , x) B ˙ H , λ (t) , (t , x) ∈ R + × R d , where ∂ t β , η denotes the Caputo tempered fractional derivative with order β ∈ (0 , 1) and tempered parameter η > 0, is the infinitesimal generator of some general time-homogeneous symmetric strong Markov process { X t } t ≥ 0 with the corresponding transition semigroup { P t , t ≥ 0 } defined by Ptf(x): = x[f(Xt)], and { B H , λ (t) } t ≥ 0 is a TFBM with Hurst index H ∈ (1 2 , 1) and tempering parameter λ > 0 which is independent of { X t } t ≥ 0 . First of all, a novel estimate of multiple stochastic integrals with respect to the inverse tempered β-stable subordinator Dβ, η(t) is presented, which greatly contributes to the stochastic analysis. Then after establishing the Fubini Theorem for the stochastic integrals with respect to TFBM BH, λ(t) and the inverse tempered β-stable subordinator Dβ, η(t), we show that the Feynman-Kac representation u (t , x) = E D [ P D β , η (t) f (x) e ∫ 0 t ∫ 0 t δ (s − r) d D β , η (s) d B H , λ (r) ] is the weak solution of the stochastic tempered fractional general diffusion equation driven by TFBM with the initial value f by means of the techniques of Malliavin calculus and convergence analysis. From the Feynman-Kac formula, several kinds of continuity of the solutions with respect to time and tempered parameter η of tempered fractional derivative are proved by using the scaling property of the inverse tempered β-stable subordinator Dβ, η(t) and the novel estimate of stochastic integrals with respect to BH, λ(t) and Dβ, η(t). In particular for the stochastic fractional general diffusion equation driven by fractional Brownian motion (FBM) but in the time fractional derivative case, we can also obtain the sharp upper and lower bounds for the Hölder regularity and the moments of the solution. [ABSTRACT FROM AUTHOR]

In this paper, we provide a probabilistic interpretation of the viscosity solution of a parabolic partial differential equation through the solution of a class of quadratic backward stochastic differential equations (BSDEs). Additionally, we demonstrate the convergence and the large deviation principle for the solutions of these quadratic BSDEs, which are linked to a family of Markov processes with diffusion coefficients that tend toward zero. [ABSTRACT FROM AUTHOR]

This paper is devoted to the stochastic travelling wave solutions of a three-species competitive–cooperative system with multiplicative noise. The existence of the travelling wave solutions can be proved by constructing upper/lower-solution and stochastic dynamical systems theory. Moreover, under the more restricted assumption on the coefficients of system, and applying Feynman–Kac formula and upper/lower solution technique, the upper/lower bounds of the asymptotic wave speed can be provided. [ABSTRACT FROM AUTHOR]

We study the bijection of the space of one-parameter families of bounded linear operators in a space of functions on a coordinate space onto the set of complex-valued finitely additive cylindrical measures in the space of trajectories in the coordinate space. From the space of measures on the space of trajectories, a set of measures representing semigroups is distinguished. A procedure is defined for constructing Feynman measures on the space of trajectories representing semigroups generated by a linear functional differential equation with delay. Using the Feynman–Kac formulas and a measure on the space of trajectories corresponding to an unperturbed semigroup, the perturbation of a semigroup by a bounded potential on the space of trajectory values is studied. A perturbed semigroup is defined by integrating the perturbation functional on the trajectory space over the Feynman cylindrical measure corresponding to the unperturbed semigroup. [ABSTRACT FROM AUTHOR]

In this paper, we concentrate on the propagation dynamics of stochastic reaction–diffusion equations, including the existence of travelling wave solution and asymptotic wave speed. Based on the stochastic Feynman–Kac formula and comparison principle, the boundedness of the solution of stochastic reaction–diffusion equations can be obtained so that we can construct a sup-solution and a sub-solution to estimate the upper bound and the lower bound of wave speed. [ABSTRACT FROM AUTHOR]

Probabilistic variants of model order reduction (MOR) methods have recently emerged for improving stability and computational performance of classical approaches. In this paper, we propose a probabilistic reduced basis method (RBM) for the approximation of a family of parameter-dependent functions. It relies on a probabilistic greedy algorithm with an error indicator that can be written as an expectation of some parameter-dependent random variable. Practical algorithms relying on Monte Carlo estimates of this error indicator are discussed. In particular, when using probably approximately correct (PAC) bandit algorithm, the resulting procedure is proven to be a weak-greedy algorithm with high probability. Intended applications concern the approximation of a parameter-dependent family of functions for which we only have access to (noisy) pointwise evaluations. As a particular application, we consider the approximation of solution manifolds of linear parameter-dependent partial differential equations with a probabilistic interpretation through the Feynman-Kac formula. [ABSTRACT FROM AUTHOR]

