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We study stability of stationary solutions for a class of nonlocal semilinear parabolic equations. To this end, we prove the Feynman–Kac formula for a Lévy processes with time-dependent potentials and arbitrary initial condition. We propose sufficient conditions for asymptotic stability of the zero solution, and use them to the study of the spatial logistic equation arising in population ecology. For this equation, we find conditions which imply that its positive stationary solution is asymptotically stable. We consider also the case when the initial condition is given by a random field. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We study the parabolic integral kernel for the weighted Laplacian with a potential. For manifolds with a pole we deduce formulas and estimates for the derivatives of the Feynman–Kac kernels and their logarithms, these are in terms of a ‘Gaussian’ term and the semi-classical bridge. [ABSTRACT FROM AUTHOR]

In this note, we use the Feynman–Kac formula to derive a moment representation for the 2D parabolic Anderson model in small time, which is related to the intersection local time of planar Brownian motions. [ABSTRACT FROM AUTHOR]

In this paper, we give a probabilistic representation of the heat equation associated with the quantum K-Gross Laplacian using infinite-dimensional stochastic calculus in two variables. Applying the heat semigroup to the particular casewhere the operator is the multiplication one,we establish a relation between the classical and the quantum heat semigroup. Finally, using a combination between convolution calculus and the generalized stochastic calculus, we give a generalization of the Feynman-Kac formula. [ABSTRACT FROM AUTHOR]

This paper studies the stochastic heat equation driven by time fractional Gaussian noise with Hurst parameter H∈(0,1/2). We establish the Feynman-Kac representation of the solution and use this representation to obtain matching lower and upper bounds for the Lp(Ω) moments of the solution. [ABSTRACT FROM AUTHOR]

In this note, we give an estimate for the dimension of the image of the unit circle under a quasiconformal mapping whose dilatation has small support. We also prove an analogous estimate for the rate of growth of a solution of a second-order parabolic equation given by the Feynman-Kac formula with a sparsely supported potential and introduce a dictionary between the two settings. [ABSTRACT FROM AUTHOR]

The growth-fragmentation equation describes a system of growing and dividing particles, and arises in models of cell division, protein polymerisation and even telecommunications protocols. Several important questions about the equation concern the asymptotic behaviour of solutions at large times: at what rate do they converge to zero or infinity, and what does the asymptotic profile of the solutions look like? Does the rescaled solution converge to its asymptotic profile at an exponential speed? These questions have traditionally been studied using analytic techniques such as entropy methods or splitting of operators. In this work, we present a probabilistic approach: we use a Feynman–Kac formula to relate the solution of the growth-fragmentation equation to the semigroup of a Markov process, and characterise the rate of decay or growth in terms of this process. We then identify the Malthus exponent and the asymptotic profile in terms of a related Markov process, and give a spectral interpretation in terms of the growth-fragmentation operator and its dual. [ABSTRACT FROM AUTHOR]

A method for approximation of the operator e−itH, where H=−12d2dx2+V(x), in the strong operator topology is proposed. The approximating operators have the form of expectations of functionals of a certain random point field. [ABSTRACT FROM AUTHOR]

The Navier-Stokes equations written in the vector potential can be recast as non-linear Schrödinger equations at imaginary times, i.e. heat equations with a potential term, using the Cole-Hopf transform. On this basis, we study two kinds of Navier-Stokes flows by means of direct numerical simulations. In an experiment on vortex reconnection, it is found that the potential term takes large negative values in regions where intensive reconnection takes place, whereas the signature of the non-linear term is more broadly spread. For decaying turbulence starting from a random initial condition, such a correspondence is also observed in the early stage when the flow is dominated by vorticity layers. At later times, when the flow features several tubular vortices, this correspondence becomes weaker. Finally, a similar set of transformations is presented for the magneto-hydrodynamic equations, which reduces them to a set of heat equations with suitable potential terms, thereby obtaining new criteria for the regularity of their solutions. [ABSTRACT FROM AUTHOR]

We provide a general criterion ensuring the exponential contraction of Feynman–Kac semi-groups of penalized processes. This criterion applies to time-inhomogeneous Markov processes with absorption and killing through penalization. We also give the asymptotic behavior of the expected penalization and provide results of convergence in total variation of the process penalized up to infinite time. For exponential convergence of penalized semi-groups with bounded penalization, a converse result is obtained, showing that our criterion is sharp in this case. Several cases are studied: we first show how our criterion can be simply checked for processes with bounded penalization, and we then study in detail more delicate examples, including one-dimensional diffusion processes conditioned not to hit 0 and penalized birth and death processes evolving in a quenched random environment. [ABSTRACT FROM AUTHOR]

