arXiv subjects

B. Rajeev

Publications and source records attributed to B. Rajeev.

4 recordsLinked to original sources

arXiv2019

Translation Invariant Diffusions and Stochastic Partial Differential Equations in ${\cal S}^{\prime}

In this article we show that the ordinary stochastic differential equations of K.Itô maybe considered as part of a larger class of second order stochastic PDE's that are quasi linear and have the property of translation invariance. We show using the `monotonicity inequality' and the Lipshitz continuity of the coefficients $σ_{ij}$ and $b_i$, existence and uniqueness of strong solutions for these stochastic PDE's. Using pathwise uniqueness, we prove the strong Markov property.

math.PR↗

arXiv2017

Translation Invariant Diffusions in the space of tempered distributions

In this paper we prove existence and pathwise uniqueness for a class of stochastic differential equations (with coefficients $σ_{ij},b_i$ and initial condition $y$ in the space of tempered distributions) that maybe viewed as a generalisation of Ito's original equations with smooth coefficients . The solutions are characterized as the translates of a finite dimensional diffusion whose coefficients $σ_{ij}\star \tilde{y},b_i\star \tilde{y}$ are assumed to be locally Lipshitz.Here $\star$ denotes convolution and $\tilde{y}$ is the distribution which on functions, is realised by the formula $\tilde{y}(r) := y(-r)$ . The expected value of the solution satisfies a non linear evolution equation which is related to the forward Kolmogorov equation associated with the above finite dimensional diffusion.

math.PR↗

arXiv2007

Probabilistic Representations of Solutions of the Forward Equations

In this paper we prove a stochastic representation for solutions of the evolution equation $ \partial_t ψ_t = {1/2}L^*ψ_t $ where $ L^* $ is the formal adjoint of an elliptic second order differential operator with smooth coefficients corresponding to the infinitesimal generator of a finite dimensional diffusion $ (X_t).$ Given $ ψ_0 = ψ$, a distribution with compact support, this representation has the form $ ψ_t = E(Y_t(ψ))$ where the process $ (Y_t(ψ))$ is the solution of a stochastic partial differential equation connected with the stochastic differential equation for $ (X_t) $ via Ito's formula.

math.PR↗

arXiv2003

Probabilistic representations of solutions to the heat equation

In this paper we provide a new (probabilistic) proof of a classical result in partial differential equations, viz. if $ϕ$ is a tempered distribution, then the solution of the heat equation for the Laplacian, with initial condition $ϕ$, is given by the convolution of $ϕ$ with the heat kernel (Gaussian density). Our results also extend the probabilistic representation of solutions of the heat equation to initial conditions that are arbitrary tempered distributions.

math.PR↗