In this paper, we first establish a version of the Feynman-Kac formula for the tempered fractional general diffusion equation$ \begin{align} \partial^{\beta, \eta}_{t} u(t,x) = \mathfrak{L}u(t,x) +b(t)u(t,x),\; \; x\in\mathcal{X},\; t\geq0, \end{align} $with initial value $ f $ belonging to a Banach space $ (\mathbb{B}, \|\cdot\|) $, where $ \partial^{\beta, \eta}_{t} $ denotes the Caputo tempered fractional derivative with order $ \beta\in(0,1) $ and tempered parameter $ \eta>0 $, $ b(t) $ is a bounded and continuous external potential on $ [0, \infty) $, $ \mathfrak{L} $ is the infinitesimal generator of a general time-homogeneous strong Markov process $ \{X_{t}\}_{t\geq0} $, and $ \mathcal{X} $ denotes a Lusin space that is a topological space being homeomorphic to a Borel subset of a compact metric space. By using the properties of the tempered $ \beta $-stable subordinator $ S_{\beta,\eta}(t) $ and the inverse tempered $ \beta $-stable subordinator $ D_{\beta,\eta}(t) $, and the stochastic calculus for the stochastic integral driven by $ D_{\beta,\eta}(t) $, we show that the Feynman-Kac representation $ u(t,x) $ defined by$ \begin{align} u(t,x) = {\mathbb{E}}^{x}\bigg[f(X_{D_{\beta,\eta}(t)}) e^{\int_{0}^{t}b(r)dD_{\beta,\eta}(r)}\bigg] \end{align} $is the unique mild and weak solutions to the tempered fractional general diffusion equation. From the Feynman-Kac formula, we further show the continuity of the solution with respect to time based on the integral properties of the Mittag-Leffler function and differential formula of covariance for $ D_{\beta,\eta}(t) $. By exploring the scaling property of $ D_{\beta,\eta}(t) $, the explicit order is also presented for the continuity of the solution with respect to tempered parameter $ \eta $. [ABSTRACT FROM AUTHOR]

In this paper, the strong existence and uniqueness for a degenerate finite system of quantile-dependent McKean-Vlasov stochastic differential equations are obtained under a weak Hörmander condition. The approach relies on the a priori bounds for the density of the solution to time inhomogeneous diffusions. The time inhomogeneous Feynman-Fac formula is used to construct a contraction map for this degenerate system. [ABSTRACT FROM AUTHOR]

In this article, based on probabilistic methods, we discuss periodic homogenization of a class of weakly coupled systems of linear elliptic and parabolic partial differential equations. Under the assumption that the systems have rapidly periodically oscillating coefficients, we first prove that the appropriately centered and scaled continuous component of the associated regime-switching diffusion process converges weakly to a Brownian motion with covariance matrix given in terms of the coefficients of the systems. The homogenization results then follow by employing probabilistic representation of the solutions to the systems and the continuous mapping theorem. The presented results generalize the well-known results related to periodic homogenization of the classical elliptic boundary-value problem and the classical parabolic initial-value problem for a single equation. [ABSTRACT FROM AUTHOR]

The Feynman–Kac formula establishes a link between parabolic partial differential equations and stochastic processes in the context of the Schrödinger equation in quantum mechanics. Specifically, the formula provides a solution to the partial differential equation, expressed as an expectation value for Brownian motion. This paper demonstrates that the Feynman–Kac formula does not produce a unique solution but instead carries infinitely many solutions to the corresponding partial differential equation. [ABSTRACT FROM AUTHOR]

This paper studies the well‐posedness of a class of nonlocal fully nonlinear parabolic systems, which nest the equilibrium Hamilton–Jacobi–Bellman (HJB) systems that characterize the time‐consistent Nash equilibrium point of a stochastic differential game (SDG) with time‐inconsistent (TIC) preferences. The nonlocality of the parabolic systems stems from the flow feature (controlled by an external temporal parameter) of the systems. This paper proves the existence and uniqueness results as well as the stability analysis for the solutions to such systems. We first obtain the results for the linear cases for an arbitrary time horizon and then extend them to the quasilinear and fully nonlinear cases under some suitable conditions. Two examples of TIC SDG are provided to illustrate financial applications with global solvability. Moreover, with the well‐posedness results, we establish a general multidimensional Feynman–Kac formula in the presence of nonlocality (time inconsistency). [ABSTRACT FROM AUTHOR]

The current paper is devoted to the stochastic N-species cooperative system with a moderately strong noise. By the theory of monotone random systems and the technique of suitable marker of wavefront, the existence of the travelling wave solution is established. By applying the Feynman-Kac formula and sup-sub solution technique, the upper and lower bounded of the asymptotic wave speed are also obtained. Finally, we give an example for stochastic 3-species cooperative systems.