We prove a local Faber-Krahn inequality for solutions u to the Dirichlet problem for Δ+V on an arbitrary domain Ω in ℝn. Suppose a solution u assumes a global maximum at some point x0∈Ω and u(x0)>0. Let T(x0) be the smallest time at which a Brownian motion, started at x0, has exited the domain Ω with probability ≥1∕2. For nice (e.g., convex) domains, but we make no assumption on the geometry of the domain. Our main result is that there exists a ball B of radius such that provided that n≥3. In the case n = 2, the above estimate fails and we obtain a substitute result. The Laplacian may be replaced by a uniformly elliptic operator in divergence form. This result both unifies and strenghtens a series of earlier results. [ABSTRACT FROM AUTHOR]

We use forward–backward stochastic differential systems to study the solution of the Navier–Stokes-equation in any dimension. For the two dimensional Navier–Stokes-equation with space periodic boundary conditions, we derive a Feynman-Kac formula associated with the vorticity equation and prove the global existence and uniqueness of the solution in a Sobolev space. For the d dimensional () case, we prove the local existence and uniqueness of the solution in a Sobolev space. [ABSTRACT FROM AUTHOR]

We study the classical bargaining problem and its two canonical solutions, ( Nash and Kalai–Smorodinsky ), from a novel point of view: we ask for stability of the solution if both players are able distort the underlying bargaining process by reference to a third party (e.g. a court). By exploring the simplest case, where decisions of the third party are made randomly we obtain a stable solution, where players do not have any incentive to refer to such a third party. While neither the Nash nor the Kalai–Smorodinsky solution are able to ensure stability in case reference to a third party is possible, we found that the Kalai–Smorodinsky solution seems to always dominate the stable allocation which constitutes novel support in favor of the latter. [ABSTRACT FROM AUTHOR]

This paper discusses an optimal investment-consumption problem in a continuous-time co-integration model, where an investor aims to maximize an expected, discounted utility derived from intertemporal consumption and terminal wealth in a finite time horizon. Using the dynamic programming principle approach, we obtain an Hamilton-Jacobi-Bellman equation related to the problem. In each of the power and logarithmic utility cases, we obtain semi-closed-form expressions of the investment-consumption strategy and the value function. We also provide some numerical examples to illustrate these theoretical results. [ABSTRACT FROM AUTHOR]

In this paper we study the linear stochastic heat equation, also known as parabolic Anderson model, in multidimension driven by a Gaussian noise which is white in time and it has a correlated spatial covariance. Examples of such covariance include the Riesz kernel in any dimension and the covariance of the fractional Brownian motion with Hurst parameter H ∈ (1/4, 1/2] in dimension one. First we establish the existence of a unique mild solution and we derive a Feynman-Kac formula for its moments using a family of independent Brownian bridges and assuming a general integrability condition on the initial data. In the second part of the paper we compute Lyapunov exponents, lower and upper exponential growth indices in terms of a variational quantity. The last part of the paper is devoted to study the phase transition property of the Anderson model. Dans cet article nous étudions l'équation de la chaleur linéaire stochastique multidimensionnelle, connue aussi comme model d'Anderson parabolique, perturbée par un bruit gaussien qui est blanc en temps et qui a une covariance corrélée en espace. Le noyau de Riesz en dimension quelconque et la covariance du mouvement Brownien fractionnaire avec paramètre de Hurst H ∈ (1/4, 1/2] en une dimension, sont des examples d'une telle covariance. D'abord, on établit l'existence d'une solution d'evolution unique et on obtient une formule de Feynman-Kac pour les moments de la solution, en utilisant une famille de ponts browniens indépendants et en supposant une condition générale d'intégrabilité sur la condition initiale. Dans la deuxième partie du travail nous calculons les exposants de Lyapunov et les exposants supérieur et inférieur de croissance exponentielle en fonction d'une quantité variationnelle. La dernière partie du travail est consacré à l'etude de la transition de phase pour le model d'Anderson. [ABSTRACT FROM AUTHOR]

We study the stochastic solution to a Cauchy problem for a degenerate parabolic equation arising from option pricing. When the diffusion coefficient of the underlying price process is locally Hölder continuous with exponent δ ∈ ( 0 , 1 ] , the stochastic solution, which represents the price of a European option, is shown to be a classical solution to the Cauchy problem. This improves the standard requirement δ ≥ 1 / 2 . Uniqueness results, including a Feynman–Kac formula and a comparison theorem, are established without assuming the usual linear growth condition on the diffusion coefficient. When the stochastic solution is not smooth, it is characterized as the limit of an approximating smooth stochastic solutions. In deriving the main results, we discover a new, probabilistic proof of Kotani's criterion for martingality of a one-dimensional diffusion in natural scale. [ABSTRACT FROM AUTHOR]