This work presents a probabilistic scheme for solving semilinear nonlocal diffusion equations with volume constraints and integrable kernels. The nonlocal model of interest is defined by a time-dependent semilinear partial integro-differential equation (PIDE), in which the integrodifferential operator consists of both local convection-diffusion and nonlocal diffusion operators. Our numerical scheme is based on the direct approximation of the nonlinear Feynman-Kac formula that establishes a link between nonlinear PIDEs and stochastic differential equations. The exploitation of the Feynman-Kac representation avoids solving dense linear systems arising from nonlocal operators. Compared with existing stochastic approaches, our method can achieve first-order convergence after balancing the temporal and spatial discretization errors, which is a significant improvement of existing probabilistic/stochastic methods for nonlocal diffusion problems. Error analysis of our numerical scheme is established. The effectiveness of our approach is shown in two numerical examples. The first example considers a three-dimensional nonlocal diffusion equation to numerically verify the error analysis results. The second example presents a physics problem motivated by the study of heat transport in magnetically confined fusion plasmas. [ABSTRACT FROM AUTHOR]

We consider a system of forward backward stochastic differential equations (FBSDEs) with a time-delayed generator driven by Lévy-type noise. We establish a non-linear Feynman–Kac representation formula associating the solution given by the FBSDEs system to the solution of a path dependent nonlinear Kolmogorov equation with both delay and jumps. Obtained results are then applied to study a generalization of the so-called large investor problem, where the stock price evolves according to a jump-diffusion dynamic. [ABSTRACT FROM AUTHOR]

Let K be an algebraic number field. With K we associate the ring of finite adeles K. In this paper we give a path integral formula for the propagator of a quantum mechanical system over the abelian group K n. Specifically, we consider matrix-valued Hamiltonian operators H K n = Δ K n ⊗ Id + V , where Δ K n is the Vladimirov operator and V is a non-negative definite potential. The free part of the Hamiltonian gives rise to a measure on the Skorokhod space of paths which allows us to prove the Feynman–Kac formula for the Schrödinger semigroup generated by − H K n . This formula is given in terms of the ordered time exponentials. [ABSTRACT FROM AUTHOR]

We consider the (unique) mild solution u (t , x) of a one-dimensional stochastic heat equation on [ 0 , T ] × ℝ driven by time-homogeneous white noise in the Wick–Skorokhod sense. The main result of this paper is the computation of the spatial derivative of u (t , x) , denoted by ∂ x u (t , x) , and its representation as a Feynman–Kac type closed form. The chaos expansion of ∂ x u (t , x) makes it possible to find its (optimal) Hölder regularity especially in space. [ABSTRACT FROM AUTHOR]

The current paper is devoted to the stochastic N -species cooperative system with a moderately strong noise. By the theory of monotone random systems and the technique of suitable marker of wavefront, the existence of the travelling wave solution is established. By applying the Feynman-Kac formula and sup-sub solution technique, the upper and lower bounded of the asymptotic wave speed are also obtained. Finally, we give an example for stochastic 3-species cooperative systems. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish a Feynman-Kac formula for the moments of the iterated Malliavin derivatives of the solution to the parabolic Anderson model in terms of pinned Brownian motions. As an application, we obtain estimates for the moments of the iterated derivatives of the solution. [ABSTRACT FROM AUTHOR]

We consider the computation of free energy-like quantities for diffusions when resorting to Monte Carlo simulation is necessary, for instance in high dimension. Such stochastic computations typically suffer from high variance, in particular in a low noise regime, because the expectation is dominated by rare trajectories for which the observable reaches large values. Although importance sampling, or tilting of trajectories, is now a standard technique for reducing the variance of such estimators, quantitative criteria for proving that a given control reduces variance are scarce, and often do not apply to practical situations. The goal of this work is to provide a quantitative criterion for assessing whether a given bias reduces variance, and at which scale. We rely for this on a recently introduced notion of stochastic solution for Hamilton–Jacobi–Bellman (HJB) equations. Based on this tool, we introduce the notion of k-stochastic viscosity approximation (SVA) of a HJB equation. We next prove that such approximate solutions are associated with estimators having a relative variance of order k − 1 at log-scale. In particular, a sampling scheme built from a 1-SVA has bounded variance as noise goes to zero. Finally, in order to show that our definition is relevant, we provide examples of stochastic viscosity approximations of order one and two, with a numerical illustration confirming our theoretical findings. [ABSTRACT FROM AUTHOR]

Brownian motion in a Euclidean space with a membrane located on a hyperplane and acting in the normal direction is constructed such that its so-called permeability coefficient can be given by an arbitrary Borel measurable function defined on that hyperplane and taking on its values from the interval [ − 1 , 1 ]. In all the publications on the topic, that coefficient was supposed to be a continuous function. A certain limit theorem for the number of crossings through the membrane by the consecutive values of the process constructed at the instants of time 0, 1 / n , 2 / n , ..., [ n t ] / n (for fixed t > 0) is proved under the assumption that n → ∞. The limit distribution in that theorem can be curiously interpreted in the case of a membrane whose permeability coefficient coincides with the indicator of a measurable subset of the hyperplane. [ABSTRACT FROM AUTHOR]

The parabolic Anderson model is the heat equation with some extra spatial randomness. In this paper we consider the parabolic Anderson model with i.i.d. Pareto potential on a critical Galton–Watson tree conditioned to survive. We prove that the solution at time t is concentrated at a single site with high probability and at two sites almost surely as t → ∞. Moreover, we identify asymptotics for the localisation sites and the total mass, and show that the solution u (t , v) at a vertex v can be well-approximated by a certain functional of v. The main difference with earlier results on Z d is that we have to incorporate the effect of variable vertex degrees within the tree, and make the role of the degrees precise. [ABSTRACT FROM AUTHOR]

Consider the following nonlocal integro-differential operator of Lévy-type Lαε,δ given by... related to stochastic differential equations driven by multiplicative isotropic α -stable Lévy noise (1 < α < 2). We study by using homogenization theory the behavior of uε,δ: R d ⟶ R of double perturbed Kolmogorov, Petrovskii and Piskunov (KPP)-type with periodic coefficients varying over length scale δ and nonlinear reaction term of scale 1 / ε,... The behavior is required as ε, δ both tend to 0. Our homogenization method is probabilistic. Since δ and ε go at the same rate, we may apply the large deviations principle with homogenized coefficients. [ABSTRACT FROM AUTHOR]

In recent years residual neural networks (ResNets) as introduced by He et al. [17] have become very popular in a large number of applications, including in image classification and segmentation. They provide a new perspective in training very deep neural networks without suffering the vanishing gradient problem. In this article we show that ResNets are able to approximate solutions of Kolmogorov partial differential equations (PDEs) with constant diffusion and possibly nonlinear drift coefficients without suffering the curse of dimensionality, which is to say the number of parameters of the approximating ResNets grows at most polynomially in the reciprocal of the approximation accuracy $ \varepsilon > 0 $ and the dimension of the considered PDE $ d\in \mathbb{N} $. We adapt a proof in Jentzen et al. [20] - who showed a similar result for feedforward neural networks (FNNs) - to ResNets. In contrast to FNNs, the Euler-Maruyama approximation structure of ResNets simplifies the construction of the approximating ResNets substantially. Moreover, contrary to [20], in our proof using ResNets does not require the existence of an FNN representing the identity map, which enlarges the set of applicable activation functions. [ABSTRACT FROM AUTHOR]

In this paper, we propose an unconditionally stable numerical technique for a multi-dimensional Black-Scholes equation to price an option with high accuracy. The proposed scheme uses the operator splitting method to reduce the multi-dimensional partial differential equation to a set of one-dimensional sub-problems. The computational domain is discretized with a uniform space and time step size to approximate the option values and asset correlation-related terms by piecewise quadratic polynomials. In order to obtain the numerical solutions of the sub-problems, we analytically integrate the polynomials to estimate the expectation of the Feynman-Kac formula. We compare our method with the implicit operator splitting method (OSM), a widely used finite difference method in the industry. Numerical experiments show that the proposed method outperforms OSM in terms of convergence in space and time directions. We also provide analysis to guarantee the unconditional stability of our method by exploiting the Feynman-Kac recursively. [ABSTRACT FROM AUTHOR]