We consider the Navier–Stokes equations written in the stream function in two dimensions and vector potentials in three dimensions, which are critical dependent variables. On this basis, we introduce an analogue of the Cole-Hopf transform, which exactly reduces the Navier–Stokes equations to the heat equations with a potential term (i.e. the nonlinear Schrödinger equation at imaginary times). The following results are obtained. (i) A regularity criterion immediately obtains as the boundedness of condition for the potential term when the equations are recast in a path-integral form by the Feynman-Kac formula. (ii) This in turn gives an additional characterisation of possible singularities for the Navier–Stokes equations. (iii) Some numerical results for the two-dimensional Navier–Stokes equations are presented to demonstrate how the potential term captures near-singular structures. Finally, we extend this formulation to higher dimensions, where the regularity issues are markedly open. [ABSTRACT FROM PUBLISHER]

In this paper, we obtain an explicit formula for the two-point correlation function for the solutions to the stochastic heat equation on $\mathbb {R}$ . The bounds for p-th moments proved in Chen and Dalang (Ann. Probab. 2015) are simplified. We validate the Feynman-Kac formula for the p-point correlation function of the solutions to this equation with measure-valued initial data. [ABSTRACT FROM AUTHOR]

In this article we introduce a new concept of an analytic operator-valued Feynman integral for functionals on Wiener space, which we then use to explain various physical phenomena. We then establish the existence of some analytic operator-valued Feynman integrals that prove useful in establishing various applications in quantum mechanics. [ABSTRACT FROM PUBLISHER]

We consider a stochastic functional delay differential equation, namely an equation whose evolution depends on its past history as well as on its present state, driven by a pure diffusive component plus a pure jump Poisson compensated measure. We lift the problem in the infinite dimensional space of square integrable Lebesgue functions in order to show that its solution is an L 2 -valued Markov process whose uniqueness can be shown under standard assumptions of locally Lipschitzianity and linear growth for the coefficients. Coupling the aforementioned equation with a standard backward differential equation, and deriving some ad hoc results concerning the Malliavin derivative for systems with memory, we are able to derive a non-linear Feynman–Kac representation theorem under mild assumptions of differentiability. [ABSTRACT FROM AUTHOR]

In this paper we first introduce the concept of a double modified analytic function space Fourier-Feynman transform using the double modified analytic function space integral. We then proceed to establish the existence of the modified analytic function space Fourier-Feynman transform for all functionals in the Banach algebra. Finally we use this double modified analytic function space transform to explain various physical phenomenon. [ABSTRACT FROM AUTHOR]

In this paper, we consider the composition of two independent processes: one process corresponds to position and the other one to time. Such processes will be called iterated processes. We first propose an algorithm based on the Euler scheme to simulate the trajectories of the corresponding iterated processes on a fixed time interval. This algorithm is natural and can be implemented easily. We show that it converges almost surely, uniformly in time, with a rate of convergence of order 1/4 and propose an estimation of the error. We then extend the well known Feynman-Kac formula which gives a probabilistic representation of partial differential equations (PDEs), to its higher order version using iterated processes. In particular we consider general position processes which are not necessarily Markovian or are indexed by the real line but real valued. We also weaken some assumptions from previous works. We show that intertwining diffusions are related to transformations of high order PDEs. Combining our numerical scheme with the Feynman-Kac formula, we simulate functionals of the trajectories and solutions to fourth order PDEs that are naturally associated to a general class of iterated processes. [ABSTRACT FROM AUTHOR]

This paper is dedicated to the study of the geometric average Asian call option pricing under non-extensive statistical mechanics for a time-varying coefficient diffusion model. We employed the non-extensive Tsallis entropy distribution, which can describe the leptokurtosis and fat-tail characteristics of returns, to model the motion of the underlying asset price. Considering that economic variables change over time, we allowed the drift and diffusion terms in our model to be time-varying functions. We used the I t o ^ formula, Feynman⁻Kac formula, and P a d e ´ ansatz to obtain a closed-form solution of geometric average Asian option pricing with a paying dividend yield for a time-varying model. Moreover, the simulation study shows that the results obtained by our method fit the simulation data better than that of Zhao et al. From the analysis of real data, we identify the best value for q which can fit the real stock data, and the result shows that investors underestimate the risk using the Black⁻Scholes model compared to our model.

A path integral Monte Carlo method (PIMC) based on a Feynman-Kac formula for the Laplace equation with mixed boundary conditions is proposed to solve the forward problem of the electrical impedance tomography (EIT). The forward problem is an important part of iterative algorithms of the inverse EIT problem, and the proposed PIMC provides a local solution to find the potentials and currents on individual electrodes. Improved techniques are proposed to compute with better accuracy both the local time of reflecting Brownian motions (RBMs) and the Feynman-Kac formula for mixed boundary problems of the Laplace equation. Accurate voltage-to-current maps on the electrodes of a model 3-D EIT problem with eight electrodes are obtained by solving a mixed boundary problem with the proposed PIMC method. • An accurate stochastic algorithm is first proposed for the Laplace equation with mixed/Robin boundary conditions. • An more accurate allocation scheme for calculating local time of Neumann and Robin boundaries improves errors from 4% to 1%. • The proposed method can be applied to the voltage-to-current mapping for the forward EIT problem. [ABSTRACT FROM AUTHOR]

This paper concerns the stochastic partial differential equation with multiplicative noise ∂ u ∂ t = L u + u W ̇ , where L is the generator of a symmetric Lévy process X , W ̇ is a Gaussian noise and u W ̇ is understood both in the senses of Stratonovich and Skorohod. The Feynman–Kac type of representations for the solutions and the moments of the solutions are obtained, and the Hölder continuity of the solutions is also studied. As a byproduct, when γ ( x ) is a nonnegative and nonnegative-definite function, a sufficient and necessary condition for ∫ 0 t ∫ 0 t | r − s | − β 0 γ ( X r − X s ) d r d s to be exponentially integrable is obtained. [ABSTRACT FROM AUTHOR]

This paper studies continuous-time Markov decision processes with a denumerable state space, a Borel action space, bounded cost rates and possibly unbounded transition rates under the risk-sensitive finite-horizon cost criterion. We give the suitable optimality conditions and establish the Feynman-Kac formula, via which the existence and uniqueness of the solution to the optimality equation and the existence of an optimal deterministic Markov policy are obtained. Moreover, employing a technique of the finite approximation and the optimality equation, we present an iteration method to compute approximately the optimal value and an optimal policy, and also give the corresponding error estimations. Finally, a controlled birth and death system is used to illustrate the main results. [ABSTRACT FROM AUTHOR]

We present an iterative scheme, reminiscent of the Multigrid method, to solve large boundary value problems with Probabilistic Domain Decomposition (PDD). In it, increasingly accurate approximations to the solution are used as control variates in order to reduce the Monte Carlo error of the following iterates—resulting in an overall acceleration of PDD for a given error tolerance. The key feature of the proposed algorithm is the ability to approximately predict the speedup with little computational overhead and in parallel. Besides, the theoretical framework allows to explore other aspects of PDD, such as stability. One numerical example is worked out, yielding an improvement between one and two orders of magnitude over the previous version of PDD. [ABSTRACT FROM AUTHOR]

We consider a critical continuous-time branching process (a Yule process) in which the individuals independently execute symmetric α-stable random motions on the real line starting at their birth points. Because the branching process is critical, it will eventually die out, and so there is a well-defined maximal location M ever visited by an individual particle of the process. We prove that the distribution of M satisfies the asymptotic relation P{M≥x}~(2/α)1/2x−α/2 as x→∞. [ABSTRACT FROM AUTHOR]

Employing recent results on stochastic differential equations associated with the standard model of non-relativistic quantum electrodynamics by B. Güneysu, J. S. Møller, and the present author, we study the continuity of the corresponding semi-group between weighted vector-valued -spaces, continuity properties of elements in the range of the semi-group, and the pointwise continuity of an operator-valued semi-group kernel. We further discuss the continuous dependence of the semi-group and its integral kernel on model parameters. All these results are obtained for Kato decomposable electrostatic potentials and the actual assumptions on the model are general enough to cover the Nelson model as well. As a corollary, we obtain some new pointwise exponential decay and continuity results on elements of low-energetic spectral subspaces of atoms or molecules that also take spin into account. In a simpler situation where spin is neglected, we explain how to verify the joint continuity of positive ground state eigenvectors with respect to spatial coordinates and model parameters. There are no smallness assumptions imposed on any model parameter. [ABSTRACT FROM AUTHOR]