Random processes with values in an infinite dimensional linear spaces are studied. These processes are presented by the random linear operators acting in the space of square integrable functions on an infinite dimensional space. For uniformly boundedness of these representations shift-invariant measures on an infinite dimensional space are used. We study a bijective mapping of a space of complex valued finitely-additive cylindrical measures on the space of trajectories with values in an infinite dimensional Hilbert space to a space of mappings of a positive real semiaxe into the space of bounded linear operators acting in the space of functions square integrable with respect to some measure on the Hilbert space. Proper topologies on the space of cylindrical measures and the space of operator valued functions are defined such that the bijection is the homomorphism. The convergence in distribution and pointwise convergence on cylindrical sets are studied for a sequence of random processes with values in the Hilbert space. Limit distribution for compositions of independent random shift operators is described by the self-adjoint Laplace–Volterra operator. The connection of the considered construction with the Feynman–Kac formula for the perturbation of the semigroup of self-adjoint contractive operator in the Hilbert space of function of an infinite dimensional argument is presented. [ABSTRACT FROM AUTHOR]

Chernoff approximations of Feller semigroups and the associated diffusion processes in Riemannian manifolds are studied. The manifolds are assumed to be of bounded geometry, thus including all compact manifolds and also a wide range of non‐compact manifolds. Sufficient conditions are established for a class of second order elliptic operators to generate a Feller semigroup on a (generally non‐compact) manifold of bounded geometry. A construction of Chernoff approximations is presented for these Feller semigroups in terms of shift operators. This provides approximations of solutions to initial value problems for parabolic equations with variable coefficients on the manifold. It also yields weak convergence of a sequence of random walks on the manifolds to the diffusion processes associated with the elliptic generator. For parallelizable manifolds this result is applied in particular to the representation of Brownian motion on the manifolds as limits of the corresponding random walks. [ABSTRACT FROM AUTHOR]

In this work, we propose a method for solving Kolmogorov hypoelliptic equations based on Fourier transform and Feynman–Kac formula. We first explain how the Feynman–Kac formula can be used to compute the fundamental solution to parabolic equations with linear or quadratic potential. Then applying these results after a Fourier transform we deduce the computation of the solution to a first class of Kolmogorov hypoelliptic equations. Then we solve partial differential equations obtained via Feynman–Kac formula from the Ornstein–Uhlenbeck generator. Also, a new small time approximation of the solution to a certain class of Kolmogorov hypoelliptic equations is provided. We finally present the results of numerical experiments to check the practical efficiency of this approximation. [ABSTRACT FROM AUTHOR]

Motivated by entropic optimal transport, we are interested in the Feynman-Kac formula associated to the parabolic equation (L + V) g = 0 with a final nonnegative boundary condition and a Markov generator L : = ∂ t + b · ∇ + Δ a / 2 . It is well-known that when the drift b , the diffusion matrix a and the scalar potential V are regular enough and not growing too fast, the classical solution g of this PDE, is represented by the Feynman-Kac formula g t (x) = E R [ exp ∫ [ t , T ] V (s , X s) d s g (X T) ∣ X t = x ] where R is the Markov measure with generator L . We do not assume that g, b and V are regular, and only require that their growth is controlled by a finite entropy condition. These hypotheses are less restrictive than the standard assumptions of the theory of viscosity solutions, and allow for instance V to belong to some Kato class. We prove that g defined by the Feynman-Kac formula belongs to the domain of the extended generator L of the Markov measure R and satisfies the trajectorial identity: [ (L + V) g ] (t , X t) = 0 , d t d P - a. e. where the path measure P is defined by P : = f (X 0) exp ∫ [ 0 , T ] V (t , X t) d t g (X T) R , with f : R n → [ 0 , ∞) another nonnegative function. We also show that the forward drift b P of P satisfies b P (t , X t) = [ b + a ∇ ~ log g ] (t , X t) , d t d P - a. e. , where ∇ ~ is some extension of the standard derivative. Our probabilistic approach relies on stochastic derivatives, semimartingales, Girsanov's theorem and the Hamilton-Jacobi-Bellman equation satisfied by log g . [ABSTRACT FROM AUTHOR]

A classical result of K. L. Chung and W. Feller deals with the partial sums S_k arising in a fair coin-tossing game. If N_n is the number of "positive" terms among S_1, S_2, ..., S_n then the quantity P(N_{2n} = 2r) takes an elegant form. We lift the restriction on an even number of tosses and give a simple expression for P(N_{2n+1} = r), r = 0, 1, 2, ..., 2n+1. We get to this ansatz by adaptating the Feynman–Kac methodology. [ABSTRACT FROM AUTHOR]