In this paper, we propose forward and backward stochastic differential equations (FBSDEs) based deep neural network (DNN) learning algorithms for the solution of high dimensional quasi-linear parabolic partial differential equations (PDEs), which is related to the FBSDEs from the Pardoux-Peng theory. The algorithms rely on a learning process by minimizing the path-wise difference between two discrete stochastic processes, which are defined by the time discretization of the FBSDEs and the DNN representation of the PDE solution, respectively. The proposed algorithms are shown to generate DNN solution for a 100-dimensional Black–Scholes–Barenblatt equation, which is accurate in a finite region in the solution space, and has a convergence rate close to that of the Euler–Maruyama scheme used for discretizing the FBSDEs. • FBSDE deep neural network (DNN) with loss comparing two stochastic processes for the PDEs from Pardoux-Peng theory. • Strong 1/2 Convergence as the Euler-Maruyama method in solving a 100-dimensional Black-Scholes-Barenblatt equation. • Use of Richardson extrapolation for accuracy enhancement. • Multiscale DNN captures temporal high frequency content of the PDEs more accurately. [ABSTRACT FROM AUTHOR]

This paper concerns the macroscopic behavior of solutions to parabolic equations with large, highly oscillatory, random potential. When the correlation function of the random potential satisfies a specific integrability condition, we show that the random solution converges, as the correlation length of the medium tends to zero, to the deterministic solution of a homogenized equation in dimension d ≥ 3. Our derivation is based on a Feynman-Kac probabilistic representation and the Kipnis-Varadhan method applied to weak convergence of Brownian motions in random sceneries. For sufficiently mixing coefficients, we also provide an optimal rate of convergence to the homogenized limit using a quantitative martingale central limit theorem. As soon as the above integrability condition fails, the solution is expected to remain stochastic in the limit of a vanishing correlation length. For a large class of potentials given as functionals of Gaussian fields, we show the convergence of solutions to stochastic partial differential equations (SPDE) with multiplicative noise. The Feynman-Kac representation and the corresponding weak convergence of Brownian motions in random sceneries allows us to explain the transition from deterministic to stochastic limits as a function of the correlation function of the random potential. [ABSTRACT FROM AUTHOR]

In this article we prove the continuity of the deterministic function u : [ 0 , T ] × D ̄ → R , defined by u ( t , x ) : = Y t t , x , where the process ( Y s t , x ) s ∈ [ t , T ] is given by the generalized multivalued backward stochastic differential equation: { − d Y s t , x + ∂ φ ( Y s t , x ) d s + ∂ ψ ( Y s t , x ) d A s t , x ∋ f ( s , X s t , x , Y s t , x ) d s + g ( s , X s t , x , Y s t , x ) d A s t , x − Z s t , x d W s , t ≤ s < T , Y T = h ( X T t , x ) . The process ( X s t , x , A s t , x ) s ≥ t is the solution of a stochastic differential equation with reflecting boundary conditions. [ABSTRACT FROM AUTHOR]

We present a method for solving the reaction–diffusion equation with general potential in free space. It is based on the approximation of the Feynman–Kac formula by a sequence of convolutions on sequentially diminishing grids. For computation of the convolutions we propose a fast algorithm based on the low-rank approximation of the Hankel matrices. The algorithm has complexity of O ( n r M log ⁡ M + n r 2 M ) flops and requires O ( M r ) floating-point numbers in memory, where n is the dimension of the integral, r ≪ n , and M is the mesh size in one dimension. The presented technique can be generalized to the higher-order diffusion processes. [ABSTRACT FROM AUTHOR]

The paper examines measurability conditions for a random process for which a general form of the Feynman–Kac formula is valid. [ABSTRACT FROM AUTHOR]

This work develops Feynman–Kac formulas for a class of regime-switching jump diffusion processes, in which the jump part is driven by a Poisson random measure associated with a general Lévy process and the switching part depends on the jump diffusion processes. Under broad conditions, the connections of such stochastic processes and the corresponding partial integro-differential equations are established. Related initial, terminal and boundary value problems are also treated. Moreover, based on weak convergence of probability measures, it is demonstrated that a sequence of random variables related to the regime-switching jump diffusion process converges in distribution to the arcsine law. [ABSTRACT FROM PUBLISHER]

We generalize the notion of renormalized solution to semilinear elliptic and parabolic equations involving operator associated with general (possibly nonlocal) regular Dirichlet form and smooth measure on the right-hand side. We show that under mild integrability assumption on the data a quasi-continuous function u is a renormalized solution to an elliptic (or parabolic) equation in the sense of our definition if and only if u is its probabilistic solution, i.e. u can be represented by a suitable nonlinear Feynman-Kac functional. This implies in particular that for a broad class of local and nonlocal semilinear equations there exists a unique renormalized solution. [ABSTRACT FROM AUTHOR]